Theoretical modelling of the electrode thickness effect on maximum power point of dye-sensitized solar cell.

Ni, Meng ; Leung, Michael K.H. ; Leung, Dennis Y.C. 等

INTRODUCTION

Dye-sensitized solar cells (DSSC) made of Ti[O.sub.2] have received increasing attention since O'Regan and Gratzel published their work in Nature in 1991 (O'Regan and Gratze1, 1991). In comparison with a conventional P-N junction-based solar cell, the impurity in the semiconductor of a DSSC causes less adverse effect on the cell performance because the important electron injection and recombination processes occur at the Ti[O.sub.2]/electrolyte interface (Gregg, 2004; Gratzel, 2005). Thus, the fabrication of DSSC can be more economical. Presently, DSSC can achieve as much as 10% energy conversion efficiency (Gratzel, 2001). The ongoing research and development focus on new DSSC materials for higher cell efficiency and theoretical modelling for insightful understanding of the basic working mechanisms (Ruffle et al., 1997; Gratzel, 2000; Kron et al., 2002, 2003a, 2003b; Rau et al., 2003; Durr et al., 2004; Fredin et al., 2005; Lenzmann et al., 2005; Gratzel, 2006; Ito et al., 2006; Kroon et al., 2006; Kumara et al., 2006; Ni et al., 2006a; Pettersson et al., 2006; Sastrawan et al., 2006).

Some studies have been conducted to determine the effect of the Ti[O.sub.2] electrode thickness on the DSSC performance. Sodergren et al. (1994) developed a steady-state mathematical model based on the electron diffusion in a porous semiconductor (Ti[O.sub.2]) thin film to obtain an explicit expression for the photocurrent as a function of the film thickness, light absorption coefficient, and light intensity. The diffusion model simplified the actual DSSC mechanisms with consideration of all the important material properties, cell design parameters, and operational parameters. Recently, Gomez and Salvador (2005) used the same diffusion model to study how the open-circuit photovoltage varied with the film thickness and the light intensity.

Nevertheless, the effect of the electrode thickness on the J-V characteristics, especially the maximum power point (NIPP), has not been adequately addressed. In practice, solar cells are normally operated in an MPP condition regulated by a maximum power point tracking (MPPT) system (Hohm and Ropp, 2003). Therefore, in this investigation, parametric analyses were conducted to study the DSSC electrode thickness effect on MPP and to optimize the thickness for the highest energy conversion efficiency.

MODELLING

Under a steady-state condition of an irradiated DSSC, the electron injection from excited dye molecules, transport in the porous semiconductor (Ti[O.sub.2]) thin film, and recombination with electrolyte at the Ti[O.sub.2]/electrolyte interface can be described by the following diffusion differential equation (Sodergren et al., 1994; Gomez and Salvador, 2005; Ni et al., 2006b):

D [[partial derivative].sub.2]/[partial derivative][chi square] - n(x) - [n.sub.0]/[tau]+[PHI][alpha][e.sup.-[alpha]x] = 0 (1)

where the x-coordinate is measured from the Ti[O.sub.2]/transparent conducting oxide (TCO) interface; n(x) is the excessive electron concentration at x; no is the electron concentration in a dark condition ([n.sub.0] = [10.sup.16] [cm.sup.-3]) (Rothenberger et al., 1992; Ferber and Luther, 2001); [tau] is the lifetime of conduction band free electrons; [PHI] is the light intensity; [alpha] is the light absorption coefficient of the porous electrode; and D is the electron diffusion coefficient. Under a short-circuit condition, electrons are easily extracted as photocurrent and none of the electrons are drawn to the counter electrode directly. Therefore, the two boundary conditions are:

n(0) = [n.sub.0] (2)

and

dn/dx[|.sub.x=d] = 0 (3)

where d is the thin film electrode thickness. The short-circuit current density [J.sub.sc] can thus be obtained as:

[J.sub.sc] = q[PHI]L[alpha]/1 - [L.sub.2][[alpha].sub.2][- L[alpha] + tanh(d/L) + L[alpha]exp(-a[alpha]/cosh (d/L)] (4)

where L is the electron diffusion length calculated by [square root of (D[tau])] and q is the charge of an electron equal to 1.60218 x [10.sup.-19] C.

If the DSSC operates under a potential difference V between the Ti[O.sub.2] Fermi level and the electrolyte redox potential, the electron density at the Ti[O.sub.2]/TCO interface (x = 0) increases to n as a boundary condition:

n(0) = n (5)

The second boundary condition at x = d remains unchanged as stated by Equation (3). Then, solving Equation (1) yields the relationship between J and V:

V = kTm/q ln [L([J.sub.sc] - J)/qD[n.sub.0]]tanh (d/L)+1] (6)

where k is the Boltzman constant and m is the ideality factor equal to 4.5 for DSSC (Lee et al., 2004; Gomez and Salvador, 2005).

Equation (6) has been validated by Sodergren et al. in their earlier work (Sodergren et al., 1994).

The power output of a DSSC can be expressed as a function of two independent variables d and J:

P = JV = JkTm/q ln [L([J.sub.sc] - J)/qD[n.sub.0] tanh (d/L) + 1] (7)

In order to achieve the highest /MPP for a given [phi], the first partial derivatives [partial derivative]P/[partial derivative]d and [partial derivative]P/[partial derivative]J should be set equal to zero and solved simultaneously for the optimal electrode thickness [d.sub.opt] and the corresponding optimal current density [J.sub.opt]. The two criteria become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Equations (8) and (9) can be easily solved by numerical method to obtain the values of [d.sub.opt] and [J.sub.opt.]

PARAMETRIC ANALYSES

The effects of electrode thickness on the J-V characteristics and MPP of DSSC were studied numerically. The practical values input for the material properties and operating conditions are listed in Table 1, unless specified otherwise. It should be mentioned that the value of light intensity (1.0 x [10.sup.17] [cm.sup.-2][S.sup.-1]) represents 1 sun condition (100 mW/[cm.sup.2]) (Lee et al., 2004; Gomez and Salvador, 2005). Fredin et al. (2005) studied the electron diffusion coefficient of porous Ti[O.sub.2] thin film and found that when the electrode porosity was about 0.5 (the most frequently reported porosity value in the literature), D was equal to 1.0 x [10.sup.-2] [cm.sup.2][s.sup.-1], which was about half of the bulk anatase Ti[O.sub.2] electron diffusion coefficient. Gomez and Salvador (2005) found D equal to 2.0 x 10-5 [cm.sup.2][s.sup.-1] in their modelling study. Other experimental and simulation studies reported the value of D to be within the range from 1.0 x [10.sup.-4] to 1.0 x [10.sup.-3] [cm.sup.2][s.sup.-1] (Dloczik et al., 1997; Lee et al., 2004). In this study, a value of D in the mid-range equal to 5.0 x [10.sup.-4] [cm.sup.2] [s.sup.-1] was selected for the parametric analyses.

Effect of Electrode Thickness on DSSC J-V Characteristics and MPP

The J-V characteristics of DSSC at different electrode thickness d are plotted in Figure 1. The variations of the short-circuit current density [J.sub.sc], open-circuit voltage [V.sub.oc], fill factor, energy conversion efficiency, and MPP with respect to d are presented in Figure 2. The results show that as d increases, [J.sub.sc] first increases abruptly, then reaches the peak, and decreases gradually afterwards (Figure 2a). Such variation in [J.sub.sc] can be explained by electron photogeneration. For given porosity and pore size, an increase in electrode thickness directly increases the internal surface area, resulting in a higher dye loading. Therefore, a thicker electrode can absorb more photons, leading to a higher [J..sub.sc]. However, if the electrode thickness is greater than the light penetration depth, the number of photons useful for electron photogeneration will reach the limit and, therefore, [J.sub.sc] cannot be increased any further. Instead, an increase in the thickness beyond the light penetration depth yields more recombination centres that cause a higher electron loss and, thus, a gradual reduction in [J.sub.sc].

Figure 2a also shows that [V.sub.oc] decreases with increasing d. This phenomenon can be explained by the electron dilution effect (Gomez and Salvador, 2005). As light is transmitted into the depth of an electrode, the intensity gradually decreases. Therefore, as d increases, the excessive electron density becomes lower resulting in a lower [V.sub.oc]. The higher series resistance of a thicker electrode also contributes to the reduction in photovoltage. The variations of fill factor and MPP with electrode thickness are shown in Figure 2b. The fill factor was found to decrease with increasing electrode thickness, indicating an increase in internal resistance of the cell. Combining the variations in current density, photovoltage, and fill factor implies that there exists an optimal thickness that yields the highest MPP for a given incident light intensity. For the present simulation analysis shown in Figure 2b, the optimal thickness [d.sub.opt] was found to be 5 [micro]m. As the energy conversion efficiency of DSSC is the ratio of MPP to the irradiation power input, the efficiency-thickness relationship follows the same variation of MPP with electrode thickness (Figure 2c). Similarly, the optimal thickness of 5 Vin was observed which maximized the efficiency of DSSC.

[FIGURE 1 OMITTED]

From the above results, it can also be seen that the efficiency of DSSC is not very high (below 5%). This is because recombination (n(x)-no/[tau] in Equation (1)) of electrons and ions in electrolyte plays an important role in the operation of DSSC. Especially, in a DSSC using solid-state electrolyte, the recombination process is more considerable and the optimal electrode thickness should be smaller than the DSSC based on liquid electrolyte. The present model can be extended to simulate the performance of DSSC, if reliable parameters are available. In addition, the present model considers single cells only. In practice, there are other important components, such as the TCO (Ni et al., 2000a) and interconnections, which cause serial resistance to the flow of electrons and ions in electrolyte. In the future, all these factors should be considered to perform more complete analyses of practical DSSC systems.

Sensitivity of Optimal Electrode Thickness to Operational Parameters

The operating temperature T and the light intensity (D are important operational parameters that affect the performance of DSSC. It is essential to study whether the optimal electrode thickness [d.sub.opt] is sensitive to T and [PHI]. In the derivation of Equations (8) and (9) that are used to solve for [d.sub.opt], T has been eliminated. Therefore, it is theoretically shown that [d.sub.opt] is independent of T. This finding is consistent with the previous experimental results (Nazeeruddin et al., 1993; Pettersson et al., 2003).

The sensitivity of [d.sub.opt] to [PHI] was tested numerically. For a constant a of 5000 [cm.sup.-1] and different [PHI] ranging from 1.0 x [10.sup.16] to 1.0 x 1017 [cm.sup.-2][s.sup.-1], the variations of MPP with d are plotted in Figure 3. As expected, the DSSC power output, i.e., MPP, increases with increasing cp. In all the tests of different [PHI], [d.sub.opt] remains relatively unchanged at about 5 [micro]m. The analysis was repeated with a low [alpha] equal to 1000 [cm.sup.-1]. As shown in Figure 4, the corresponding [d.sub.opt] equal to about 17 [micro]m is also rather insensitive to [PHI]. It is concluded that over a wide range of light absorption coefficient, the light intensity has negligible effect on the selection of optimal electrode thickness.

The above analytical results show that the optimal electrode thickness is not sensitive to either the operating temperature or the light intensity under practical operating conditions. The findings are very important as they imply that the design optimization of DSSC should not be affected by the geographical, seasonal, and solar hour factors. In other words, a DSSC designed with the optimal electrode thickness should always produce power at a highest MPP condition anywhere, anytime.

Sensitivity of Optimal Electrode Thickness to Material Properties

Equations (4) and (9) clearly express that the short-circuit current [J.sub.sc] and the optimal current density [J.sub.opt] are functions of the electrode material properties, including the light absorption coefficient [alpha], electron diffusion coefficient D, and electron lifetime [tau]. The relationships between the optimal electrode thickness [d.sub.opt] and the material properties are discussed in this section. In Figure 5, the MPP versus d curves are plotted for [alpha] equal to 5000, 3000, and 1000 [cm.sup.-1]. It can be seen that at a higher [alpha], the power output is higher and [d.sub.opt] becomes smaller. It is because more photons are absorbed near the surface of the electrode and, thus, the photogeneration and electron collection are more efficient.

[FIGURE 2 OMITTED]

The effect of D on [d.sub.opt] is illustrated in Figure Ga. When D increases, the diffusion length increases. Therefore, more electrons can be collected resulting in a higher current density. The higher electron extraction implies a lower electron density and, thus, a lower photovoltage (Figure Gb). The combined effect results in an overall increase in MPP. It should be mentioned that when D is further increased, there is no further increase in MPP, as all available electrons (besides recombination) have been collected (it is noted that D is not directly related to recombination as indicated by Equation (1)).

[FIGURE 3 OMITTED]

The variation in MPP due to the change in [tau] is presented in Figure 7a. Similar to D, a higher [tau] implies longer electron diffusion length. Thus increasing [tau] increases the number of electrons to be collected, resulting in a higher current density. When the diffusion length is higher than required, a further increase in [tau] does not further increase the current density (Figure 7b). However, different from D, [tau] is directly related to recombination of electrons with ions in electrolyte, as indicated by the formula n(x)-[n.sub.0]/[tau] in Equation (1). An increase in [tau] directly decreases the rate of recombination, resulting in higher electron density in the Ti[O.sub.2] electrode and thus a higher [V.sub.oc] (Figure 7b). Thus, the power output, which is equal to the multiplication of current density and voltage, increases with increasing [tau].

The ideality factor (m) of value 4.5 was derived by Lee et al. (2004) from experimental measurements. In this study, the effect of m on MPP was investigated with varying electrode thickness, as shown in Figure 8. It was found that MPP increased with increasing m. However, for all the tested values of m, the optimal electrode thickness remained to be about 5 [Lm (Figure 8). With an increase in m, the fill factor remains unchanged, while the open-circuit voltage increases with increasing m (Lee et al., 2004), which results in a higher power output and thus a higher MPP.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

COMPARISON WITH PUBLISHED EXPERIMENTAL DATA

Although a lot of research work on DSSC has been published in the literature, studies on the electrode thickness effect are limited. Some previous relevant experimental works are presented in this section for comparison with the modelling results of this investigation. Hara et al. (2000) studied the mercurochrome-sensitized solar cells using different photoelectrode materials, i.e., Ti[O.sub.2], [Nb.sub.2][O.sub.5], ZnO, Sn[O.sub.2], and [In.sub.2][O.sub.3]. It was found that as the thickness of Ti[O.sub.2] electrode increased from 4.5 to 28 [Lm, the [V.sub.oc] decreased from 0.58 to 0.48 V. The predicted [V.sub.oc] by the present electron diffusion model agreed well with Hara et al.'s experimental data as shown in Figure 9. Park et al. (2000) studied N719-sensitized rutile solar cells and found a similar trend that when the rutile thickness increased from 5 to 11.5 [micro]m, [V.sub.oc] decreased from 0.759 to 0.727 V.

Some measured values of DSSC energy conversion efficiency, defined as MPP divided by irradiation power, are presented in Figure 10. The curves are smooth fits of the experimental data (symbols). Keis et al. (2002) investigated the DSSC performance using ZnO sensitized by ruthenium bipyridyl complex as photoelectrode materials and the optimal thickness for the highest energy conversion efficiency was found to be 8 [Lm. Dai et al. (2004) reported the performance of Ti[O.sub.2] DSSC samples (sensitized by N3) prepared by sol-gel processes using precursors with different pH values and autoclaving at different temperatures. When the pH value of the precursor solution was 1.2 and the autoclaving temperature was 250[degrees]C, the optimal electrode thickness was found to be about 9 [micro]m. Wang et al. (2000) studied Ti[O.sub.2] DSSC sensitized by Hemicyanine derivatives and reported that the optimal Ti[O.sub.2] thickness was 5 [micro]m. Since the MPP is proportional to the energy conversion efficiency, the variation of efficiencies with electrode thickness shown in Figure 10 is equivalent to the variation of MPP with electrode thickness.

[FIGURE 8 OMITTED]

Figure 11 shows how the measured variation of short-circuit current density with electrode thickness by Fukai et al. (2007). Sn[O.sub.2] was used as photoelectrode. The commonly used N719 dye was employed to photon absorbers. In consistence with the present modelling results, the measured Jsc increased significantly with electrode thickness and reached the maximum at an electrode thickness of about 11 [micro]m. Further increase in thickness would cause a slight decrease in [J.sub.sc].

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

In comparison between Figures 2 (modelling results), 10, and 11 (experimental data), it can be seen that the theoretical predictions follow the same trend as the experimental measurements.

The different values of optimal electrode thickness found by individual research groups are possibly due to different dye molecules of various light absorption coefficients. The microstructure parameters, such as porosity, pore size, Ti[O.sub.2] particle size, and roughness factor, have significant effects on the electron lifetime, electron diffusion coefficient, specific surface area, as well as light absorption coefficient, resulting in apparent difference in the cell performance (Park et al., 1999; van de Lagemaat et al., 2001; Benkstein et al., 2003; Nakade et al., 2003; Cass et al., 2005). Therefore, it is important to be able to model the performance of a DSSC with respect to the microstructure of its electrode (Park et al., 1999; Srikanth et al., 2001; Lindstrom et al., 2002; Nakade et al., 2003; Saito et al., 2004; Au et al., 2005). Previously, an extended diffusion model has been used to study the porosity effect on the DSSC performance (Ni et al., 2006b). Similarly, the microstructure characteristics can be included in the present simple electron diffusion model to study the effects of electrode thickness and microstructure for a thorough optimization of the DSSC cell design.

[FIGURE 11 OMITTED]

CONCLUSIONS

A simple model based on electron diffusion in semiconductor thin film was developed to study the effect of DSSC electrode thickness on the MPP. It was found that the open-circuit voltage decreased with increasing electrode thickness. The optimal electrode thickness for the highest MPP was obtained and the value was reasonably consistent with the experimental results reported in the literature. The optimal thickness was mostly independent of the cell operating parameters, but closely related to the physical properties of the photoelectrode. The results imply that to obtain the highest energy conversion efficiency, it is important to size the electrode thickness to its optimal value. The present model can be extended to cover the microstructure parameters, such as porosity, pore size, and particle size of the DSSC electrode for a complete optimization analysis.

ACKNOWLEDGEMENTS

The work described in this paper was partly supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 7150/05E).

Manuscript received October 11, 2006; revised manuscript received July 14, 2007; accepted for publication August 9, 2007.

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Michael K. H. Leung and Dennis Y C. Leung

Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

* Author to whom correspondence may be addressed. E-mail address: [email protected]

DOI 10.1002/cjce.20015

Table 1. Input values used in parametric analyses

Parameter Value

Light intensity, [PHI] ([cm.sup.-2] 1.0 x [10.sup.17]
 [s.sup.-1])
Light absorption coefficient, 5000
 [alpha]([cm.sup.-1])
Electron lifetime, [tau] (ms) 10
Ideality factor, m 4.5
Operating temperature, T (K) 300

Parameter References

Light intensity, [PHI] ([cm.sup.-2] Lee et al. (2004);
 [s.sup.-1]) Gomez and Salvador (2005)
Light absorption coefficient, Lee et al. (2004);
 [alpha]([cm.sup.-1]) Gomez and Salvador (2005)
Electron lifetime, [tau] (ms) Gomez and Salvador (2005);
 Dloczik et al. (1997)
Ideality factor, m Lee et al. (2004); Gomez
Operating temperature, T (K) and Salvador (2005)

Note: The value of [PHI] (1.0 x [10.sup.17][cm.sup.-2][s.sup.-1])
represents 1 sun condition (100 mW/[cm.sup.2]) (Gomez and Salvador,
2005; Lee et al., 2004)