Charge transport in dye-sensitized solar cell^*

The schematic structure of DSCs is shown in figure 1. The J_g is expressed as the photo-generated electron flux in TiO₂ including charge injection from dye to TiO₂, electron transfer from I⁻ to oxide dye, the electron diffusion in TiO₂. The J_dif is expressed as the diffusion limited current including transports of I⁻ and I₃⁻. The reaction rate at counter electrode also influences J_dif. However, the rate can be neglected because it is faster than ion transports. The thicknesses of spacer for DSCs and the TiO₂ film are defined as T and d, respectively.

In the case of J_dif > J_g, the current in DSCs is explained by the diffusion equation of the electrons in TiO₂ as the following equation [16–20]

$$\begin{eqnarray}&&\frac{\partial n\left( x,t \right)}{\partial t}=D\frac{{{\partial }^{2}}n\left( x,t \right)}{\partial {{x}^{2}}}-\frac{n\left( x,t \right)}{\tau }+\alpha {{I}_{0}}{{e}^{-\alpha x}}.\end{eqnarray} \tag{ 1 }$$

The current of DSCs is dominated by J_g. The fast ion transports can compensate the ions for electron transfer at the interface. The J_sc is generally linear related to the light intensity below 1 sun. The response of the J_sc depends on the electron transport.

In the case of J_g > J_dif, the surface concentration of I⁻ and I₃⁻ at interface becomes 0 because the ion transport cannot compensate the ions for electron transfer at the interface. The current depends on the ion transports. The distribution of ions in DSCs has to be considered. Papageorgiou et al [6, 7] had investigated the relationship between the charge (electron, I⁻ and I₃⁻) transport in the electrolyte and the photovoltaic performance by the ion transport model. The ion transport model [6, 7, 9] was modified by the followings to explain $J_{{\rm sc}}^{{\rm s}}$ against physical parameters in DSCs and determined the suitable physical parameters for favorable ion transport in ionic liquid. (i) The light absorption profile is introduced into the ion diffusion equation. (ii) The diffusion coefficient (D_OX) in porous TiO₂ is different from that ($D_{{\rm OX}}^{{\rm b}}$) in bulk electrolyte solution. The $D_{{\rm OX}}^{{\rm b}}$ can be expressed by βD_OX. The β value of in the electrolyte of MPImI was 5 when the diffusion limited currents of TiO₂ sandwiched by Pt electrode were measured. (iii) The current is influenced by the distribution of the concentration of ${{{\rm I}}_{3}}^{-}$ since the concentration of I⁻ is ten times larger than that of ${{{\rm I}}_{3}}^{-}$. (iv) The decrease of ${{{\rm I}}_{3}}^{-}$ due to charge recombination in porous TiO₂ can be neglected. (v) the concentration of ${{{\rm I}}_{3}}^{-}$ at T needs to be 0 or over.

In porous TiO₂ ($0{\mkern 1mu} \leqq {\mkern 1mu} x{\mkern 1mu} \leqq {\mkern 1mu} d$), the concentration of ${{{\rm I}}_{3}}^{-}$ (C_OX) can be expressed by

$$\begin{eqnarray}&&\frac{\partial {{C}_{{\rm OX}}}}{\partial t}={{D}_{{\rm OX}}}\frac{{{\partial }^{2}}{{C}_{{\rm OX}}}}{\partial {{x}^{2}}}+\frac{\alpha \phi {{I}_{0}}}{2\rho }{{e}^{-{\alpha }x}},\end{eqnarray} \tag{ 2 }$$

where x is distance from TCO, t is time, ρ is porosity of porous TiO₂ I₀ is the incident photon flux corrected for reflection loss, ϕ is the electron transfer and transport yield, and α is the absorption coefficient. In the bulk electrolyte solution layer of DSCs ($d{\mkern 1mu} \leqq {\mkern 1mu} x{\mkern 1mu} \leqq {\mkern 1mu} T$), the concentration (C_OX) of ${{{\rm I}}_{3}}^{-}$ can be expressed by

$$\begin{eqnarray}&&\frac{\partial {{C}_{{\rm OX}}}}{\partial t}=D_{{\rm OX}}^{{\rm b}}\frac{{{\partial }^{2}}{{C}_{{\rm OX}}}}{\partial {{x}^{2}}}\cdot \end{eqnarray} \tag{ 3 }$$

The flux (J_flux) of ion transport can be expressed by the followings. The flux ($J_{{\rm flux}}^{{\rm Ti}{{{\rm O}}_{{\rm 2}}}}$) in porous TiO₂ ($0{\mkern 1mu} \leqq {\mkern 1mu} x{\mkern 1mu} \leqq {\mkern 1mu} d$) is

$$\begin{eqnarray}&&J_{{\rm flux}}^{{\rm Ti}{{{\rm O}}_{{\rm 2}}}}=\rho {{D}_{{\rm OX}}}\frac{\partial {{C}_{{\rm OX}}}}{\partial x}\cdot \end{eqnarray} \tag{ 4 }$$

The flux ($J_{{\rm flux}}^{{\rm b}}$) in the bulk electrolyte solution ($d{\mkern 1mu} \leqq {\mkern 1mu} x{\mkern 1mu} \leqq {\mkern 1mu} T$) is

$$\begin{eqnarray}&&J_{{\rm flux}}^{{\rm b}}=D_{{\rm OX}}^{{\rm b}}\frac{\partial {{C}_{{\rm OX}}}}{\partial x}\cdot \end{eqnarray} \tag{ 5 }$$

The concentration gradient of C_OX at TCO equals 0 as the boundary condition at x = 0

$$\begin{eqnarray}&&{{\left( \frac{\partial {{C}_{{\rm OX}}}}{\partial x} \right)}_{{\rm x}={\rm 0}}}=0\cdot \end{eqnarray} \tag{ 6 }$$

I assumed that two boundary condition can be fixed at x = d. Because the electrolyte solution is continuous at x = d, the C_OX value calculated from equation (2) must equal to that from equation (3) at x = d

$$\begin{eqnarray}&&{{C}_{{\rm OX}}}(x=d,\;0\leqslant x\leqslant d)={{C}_{{\rm OX}}}(x=d,\;d\leqslant x\leqslant T)\cdot \end{eqnarray} \tag{ 7 }$$

The flux of equation (4) must be same as that of equation (5) at x = d

$$\begin{eqnarray}&&\rho {{D}_{{\rm OX}}}{{\left( \frac{\partial {{C}_{{\rm OX}}}}{\partial x} \right)}_{x=d}}=D_{{\rm OX}}^{{\rm b}}{{\left( \frac{\partial {{C}_{{\rm OX}}}}{\partial x} \right)}_{x=d}}\cdot \end{eqnarray} \tag{ 8 }$$

The total amount of ${{{\rm I}}_{3}}^{-}$ has to be conserved in DSCs

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \int \limits_{x=0}^{x=d} \rho {{C}_{{\rm OX}}}\left( x \right){\rm d}x+\int \limits_{x=d}^{x=T} {{C}_{{\rm OX}}}\left( x \right){\rm d}x \\ \quad =C_{{\rm OX}}^{{\rm 0}}\left( \rho d+T-d \right). \\ \end{array}\end{eqnarray} \tag{ 9 }$$

In steady state, the left side of equations (2) and (3) equals 0. The distribution of C_OX can be solved from equations (2)–(9). The C_OX is expressed as follows

$$\begin{eqnarray}&&{{C}_{OX}}\left( x \right)=-\frac{\phi {{I}_{0}}}{2\rho {{D}_{OX}}}\left( x+\frac{1}{\alpha }\left( {{e}^{-\alpha x}}-1 \right) \right)+{{C}_{OX}}\left( 0 \right)\end{eqnarray} \tag{ 10 }$$

at $0{\mkern 1mu} \leqq {\mkern 1mu} x{\mkern 1mu} \leqq {\mkern 1mu} d$, and

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{C}_{OX}}\left( x \right)=\frac{\phi {{I}_{0}}}{2D_{OX}^{b}}\left( {{e}^{-\alpha d}}-1 \right)\left( x-d \right)-\frac{\phi {{I}_{0}}}{2\rho {{D}_{OX}}} \\ \quad \quad \;\times \left( d+\frac{1}{\alpha }\left( {{e}^{\alpha d}}-1 \right) \right)+{{C}_{OX}}\left( 0 \right) \\ \end{array}\end{eqnarray} \tag{ 11 }$$

at $d{\mkern 1mu} \leqq {\mkern 1mu} x{\mkern 1mu} \leqq {\mkern 1mu} T$.

When $C_{{\rm OX}}^{0}$ is defined as the initial concentration of ${{I}_{3}}^{-}$, C_OX(0) can be expressed by the following equation

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{C}_{{\rm OX}}}(0)=C_{{\rm OX}}^{0}+\phi {{I}_{0}}\left( \frac{1}{{{D}_{{\rm OX}}}}\left\{ d\left( \frac{d}{2}-\frac{1}{\alpha } \right)-\frac{1}{{{\alpha }^{2}}} \right. \right. \\ \quad \quad \quad \quad \left. \times \left( {{e}^{-{\alpha }d}}-1 \right) \right\}-\frac{1}{2\rho D_{{\rm OX}}^{{\rm b}}}\left( {{e}^{-{\alpha }d}}-1 \right) \\ \quad \quad \quad \quad \times {{(T-d)}^{2}}+\frac{1}{\rho {{D}_{{\rm OX}}}}(T-d) \\ \quad \quad \quad \quad \times \left. \left\{ d+\frac{1}{\alpha }\left( {{e}^{-{\alpha }d}}-1 \right) \right\} \right)/\left( 2(\rho d+T-d) \right)\cdot \\ \end{array}\end{eqnarray} \tag{ 12 }$$

The concentration distribution (C_OX(0)) from TCO to the counter electrode at various light intensity is shown in figure 5(a). The T is 30 μm, d is 13 μm, ϕ is 0.65, $C_{{\rm OX}}^{0}$ is 0.03 M, $D_{{\rm OX}}^{{\rm b}}$ is 3.5 × 10⁻⁷ cm² s⁻¹, β is 5, α at 650 nm is 1200 cm⁻², respectively. As the light intensity increases, the C_OX(0) increases and C_OX(T) decreases.

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In the case of J_dif > J_g, the photocurrent density of DSCs can be expressed as ion flux at x = T

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{J}_{{\rm SC}}} & = & -nFJ_{{\rm flux}}^{{\rm b}}=-2FD_{{\rm OX}}^{{\rm b}}{{\left( \frac{\partial {{C}_{{\rm OX}}}}{\partial x} \right)}_{{\rm x}=T}} \\ {} & = & F\phi {{I}_{0}}\left( 1-{{e}^{{\alpha }d}} \right), \\ \end{array}\end{eqnarray} \tag{ 13 }$$

where n is electrons per this reaction and F is Faraday constant, respectively.

The linear relationship between photocurrent and light intensity can be explained by equation (13).

C_OX(T) becomes minus at light intensity of 3.1 mW cm⁻² as shown in figure 5(a). In a real device, C_OX(T) of equation (11) has to be 0 or over as described as (v). The model needs to be modified to explain the ion transport limitation. When C_OX(T) is calculated to be under 0, the C_OX(T) must keep being 0, and equation (8) is neglected. This assumption means that (dC_OX/dx)_x=T becomes constant against I₀. J_SC is dominated by the diffusion of ${{{\rm I}}_{3}}^{-}$ and becomes constant against I₀ defined as diffusion limited current ($J_{{\rm SC}}^{{\rm s}}$). The constant $J_{{\rm SC}}^{{\rm s}}$ is due to (dC_OX/dx)_x=T of equation (13). The relationship between the calculated J_sc and the light intensity (I₀) is shown in figure 5(b). The calculated $J_{{\rm sc}}^{{\rm s}}$ from figure 5(b) was 1.3 mA cm⁻², corresponding to the experimental results in figures 3(a) and 4.

$J_{{\rm sc}}^{{\rm s}}$ has to be over 30 mA cm⁻² which is calculated from the external quantum efficiency (EQE) of 95% from 400 nm to 900 nm in solar light. Otherwise, the solar to energy conversion efficiency of DSCs is limited by the ion transport. The diffusion limited current density (J_dif) can be generally expressed by the following formula [15]

$$\begin{eqnarray}&&{{J}_{{\rm dif}}}=\frac{2nFDC}{d}\cdot \end{eqnarray} \tag{ 14 }$$

J_dif depends on the diffusion coefficient of ions (D), carrier density (C), and the distance (d) of the ion transport, which respectively corresponds to the diffusion coefficient($D_{{\rm OX}}^{{\rm b}}$) of ${{{\rm I}}_{3}}^{-}$, the $C_{{\rm OX}}^{0}$ value of ${{{\rm I}}_{3}}^{-}$, and the d value of the TiO₂ film. Therefore, each parameter needs to be discussed to improve $J_{{\rm sc}}^{{\rm s}}$.

The relationship between the calculated $J_{{\rm sc}}^{{\rm s}}$ and $D_{{\rm OX}}^{{\rm b}}$ or $C_{{\rm OX}}^{{\rm 0}}$ is shown in figure 6. $J_{{\rm sc}}^{{\rm s}}$ increases with increase of $D_{{\rm OX}}^{{\rm b}}$ as shown in figure 6(a). The relationship between $J_{{\rm sc}}^{{\rm s}}$ and $D_{{\rm OX}}^{{\rm b}}$ is related to the $C_{{\rm OX}}^{{\rm 0}}$. The $J_{{\rm sc}}^{{\rm s}}$ also increases with increase of $C_{{\rm OX}}^{0}$ as shown in figure 6(b). In case of $C_{{\rm OX}}^{{\rm 0}}$ = 0.05 M, the $D_{{\rm OX}}^{{\rm b}}$ needs to be over 1 × 10⁻⁵ cm² s⁻¹. The typical $C_{{\rm OX}}^{{\rm 0}}$ value is 0.05 M in the electrolyte solution based on acetonitrile. The $D_{OX}^{b}$ value of this electrolyte solution is about 1 × 10⁻⁴ cm² s⁻¹. Therefore, the electrolyte solution based on acetonitrile can fully compensate the photon flux of the solar light. On the other hand, the $D_{OX}^{b}$ based on a typical ionic liquid is 5 × 10⁻⁷ cm² s⁻¹, about 100 times smaller than that of the electrolyte solution based on acetonitrile. $J_{sc}^{s}$ becomes about 1% of the electrolyte solution based on acetonitrile.

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The electrolyte solution with large $C_{OX}^{0}$ value needs to be prepared to improve $J_{sc}^{s}$. In the case of MPImI, the $C_{OX}^{0}$ value has to be over 0.26 M. However, the I₂ cannot be dissolved over 0.26 M in this electrolyte. And the strong light absorption of ${{{\rm I}}_{3}}^{-}$ reduces the light harvesting efficiency of the dye on TiO₂ in DSCs. The way to increase $C_{OX}^{0}$ is limited by the solubility and the light absorption of ${{{\rm I}}_{3}}^{-}$.

Molecular design has been used to try to decrease the viscosity of an ionic liquid [3, 5] because $D_{OX}^{b}$ increases with decrease of viscosity. A suitable $D_{OX}^{b}$ value for DSCs needed to be shown for the design of the ionic liquid. $D_{OX}^{b}$ needs to be over 1.5 × 10⁻⁶ cm² s⁻¹ from figure 6(b) and the $C_{OX}^{0}$ needs to be over 0.1 M to certainly enhance $J_{sc}^{s}$. The high solubility (>0.3 M) and low visible absorption properties of a redox mediator also need to be designed for DSCs.

The effect of porous TiO₂ on the ion transport also needs to be discussed to improve $J_{sc}^{s}$. The plots of calculated $J_{sc}^{s}$ versus $D_{OX}^{b}$ or $C_{OX}^{0}$ again are shown in figure 7 to compare with the each diffusion limited current density (J_dif) of the porous TiO₂ film and the bulk electrolyte in the DSC. The solid line in figure 7(a) is due to J_dif of the porous TiO₂ layer and is calculated from equation (14) at C = $C_{OX}^{0}$ = 0.05 M, d = 13 μm, and D = D_OX, respectively. The dashed line is due to J_dif of the bulk electrolyte and is calculated from equation (14) at C = $C_{OX}^{0}$= 0.05 M, d = 17 μm, and D = $D_{OX}^{b}$, respectively. The slope of $J_{sc}^{s}$ versus $D_{OX}^{b}$ is similar to that of J_dif versus $D_{OX}^{b}$ of the porous TiO₂ layer. The solid line in figure 7(b) is due to J_dif of the porous TiO₂ layer and is calculated from equation (14) at C = $C_{OX}^{0}$, d = 13 μm, and D = D_OX = 7 × 10⁻⁸ cm² s⁻¹, respectively. The dashed line is due to J_dif of the bulk electrolyte and is calculated from equation (14) at C = $C_{OX}^{0}$, d = 17 μm, and D = $D_{OX}^{b}$ = 3.5 × 10⁻⁷ cm² s⁻¹, respectively. The slope of $J_{sc}^{s}$ versus $C_{OX}^{0}$ is also similar to that of J_dif versus $C_{OX}^{0}$ of the porous TiO₂ layer. The results show that the ion transports in the porous TiO₂ layer influence the total ion transport in DSCs.

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The β is defined as $D_{OX}^{b}/{{D}_{OX}}$. The D_OX is usually smaller than $D_{{\rm OX}}^{{\rm b}}$ (β > 1). The mechanism of β > 1 is under investigation. The β value seems to depend on the viscosity. The ions can move more randomly when the viscosity is low. Ions easily collide with the surface of porous structure. Therefore, the diffusion rate of ions slows down in porous TiO₂. On the other hand, the β value seems to decrease with increase of viscosity of the electrolyte. In the case of high viscosity, ions cannot quickly move in the electrolyte solution. The probability of the collision between ions and the surface of porous structure must become smaller.

The relationship between the calculated $J_{sc}^{s}$ and thickness (d) of TiO₂ is shown in figure 8. The results can be explained by using equation (14). The ion can be easily supplied with decrease of the distance. $J_{sc}^{s}$ is exponentially influenced by the d value. The thin TiO₂ films are favorable for the ion transport, especially in the case of small $D_{OX}^{b}$. However, the light harvesting efficiency of dye on TiO₂ decreases with decrease of d. The high light absorption ability of dyes needs to be designed for DSCs based on ionic liquid. In the case of $C_{OX}^{0}$ = 0.05 M, $D_{OX}^{b}$ needs to be 5 × 10⁻⁵ cm² s⁻¹, corresponding to the condition of figure 6(a).

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