Recombination resistance

The thickness of the nanocrystalline layer required to satisfy the last conditions is typically of the order of a few microns depending on the optical cross section of the sensitizer and its concentration in the film as discussed earlier. A simple consideration shows that the electron collection efficiency is related to the electron transport (fx) and recombination time (tr) or the respective electron transfer and recombination resistances (R and/ ct) by the equation [55] ... 

AC impedance spectroscopy also has seen extensive utility in the study of the hole injection or recombination process depicted in Fig. 14. An equivalent circuit for this process is illustrated in Fig. 17 it does resemble the circuit in Fig. 16(a), except for the Warburg component [84]. Early studies [85-88] utilized the recombination resistance parameter, / r, that was extracted from model fits of the measured AC impedance data. This parameter was seen to be inversely related to the hole injection current, thus signifying that it is indeed related to the recombination process. However, the challenge is to differentiate... 

Fig. 4 (a) Current density-voltage curves of a set of DSCs with different electrolytes. Points are obtained from Impedance Spectroscopy measurements and lines by an integration procedure, (b) Current density-voltage curves with respect to Fermi level voltage V-p, in which the voltage drop due to internal series resistance has been corrected, (c) Recombination resistance between the semiconductor and the acceptor species in the electrolyte, (d) Chemical capacitance of the Ti02. 

To avoid the common problem of uncertainty of jV curve modeling, it has become widely accepted in the DSC area to use the technique of impedance spectroscopy (IS) that separates different resistive components [28, 29] by means of an equivalent circuit analysis that takes into account the spectral shapes. One key point about IS applicable to DSC is that it provides a direct probe of recombination via the recombination resistance,... 

It has been established that we are able to measure recombination with great accuracy, and it is a challenge of great significance, from both fundamental and applied stances, to be able to describe these observations by a comprehensive theory. We wiU carry out this discussion below, but first we need to develop a number of points concerning the disorder in the electron subsystem. It should be pointed out, however, that the exponential dependence of the recombination resistance on voltage, which is characteristic of high performance liquid electrolyte... 

Another important method of application of the recombination resistance is that, by integration of (20), we can recover j. In addition, for the simple model of (16), integration is not even needed as the following expression [27] can be used ... 

We have already mentioned that modification of electrolytes and other factors produce a shift in the position of the TiOz conduction band, which is an important property of the DSC. Ec affects charge injection from the dye. In addition, the change of the position of the conduction band produces a strong variation of the recombination resistance, because the parameter /o is modified. Indeed note that by (2) and (14)... 

These variations are well illustrated in Fig. 4c, in which part of the shift of the recombination resistance is due to the change of the conduction band position that is visible in the chemical capacitance of Fig. 4d. An important tool to evaluate correctly the recombination rate is therefore to plot the recombination resistances of different devices at the same equivalent value of the position of the conduction band. Therefore we define a suitable potential,... 

Therefore, when plotted with respect to 5, as shown in Fig. 4e, differences in the recombination resistance correspond exactly to the variation of the reciprocal of the kinetic parameter 7ok> removing the influence of change conduction band position. The analysis is more complicated if the samples present different values of the DOS parameter a or recombination exponent p. 

Here C is the chemical capacitance. is the macroscopic recombination resistance of the layer that was introduced in (20) and will be the object of detailed analysis in the remaining sections of this chapter. The impedance model for diffusion-recombination adopts the form [44]... 

These features occurring at the semiconductor/dye/hole conductor system can be analyzed using a variety of techniques [60,176]. We have already commented on how to measure recombination properties, via electrons lifetimes and recombination resistance, among a variety of available methods [126, 127, 155, 177-179], In the final sections of this chapter we aim to provide deeper insight into the fundamental mechanisms of recombination in a DSC. 

In this section we aim to obtain the model calculation of the recombination flux, recombination resistance, and the electron lifetime in a DSC associated with the electron transfer by a combination of stuface states that form an exponential distribution, with a DOS gss(E), as shown in Fig. 12b, also considering the direct transfer from the conduction band [16, 50, 52, 180, 181, 206, 214]. 

Equation (142) states that the reciprocal charge-transfer resistance is proportional to the product of the density of surface states at the Fermi level, and the probability of electron transfer from such states. The calculation of the recombination resistance involves a small displacement of the Fermi level that fills the surface states precisely at the Fermi level, and hence the resistance detects only those states. In summary we have... 

There is no functional difference between t and Tf. The significance of this result must be emphasized. In a DSC we measure recombination resistance, chemical capacitance, and electron lifetime. However, when we wish to study fundamental charge transfer questions we really want to determine i/ei. Equation (146) states that we have access to the electron transfer probability. 

The expression (148) can be generalized very easily to include additional parallel recombination pathways. For example we can incorporate the direct transference from the conduction band that has a recombination resistance... 

We observe that the denominator of (148) is the reciprocal of the total recombination resistance, (151), and the numerator is the total chemical capacitance. Since the reciprocal parallel resistances are added to obtain a total resistance, we have more generally [50]... 

Alternatively we may analyze the recombination resistance dependence on voltage. The resistance in (142) takes the form [16]... 

Fig. 19 Recombination in a DSC according to the Marcus model of charge transfer in an exponential distribution of surface states. Horizontal axis is the voltage or equivalently the electron Fermi level. The position of the conduction band, Eq, is indicated. Plots at different values of reorganization energy as indicated, (a) Electron transfer probability, (b) Electron recombination resistance. Simulation parameters are T = 300 K, Tq = 1,200 K, a = 0.25, / = 0.75, e = 1 eV vs E edox, = 1 x 10 cm s Rqx = 10 cm ...

Summarizing the model results, we find that despite many simplifications the model of Fig. 12b provides a detailed description of the lifetime and recombination resistance, quantities that can be measured as a function of the voltage in a DSC. The main feature is that the parabola in the exponential of the Marcus model electron transfer rate between the semiconductor surface and redox acceptor in the electrolyte translates in curvatures of Tn and R ec- These quantities have been reported in hundreds of publications, but usually a linear behavior in semilogarithmic plot is observed, as reported in representative measurements of Figs. 4, 5, and 14. The absence of curvature could be evidence of a large reorganization energy in the DSC with redox couples, and the value X = leV is often used [213]. 

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