As mentioned in the introduction, several authors have reported a power law depen-dence of Jsc upon light intensity I, i.e. Jsc ∝ Iα, where α ranges typically from 0.85 to 1.[1–7]Thus far, the deviation from α=1 has been conjectured to arise from a small loss of carriers via bimolecular recombination.[2,4,7]It has been argued that pure bimolecular recombination would lead to a square-root dependence of Jscon intensity, i.e., α =0.5.
The argument is based on the assumption that generation and recombination of carriers cancel and that n=p. Under these assumptions, it follows from Eq. (3.10) that
n=p ∝pGe−h. (4.4)
When only drift of charge carriers is taken into account, one has
JL∝pGe−hF, (4.5)
which implies that α=0.5. However, as we have seen in section 3.2, Eq. (3.10) only holds at open-circuit, i.e., when JL =0, clearly this does not apply to short-circuit conditions.
70 80 90 100 110 120 0.92
0.96 1.00
Annealing temperature [ o
C]
Figure 4.3: The experimentally determined exponent α as a function of annealing temperature (open symbols) and the prediction by the numerical model (filled symbols), the solid lines are drawn as a guide to the eye, while the dashed line corresponds to α=1.
Therefore, the influence of bimolecular recombination on the intensity dependence of Jsc has not yet been established. In this section, it is shown that bimolecular recombi-nation does not account for the observed values of α, but that the true cause lies in the build-up of net space charge due to imbalanced transport of charge carriers.
How can one put this hypothesis to the test? P3TH/PCBM devices are highly suited for this purpose, since the ratio of electron to hole mobility can be changed over several orders of magnitude by a simple post-production annealing treatment, see Fig. 1.3. This enables one to verify whether the occurrence of space-charge-limited photocurrents is accompanied by α < 1. Figure 4.3 shows the experimentally obtained exponents α, ranging from 0.94 (device annealed at 70°C) to 1.00 (device annealed at 120°C). Clearly, α is close or equal to unity for a small difference between the electron and hole mobil-ity (corresponding to high annealing temperatures), while α deviates significantly from unity for large differences in mobility (low annealing temperatures).
In order to check whether the photocurrent is space-charge-limited, let us take a closer look at the voltage dependence of Jph. Figure 4.4(a) shows Jph versus effective applied voltage V0−Va of a device annealed at 70°C under 1.15 kW/m2illumination.
Such a plot typically shows three regimes: a linear dependence of Jphon V0−Vafor small fields, a square-root part (the space-charge-limited regime due to the large mobility dif-ference), and a gradual transition to saturation of Jphat high fields corresponding to high reverse-bias (where Jph = qGL). Figure 4.4(b) shows the intensity dependences of the photocurrent at three different V0−Vacorresponding to the various regimes; clearly, the intensity dependence changes when going from low to high V0−Va. The extent of the space-charge-limited regime depends on the thickness of the active layer and on light in-tensity: when the light intensity is increased, the space-charge-limited regime grows and extends to higher V0−Va. In the present case the photocurrent is space-charge-limited
100 1000
Figure 4.4: (a) Experimental photocurrent density Jph as a function of effective applied voltage V0−Va for a device annealed at 70°C under illuminated at various intensities (symbols). The dashed line corresponds to a square-root dependence of Jphon V0−Va, while the arrows indicate the intensity dependence at 0.1 V, short-circuit, and 1.5 V, respectively. The corresponding current densities as a function of light intensity are shown in (b) (symbols), together with linear fits to the data (lines).
for the effective applied voltage range 0.1–0.35 V (the square-root regime with Jph∝I0.8), as shown in Fig. 4.4.
Furthermore, Fig. 4.4 shows that Jphindeed saturates at high enough (reverse bias) voltages as indicated by the linear dependence on intensity at V0−Va=1.5 V. It should be noted that as V0is typically 0.03 V larger than the open-circuit voltage, short-circuit conditions correspond to V0−Va =0.64 V. Therefore, the short-circuit current, at V0− Va=0.64 V, corresponds to a regime where the transition from the space-charge-limited regime to the saturation regime occurs and, as a consequence, 0.8<α<1.
4.2.1 Numerical results
The numerical model introduced in chapter 2 enables one to investigate theoretically the intensity dependence of Jsc. Figure 4.5 shows fits to the current-voltage characteristics of devices annealed at different temperatures illuminated at 1.15 kW/m2(no filter). The values of the parameters used in obtaining these fits are listed in Table 4.1. The modeling of P3HT/PCBM devices is discussed in greater detail in Ref. [11]. By decreasing the generation rate G proportionally to the intensity, the value of α predicted by the model can be determined. The filled symbols in Fig. 4.3 denote the simulation results; clearly these results are in good agreement with the experimental data, showing that the MIM model describes the intensity dependence of Jsccorrectly.
What about the influence of bimolecular recombination on α? The numerical device model enables one to address this influence by increasing the recombination strength in
1 10 100
Figure 4.5: (a) Current density under illumination JLas a function of applied bias Vaof devices annealed at various temperatures (symbols). The lines denotes the fits made with the numerical model. (b) The exponent α as a function of recombination strength krnormalized to the value used in the fit to the experimental data [Eq. (2.24)], showing that α is only weakly dependent on kr.
Table 4.1:Overview of parameters used in the fits to the data of Fig. 4.5(a).
parameter unit 70°C 90°C 100°C 120°C
µn m2/Vs 1.1×10−7 1.5×10−7 2.0×10−7 2.0×10−7
0.0 0.2 0.4 0.6 0.8 1.0
Figure 4.6: Numerical results for devices annealed at two different temperatures at a bias of Va= 0.3 V. Part (a) shows the electron (gray lines) and hole (black lines) densities, while part (b) shows the potential.
the numerical calculations for the device annealed at 70°C. Figure 4.5 shows the result-ing α when the recombination strength kr is increased up to two orders of magnitude. It appears that α is only weakly dependent on kr; even increasing the bimolecular recom-bination strength by a factor of 100 does not change α. This observation confirms that bimolecular recombination does not account for the observed sublinear dependence of Jscon intensity. Note, however, that this does not imply that bimolecular recombination is not an important loss mechanism with respect to the performance.[12]
To illustrate the effects of space charge build-up, Fig. 4.6 shows the simulated carrier densities and potential of devices annealed at 70°C and 120°C at a bias Va= 0.3 V. At this bias the former device is space-charge-limited (see Fig. 4.4(a) with V0−Va= 0.34 V), while the latter is not. As discussed in subsection 4.1.2, there exist three regimes in the space-charge-limited device, see Fig. 4.6(a): Near the cathode the electron density is much larger than the hole density. Next to this region, there exists a balance between electron and hole density, yielding a neutral region. Near the anode, the holes dominate the device, resulting in a large net space charge and concomitant large voltage drop, as indicated in Fig. 4.6(b). The field distribution in this device bears the characteristics of space charge effects as shown in Fig. 4.2(b).∗ In the case of more balanced charge transport, as for the device annealed at 120°C, the extraction of holes is not the limiting factor, at least at normal (1 Sun) intensities. Therefore, the field in the device is much more homogeneous and the absolute difference between electron and hole density is not as pronounced.
∗Note that Fig. 4.2(b) shows the electron energy, while Fig. 4.6(b) shows the potential.
4.3 Conclusions
In this chapter, two simple analytical models have been introduced. It follows that the short-circuit current should show a power law behavior on intensity, Jsc ∝ Iα, with 0.75 < α < 1, depending on the mobility of both charge carriers: When both types of charge carriers have a sufficiently high mobility, α will be close to unity. On the other hand, when (only) one of the charge carriers has a very low mobility, the solar cell will suffer from a build-up of net space charge, resulting in α<1.
The application of the numerical model, as outlined in chapter 2, confirms that the experimentally observed intensity dependence is indeed caused by space charge effects.
Moreover, increasing the bimolecular recombination strength does not change α, hence, recombination losses per se do not account for the intensity dependence.