The polymer-based photovoltaic (PV) cells consisting of conjugated polymer as electron donor (D) and nanocrystals as electron acceptor (A) are of great interest due to their advantages over conventional Si-based cells, such as low cost, easy-processability, and capability to make flexible devices 1 2 3 . Generally, the p-type conducting polymer acts as both electron donor and hole conductor in the photovoltaic process of the device, while the n-type semiconductor serves as both electron acceptor and electron conductor. The electron donor and acceptor can be intermixed into bulk architecture or cast into a bilayer structure in the PV devices 4 5 6 7 8 9 10 11 12 13 . The latter architecture is attractive for efficient devices, because the photogenerated electrons and holes are, to a great extent, confined to acceptor and donor sides of the D/A interface, respectively, where the spatial separation of electrons and holes will minimize the interfacial charge recombination and facilitate the transport of charge carriers toward correct electrodes with greatly reduced energy loss at wrong electrodes 1 2 3 .

The primary processes involved in the photocurrent generation in a polymer-based PV cells include the exciton generation in the polymer after absorption of light, exciton diffusion toward the D/A interface, exciton dissociation at the D/A interface via an ultrafast electron transfer. The kinetics of the charge-carrier separation and recombination at the D/A interface imposes a great effect on the cell efficiency, and modeling the kinetics of the interfacial charge separation and recombination will offer a good way to understand the efficiency-limiting factors in the devices and to inform experimental activities. For this purpose, several theoretical models dealing with the interfacial charge separation and recombination have been developed in the past years. However, most of them are based on either Monte Carlo (MC) simulation 14 15 16 17 18 19 20 21 or numerical calculations 22 23 , and only a few models offer analytical expressions 5 24 25 26 . Furthermore, the previous studies mainly focused on understanding the influences of interfacial dipoles 14 20 , energetic disorder 15 20 , light intensity 17 , interface morphologies 18 19 20 21 22 , and electrostatic interactions 20 , on the interfacial charge separation and recombination at the organic/organic interfaces. The quantitative analysis of the charge transfer mechanism at the organic/inorganic interfaces in the polymer-based PV cells has been scarcely explored so far. Commonly, the photoinduced interfacial charge transfer from the polymers to inorganic semiconductors is explained by the exciton dissociation at the D/A interface due to the favorable energy match between the D and A components, without considering the role of the interfacial electric field 16 27 28 29 30 31 . Breeze et al 5 proposed an analytical expression including the interfacial electric field for the exciton dissociation efficiency in bilayer MEH-PPV/TiO₂ photovoltaic device, which only expresses the dependence of exciton dissociation efficiency on the polymer layer thickness, not on the TiO₂ layer thickness. To understand the influence of TiO₂ layer thickness on the exciton dissociation efficiency, one needs to consider the electrical properties of the system. In other words, more factors, such as voltage drop across the TiO₂ layer, field-dependent mobility, field-dependent exciton dissociation, and charge recombination at the D/A interface, are necessarily to be incorporated into the model.

In this article, we propose a simple analytical model to describe the exciton dissociation and charge recombination rates at the D/A interface for the bilayer MEH-PPV/TiO₂ cells by modeling the photocurrent-voltage characteristics of the devices. Not only this model is successful in describing the effect of the internal electric field at the D/A interface on exciton dissociation efficiency, but also describes the dependence of the exciton dissociation efficiency on the polymer and TiO₂ layer thicknesses. We verify our model by fitting the measured experimental data on bilayer MEH-PPV/TiO₂ devices under different conditions. The results obtained from the model show that the photocurrent of the devices is determined by the competition between the exciton dissociation and the charge recombination at the D/A interface; the exciton dissociation efficiency increases with either the increase in the forward electric field or the decrease in the thicknesses of polymer and/or TiO₂ layers. In addition, it is found that a higher incident light intensity leads to a higher photocurrent density, but a lower exciton dissociation efficiency.

Poly[2-methoxy-5-(2-ethylhexyloxy)-1,4-phenylenevinylene] (MEH-PPV) (Avg. M _n = 40000-70000) was purchased from Aldrich (product of USA). Titanium tetraisopropoxide [Ti(O ^i- Pr)₄] (Acros, 98+%) was used as TiO₂ precursor. The bilayer PV devices with a structure of ITO/TiO₂/MEH-PPV/Au, as shown in Figure 1, were constructed by spinning down first a nanostructured titanium dioxide (TiO₂) layer and then a MEH-PPV layer over indium tin oxide (ITO, ≤15 Ω/∀, Wuhu Token Sci. Co., Ltd., Wuhu, China) sheet glass, as described elsewhere 11 . The current-voltage (J-V) characteristics were measured on a controlled intensity modulated photo spectroscopy (CIMPS) (Zahner Co., Kronach, Germany) in ambient conditions. The devices were illuminated through ITO glass side by a blue light-emitting diode (LED) as light source (BLL01, λ _max = 470 nm, spectral half-width = 25 nm, Zahner Co., Kronach, Germany). A reverse voltage sweep from 1 to -1 V was applied and the current density under illumination (J _L) was recorded at 300 K. In order to determine the photocurrent, the current density in the dark (J _D) was also recorded, and the experimental photocurrent is given by J _ph = J _L - J_D 24 26 32 , as shown in Figure 2. From the resulting J _ph-V characteristics the compensation voltage (V ₀) was determined as the bias voltage where J _ph = 0 (inset to Figure 2). During all measurements, the gold and ITO contacts were taken as negative and positive electrodes, respectively, and the effective illumination area of the cells was 0.16 cm².

In order to calculate the electric fields E _p and E _n, the accumulated charge density at the D/A interface is assumed to be a constant and Q=1.0 × 10^-4 C/m² 33 . We find that Q has a weak influence on the calculated results by our model, for which the reason may be that the internal electric field in the devices is only slightly modified due to the band bending created by the accumulation of the charge carriers at the D/A interface 24 . Therefore, it is reasonable that we simply assume Q is a constant. In spite of the parameters ε _p = 4ε ₀ which is comparable to ε _p = 3ε ₀ 45 , ε _n = 55ε ₀ 49 , α _p(λ = 470 nm) = 10⁵ cm^-1, and L _p = 15 nm 12 13 50 , there are still three parameters (i.e., λ, k ₀/v ₀, and β) needed to obtain J _ph by Equation 15. Our calculated data revealed that the shape of J _ph-V curve is strongly dependent on the values of λ, but less dependent on the values of k ₀/v ₀ and β. Therefore, the parameter λ can be first obtained by curve fitting taking the order of magnitude of 10^-5 for k ₀/v ₀ and that of 10^-3 for β; then, the values of k ₀/v ₀ and β can be obtained by the best fit. In our model, we take λ = 3 and β is a constant with a value of 5 × 10^-3 V^-2. Finally, the ratio k ₀/v ₀ is the only adjustable fit parameter in fitting the experimental photocurrent. Since k ₀ and v ₀ are zero-field recombination rate constants, the ratio k ₀/v ₀ is independent of the electric field. However, the ratio k ₀/v ₀ depends on the used materials or the geometry of the devices 48 such as the TiO₂ (polymer) film thickness as shown in Figure 4.

Note that, all the following theoretical curves were obtained by considering the experimentally determined compensation voltage V ₀. As shown by the solid lines in Figure 4, the excellent fits to the photocurrent-voltage characteristics of three types devices are obtained using the parameters described above. During the calculations, we use different k ₀/v ₀ values to fit the photocurrent-voltage characteristics of the differently structured devices (Figure 4a,b,c) and the same cell under the varied illumination intensities (Figure 4c,d,e). In Figure 4, it can be seen that the photocurrent increases as the applied voltage turns from reverse to forward direction, and subsequently tends to saturate at higher forward voltages. This phenomenon can be attributed to the dependence of the exciton dissociation efficiency η _CT on the internal electric field (Equation 14), since the efficiency η _ED of exciton dissociation by charge transfer at the D/A interface is constant and the efficiency η _CC of charge collection at electrodes is equal to 1 (Equation 4) 39 . As suggested from Figure 3a, the exciton dissociation efficiency at the D/A interface increases with increasing the forward electric field strength (i.e., the forward applied voltage), and finally approach unit when the forward electric field strength is large enough. In order to examine the dependence of η _CT on the applied voltage V, the TiO₂ and polymer film thicknesses and illumination intensity, we plot the expression η _CT from Equation 14 for all devices, as shown in Figure 5.

Figure 5a shows that, for the devices with different TiO₂ thicknesses (d), when V - V ₀ > 0, i.e., E _p(E _n) > 0, η _CT increases with the increasing forward applied voltage, indicating that the forward electric field is beneficial to the exciton dissociation efficiency as indicated in Figure 3a. When the forward electric field is large enough (V > -0.4 V here), η _CT for the device with d = 65 nm is larger than the calculated one for the device with d = 120 nm, which is in agreement with the result that a thicker TiO₂ film leads to a higher series resistance and a lower photocurrent 11 .

As for the devices with different polymer thicknesses (l) (Figure 5b), the similar dependence of the dissociation efficiency η _CT on the applied voltage is obtained, i.e., a higher the forward electric field results in a larger exciton dissociation efficiency η _CT. However, the thicker polymer film leads to a much smaller exciton dissociation efficiency in the whole applied voltage region. It is very likely due to the slower hole transfer rate in the polymer film as a result of the weakened internal electric field by the increased polymer film thickness, which leads to the smaller exciton dissociation rate at the D/A interface and further the lower exciton dissociation efficiency 5 51 .

Figure 5c shows the influences of various incident intensities on the exciton dissociation efficiency η _CT. It is found that η _CT decreases with increasing the incident intensity at same applied voltage. The similar phenomenon that the efficiency of charge separation per incident photon decreases with increasing the incident light intensity has also been observed in bilayer TiO₂/PdTPPC 16 and TiO₂/P3HT 40 cells in the absence of internal electric field, and was attributed to the occurrence of exciton-exciton annihilation within the polymer layer. In our case, this phenomenon can be understood as follows. Although a higher incident intensity creates more excitons in the polymer layer and generates higher free electron and hole densities at the D/A interface, the higher densities of the charge carriers at the interface increases the charge recombination probability at the same time; moreover, as discussed above, the increasing forward applied voltage will enhance the exciton dissociation efficiency at the D/A interface. In other words, there is a competition between exciton dissociation and charge recombination at the D/A interface and the last result is that the exciton dissociation efficiency η _CT decreases as shown in Figure 5. This important result indicates that the increase in the photocurrent density under a higher incident light intensity is due to the increase in exciton density rather than the increase in the exciton dissociation efficiency, which is useful to optimize device performance.

An analytical model for the photocurrent-voltage (J _ph-V) characteristics of the bilayer polymer/TiO₂ photovoltaic cells is developed, where the generation of free charges takes place via dissociation of photogenerated excitons. The model describes the dependence of photocurrent generation on the device geometry and gives an analytical expression for the exciton dissociation efficiency. The experimental J _ph-V data of the MEH-PPV/TiO₂ devices are satisfactorily fitted to the model. Results show that increasing TiO₂ or the polymer layer in thickness will reduce the exciton dissociation efficiency η _CT in the device and further the photocurrent. It is found that the photocurrent is determined by the competition between the exciton dissociation and charge recombination at the D/A interface, and the increase in photocurrent under a higher incident light intensity is due to the increased exciton density rather than the increase in the efficiency η _CT. Our results indicate that a thinner polymer layer combined with a thinner TiO₂ layer favors the higher exciton dissociation efficiency in the bilayer devices. The model will provide information on optimization of device performance by investigating the effects of material parameters on device characteristics.

A: acceptor; CIMPS: controlled intensity modulated photo spectroscopy; D: donor; HOMO: highest-occupied molecular orbital; ITO: indium tin oxide; LED: light-emitting diode; MC: Monte Carlo; PV: photovoltaic; TiO₂: titanium dioxide.

The authors declare that they have no competing interests.

CC performed the experiments, developed the theory model, and drafted the manuscript. FW participated the theoretical analysis. HG and WS participated the device preparation. MW conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.