1. Introduction

Organic photovoltaic (OPV) devices have become increasingly popular in recent years, largely because of their flexibility, lightweight, low cost, and ease of production [1]. Power conversion efficiency (PCE) of these devices has recently reached 18% for single layer OPVs [2] and 18.6% for tandem devices [1]. The remarkable progress in PCE is achieved through the continuous development of new materials and improvement of device fabrication technologies. To fully exploit the potentials of OPVs a detailed knowledge of active layer morphology and a complete understanding of device physics is required.

The active layer thickness (ALT) dependent performance of OPVs is one of the unsolved issues which hides the answers to many key questions related to these devices. Therefore, great attention should be paid to this topic in terms of the systematization of existing research results as well as in the form of new experiments and theoretical approaches that can lead to important findings.

In 1998, considering bilayer PEOPT(poly(3-(4′-(1^″,4^″),7^″-trioxaoctyl)phenyl)thiophene)/C₆₀ organic photodiode, Roman et al. [3] proposed that optical interference has a significant influence on the efficiency of the device. The authors stated that the incident photon to electron conversion efficiency (IPCE) is strongly affected by the amplitude of the optical electric field on the PEOPT/C₆₀ heterojunction, which is, according to them, the result of the multiple interference in the multilayer OPV structure. By using the transfer matrix model (TMM) they calculated the square of the optical electric field |E|² at the PEOPT/C₆₀ heterojunction for devices with different PEOPT and C₆₀ film thicknesses. They determined the ratio of calculated |E|² for two devices with different PEOPT layer thicknesses and compared it with the measured IPCE ratio. The procedure was repeated for two devices with different C₆₀ layer thicknesses. The agreement was reasonably good. They concluded that optical interference plays a significant role in OPVs and that layer thicknesses are crucial in optimizing the device's performance. In their following research [4] again using the TMM, they have modeled the photocurrent spectra (PCS) of PEOPT/C₆₀ bilayer devices with different C₆₀ layer thicknesses but the agreement with the experimental data was not convincing and additional explanations had to be introduced. The dissociation of excitons on the C₆₀/Al interface was included in the modeling in order to obtain a reasonable fit of modeled data to experimental data. In 2006, Douglas et al. used the TMM together with the drift-diffusion model (DDM) developed for organic solar cells (OSCs) to predict the short-circuit current (I_sc) ALT dependence in MEH-PPV(poly-2-methoxy-(2′-ethylhexyloxy)-1,4-phenylene):PCBM([6,6]-phenyl-C₆₁-butyric acid methyl ester) based solar cells [5,6]. Although the experimentally obtained I_sc changed oscillatory with the ALT, measured and calculated data did not show the exact match. In 2007, Monestier et al. [7] used a similar model to reproduce the thickness-dependent I_sc behavior of P3HT(poly(3-hexyl-2,5-dimethylthiophene):PCBM solar cells. The I_sc dependence on the ALT again showed oscillatory character but the simulations did not fully follow the change. In 2008, Moule et al. [8] compared the TMM-calculated external quantum efficiency (EQE) ALT dependencies at different incident light wavelengths to the measured ones for P3HT:PCBM based OPV devices. Two essential types of deviation were noticed. The simulated and measured EQEs did not match either by value or by maxima position. It was further found that the EQE is anti-correlated to the optical field intensity determined by the TMM in the vicinity of the PEDOT:PSS(poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate))/P3HT:PCBM interface, which led the authors to assume that the reduced generation zone (RGZ) is formed at this junction. To perform better matching between the experiment and theory, a half-Gaussian shape of RGZ was suggested. Various loss mechanisms in the RGZ were considered (vertical segregation, bimolecular and geminate recombination, reduced exciton dissociation), and reduced exciton dissociation was adopted as dominant. The RGZ model of EQE showed a better agreement with the experiment, but the match was still not good enough, especially for the 500 nm and 600 nm incident light wavelengths. Since it has been established that the thickness-dependent behavior of OSCs cannot be explained solely by optical interference, many processes and physical quantities in OSCs were reported to be ALT dependent. In 2012, Kirchartz et al. [9] indicated that the space charge in thicker OPV devices makes electric field-free regions with a low charge carrier collection efficiency (CCCE), thus, the CCCE was found to be ALT dependent. One year later, it was established by Namkoong et al. that ALT has a significant impact on recombination processes in OSCs [10]. They argued that the defect density increases with PCDTBT(poly[N-9″-hepta-decanyl-2,7-carbazole-alt-5,5-(4′,7′-di-2-thienyl-2′,1′,3′-benzothiadiazole)]):PCBM thin film thickness and that recombination mechanism undergoes the change from a single-level trap assisted recombination in thinner devices to a coupled trap-assisted recombination in thicker devices. Optical simulations based on the TMM were included in their research. In 2015, Park et al. [11] investigated the internal quantum efficiency (IQE) ALT dependence in PTB7(poly[(4,8-bis-(2-ethylhexyloxy)-benzo(1,2-b:4,5-b′)dithiophene)-2,6-diyl-alt-(4-(2ethylhexyl)-3fluorothieno [3,4-b]thiophene-)-2-carboxylate-2-6-diyl)]):PC71BM([6,6]-phenyl-C₇₁-butyric acid methyl ester) based solar cells. They calculated the IQE by dividing the EQE with the ratio of the number of photons absorbed in the active layer determined by the TMM to the number of incident photons. They found that both the IQE and the CCCE in OSCs depend on the ALT. In 2019, Sharma et al. [12] investigated the inverted PTB7:PC70BM([6,6]-phenyl-C₇₁-butyric acid methyl ester) based OPV and again concluded that thicker devices exhibit reduced CCCE. Dark current-voltage (I-V) measurements revealed a large defect density in thicker devices, while the impedance spectra analysis indicated restrictions in the charge carrier transport in these devices. Finally, in 2020, the ALT dependent charge carrier mobility and injection barrier heights were reported in P3HT:PCBM solar cells [13]. A non-monotonic change of the charge carrier mobility with the ALT was observed based on the dark I-V measurements.

It is interesting to point out that the absorption coefficient of organic thin films (OTFs) determined by the application of the Beer-Lambert law was also found to be thickness dependent [14,15]. The absorption coefficient of MEH-PPV thin films on quartz substrates changes its value with the film thickness in an oscillatory manner [15]. The thickness-dependent absorption coefficient was explained by optical interference effects [14].

Another feature that depends on the ALT in bulk heterojunction OPVs is an optimal blend ratio. It was shown that the blend ratio which provides the best performance of P3HT:PCBM and OC1C10–PPV(poly(2-methoxy-5-(3′,7′-dimethyloctyloxy)-p-phenylenevinylene)):PCBM based OPVs is not static but strongly depends on the ALT [16]. Not only the blend ratio but also the optimal domain size varies significantly with the ALT [17].

There is another feature that can be crucial for a complete understanding of ALT dependent behavior of OPVs. It is a relation between ALT and its 3D morphology of only spin-coated films and thermally annealed spin-coated films. It is well known that OTFs of different thicknesses contain morphological differences arising from the fact that they cannot be produced under the same fabrication conditions (solvent concentrations and/or spin-coating speeds have to be different to produce different ALT) [18]. However, these differences are reported not to be substantial as they can be deduced from the absorption spectra of OTFs with the same thickness but produced from solutions of different concentrations and by different spin-coating speeds [14]. The work of Bavel et al. [19] brings very interesting and significant results. In their research, the crystallinity of 50, 100, and 200 nm thick P3HT:PCBM films were examined for only spin-coated films, and after the application of thermal annealing. The transmission electron microscope (TEM) was used to record the 3D morphology of P3HT:PCBM films and the crystallinity was determined by the electron diffraction measurements. Extraordinary, it was shown that the degree of the P3HT crystallization changes with the film thickness in a non-monotonic fashion in both cases. In the case of only spin-coated films, the variation of the P3HT crystallization degree with OTF thickness was not pronounced. After thermal annealing P3HT crystallinity dramatically changed, and the differences between films became much more significant. The best crystallization degree was achieved in a 100 nm thick P3HT:PCBM film. It was also shown that the distribution of crystallized P3HT domains (P3HT nanowires) in the annealed P3HT:PCBM film is not uniform and is strongly affected by the film thickness. The results presented in the work of Bavel et al. [19] imply that there is a strong correlation between the degree and the form of crystallization in OTFs and their thickness. This correlation produces a non-monotonic change of crystallinity with film thickness and it is much more pronounced in the heated OTFs. It should be stressed that heating is an integral part of OTFs fabrication. Even if they are not thermally annealed, spin-coated films are regularly heated to dry [20].

To properly perceive and uncover the intrinsic impact of OTF thickness in OPVs, we need to know whether or not optical interference plays an important role in these devices. If the optical interference does not have a significant effect on OPVs operation, then the research results that take it into account are misinterpreted.

In this paper, in order to understand the thickness-dependent physics in the P3HT:PCBM based OPVs, different optical and electrical measurements were performed. At first, the optical measurements on the thin P3HT:PCBM films of eight different thicknesses spin-coated from the solvent with the same P3HT concentration were conducted and the corresponding optical parameters were determined from the measured transmittance spectra. Although the optical interference was taken into account, the determined refractive index n and the extinction coefficient k of P3HT:PCBM films on quartz substrates were not unique. Both n and k showed non-monotonic change with the OTF thickness. The electrical OPV characterization involved the investigation of the ITO(indium tin oxide)/PEDOT:PSS/P3HT:PCBM/Al devices with six different P3HT:PCBM ALTs under photodetector and solar cell working conditions. In the photodetector operation mode, the PCS under monochromatic visible light irradiation were measured. In the solar cell working regime, the current density-voltage (J-V) characteristics were recorded under solar simulator light. The simulations of the PCS and J-V characteristics were performed using the DDM. When the standard DDM, which uses the TMM to determine the charge carrier photogeneration rate profile in the active layer, was applied, a poor agreement between the measured and simulated PCS was achieved. A detailed analysis of the measured PCS and a comparison with the simulation indicated that interference effects are not significant in the ITO/PEDOT:PSS/P3HT:PCBM/Al devices. The constant profile photogeneration rate calculated from the measured absorption spectra of P3HT:PCBM thin films with different thicknesses gave a good prediction of PCS peak positions for thinner devices showing a symbatic photocurrent response with the absorption coefficient [21]. To completely match the calculated and measured PCS and J-V characteristics, it was necessary to introduce the electric field dependent IQE which follows the Poole-Frenkel (P-F) expression with thickness-dependent parameters and to let the electron and hole mobilities be thickness dependent too. Although surprising, according to our analysis, it seems that the P3HT:PCBM films of different thicknesses have a distinct morphology, and thus, their parameters are different. The parameter values change non-monotonically with the film thickness.

2. Experiment

All fabrication and experiments were done in the Institute for Micromanufacturing (IfM) of Louisiana Tech University (LaTech), Ruston, LA, USA.

2.1 Materials

Regioregular electronic grade P3HT and PCBM, chlorobenzene, and HCl were purchased from Sigma Aldrich. PEDOT:PSS was purchased from Heraeus. Indium tin oxide-coated boro-aluminosilicate glass substrates 25 × 25 × 1.1 mm with 4-10 Ω/sq were purchased from Delta Technologies.

2.2 Device fabrication

The glass/ITO/PEDOT:PSS/P3HT:PCBM/Al devices were manufactured inside the class 1000 cleanroom in the IfM of LaTech with the fabrication process similar to the air process published by Nam et al. [22]. The ITO substrates were cleaned with acetone, isopropanol, and rinsed with DI water. Each substrate was patterned with the standard photolithography technique (Shipley PR 1813 positive photoresist and MF-319 developer were used) to define a shared anode on the substrate. The ITO was etched with the 20% HCl heated to 75 °C for about 2 minutes. PEDOT:PSS was ultrasonicated for 10 minutes, then filtered with a 0.45 μm PVDF filter and statically spin-coated onto the substrates for 30 s at 4000 RPM, where RPM stands for rounds per minute. P3HT and PCBM were mixed with chlorobenzene separately at 50 °C and allowed to stir overnight with stir bars. The next morning, the solutions were combined and filtered through a 0.45 µm PTFE filter and stirred for one hour. The P3HT:PCBM ratio by weight was 1:1. The devices with active layer thicknesses of 80, 90, 130, and 190 nm were fabricated from the P3HT:PCBM chlorobenzene solutions of 12 mg/ml concentration with the dynamic spin-coating with speeds of 2000, 1500, 1000, and 600, respectively, for 50 s. For the 230 nm and 300 nm devices, the solution concentration was 18 mg/ml to enable thicker films, and the spin-coating speeds were 1000 RPM and 800 RPM, respectively. The 100 nm Al electrodes (6 per substrate) were fabricated through a shadow mask by thermal evaporation (Denton) at a rate of 0.4 nm/s and pressure of 1 µTorr. The devices were annealed for 15 min at 150 °C. The active area of each device was 10.5 mm².

2.3 Optical sample fabrication

The P3HT:PCBM thin films on fused quartz were fabricated inside the class 1000 cleanroom in the IfM of LaTech. The 1 × 1 × 1 mm quartz samples from Ted Pella were cleaned with acetone, isopropanol, and rinsed with DI water. The fused quartz samples for optical testing were processed from the same P3HT:PCBM solutions and with the same spin-coating speeds as their photovoltaic device counterparts. The samples with additional ALTs of 105, 145, 160, and 180 nm were processed from the P3HT:PCBM solution in chlorobenzene with 12 mg/ml concentration. Each sample was dried on a hot plate at 70 °C for 5 min before optical tests.

2.4 Optical measurements

Optical transmittance and reflectance of P3HT:PCBM on fused quartz samples were measured with the thin-film analyzer (Filmetrics F10-RT). The measurement procedure includes the baseline measurement that serves as a calibration of the optical system, and then the measurement of the sample under test. Measured films must be optically smooth and flat to be measured. FILMeasure software determines the thin-film thickness and refractive index n and extinction coefficient k spectra from the optical data. Filmetrics Inc. specifically analyzed several of our P3HT:PCBM thin film samples on fused quartz and made a specific-purpose model called “P3HT:PCBM Recipe” for FILMeasure based on a Bridge-Lorentzian model. This P3HT:PCBM Recipe was used to calculate the film thickness and optical “constants” (n and k) spectra of each P3HT:PCBM on a fused quartz sample. The P3HT:PCBM thin film thicknesses on fused quartz were also measured with the surface profilometer (Veeco).

2.5 Optoelectronic measurements

The solar cells were tested under a solar simulator (Spectra Physics) 1000 W/m² (corresponding to the AM 1.5 spectrum) optical power density and J-V characteristics were measured with the Keithley 2400 SourceMeter. The output power density from the solar simulator was measured with the calibrated Oriel 91150V reference solar cell. The electrical characterization of polymer photodetectors was carried out with the monochromator (Oriel Cornerstone 260), which illuminated the polymer device with a selected single wavelength in a range of 350 nm to 750 nm. The monochromator output optical power density was measured with the calibrated Newport 818-UV silicon photodiode and the Newport 1936-R Power Meter. The polymer photodetector was connected to a Keithley 6487 Picoammeter which provided bias voltage and measured the resulting current.

For considered OPV devices of different ALTs, the six samples for each ALT value were fabricated and tested in both photodetector and solar cell regime. The OPV devices which showed the best performance (the highest power conversion efficiency) in the solar cell regime were chosen to be analyzed by the DDM. The average parameters for solar cells with different ALTs are presented in Table 1.

Table 1. The average parameters for solar cells with different ALTs

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3. Results and discussion

3.1. Thickness dependent optical parameters of P3HT:PCBM thin films

Optical measurements of P3HT:PCBM thin films with eight different thicknesses of 80, 90, 105, 130, 145, 160, 180, and 190 nm were conducted. The thin P3HT:PCBM films were cast from a solution with the same P3HT concentration (12 mg/ml) by changing only spin-coating speed. This was done because the samples prepared with different solution concentrations may exhibit different microphase separation and, consequently, may no longer have the same n and k [23]. The thicknesses 80, 90, 130, and 190 nm are the same as ALTs in four (out of six) considered OPV devices, and the remaining two ALTs (230 and 300 nm) couldn’t be prepared from the solution with the same concentration. The P3HT:PCBM thin films with thicknesses 105, 145, 160, and 180 nm were added in order to better comprehend the effect of thickness on P3HT:PCBM optical parameters. For each P3HT:PCBM thin film the refractive index and extinction coefficient spectra were extracted from measured optical transmittance spectrum with FILMeasure software which accounts for multiple interference (see Experimental section). The transmittance spectra were excellently reproduced by the software that derived n(λ) and k(λ) which can be seen from Fig. 1(d). Also, the film thickness values determined by FILMeasure software were very close to those measured with the profilometer. The profilometer thicknesses and FILMeasure thicknesses are given in Table 2 for comparison. The determined n(λ) and k(λ) for considered P3HT:PCBM films are presented in Figs. 1(a) and (b), respectively. Although the obtained results are in conformity with the literature [7,23–25], it can be noticed from Figs. 1(a) and (b) that n(λ) and k(λ) change to some extent with the film thickness. To better perceive and characterize the noticed variation of optical parameters with the film thickness k(λ) peak value k_max and n(λ) saddle point value n_sd were depicted in Fig. 1(c) as functions of the film thickness. The non-monotonic thickness dependence of k_max and n_sd can be observed from Fig. 1(c). It should be stressed that the existing variation of k_max and n_sd can in no way be interpreted as the scatter of data. Repeated measurements for one P3HT:PCBM thin film thickness showed the standard deviation of 0.004 for n and 0.002 for k.

 

Fig. 1. Refractive index n (a) and extinction coefficient k (b) for 80, 90, 105, 130, 145, 160, 180, and 190 nm-thick P3HT:PCBM thin films. (c) Peak values of k(λ) and saddle point values of n(λ) as functions of P3HT:PCBM thin film thickness, and (d) The measured and Filmetrics simulated transmittance spectra for 80, 130, and 190 nm-thick P3HT:PCBM thin films.

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Table 2. Comparison of the profilometer thicknesses and FILMeasure thicknesses

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A comprehensive study of various methods for extraction of P3HT:PCBM optical constants based on spectroscopic ellipsometry and transmittance measurements is given in the work of Ng et al. [23] The fitting procedures based on different optical models (isotropic and anisotropic models using generalized oscillator model or a model incorporating a Huang−Rhys vibronic envelope) were applied to single films or simultaneously to multiple films of different thicknesses to get a unique set of optical constants. The surface roughness was also modeled. However, the comparison of the film thicknesses derived from fitting and their values determined by any non-optical method (profiler or scanning electron microscopy) was omitted. Also, in the work of Gaudin et al. [14] a variable angle spectroscopic ellipsometry (VASE) was used to determine a unique set of optical constants for MEH-PPV films with thicknesses in the range of 18–178 nm. Again, all thicknesses were derived from VASE measurements and they were not compared to their values obtained in any other way. Certainly, if the film thickness is a fitting parameter in the procedure for extracting optical constants, then for getting a correct result, a condition must be set that the thickness obtained by fitting must be approximately equal to the actual one. In this paper a very good reproduction of the transmittance spectra of the P3HT:PCBM thin films with different thicknesses is achieved with realistic thickness values obtained from fitting. The determined n and k vary with the film thickness in a non-monotonic fashion.

3.2. Electrical characterization of P3HT:PCBM based photovoltaic devices

3.2.1 DDMs used in simulations

The DDM developed for inorganic semiconductor devices is often used for modeling the OPVs [6,26]. The inherent physics of organic materials is involved through the charge carrier photogeneration, transport, and recombination processes. Also, the DDM is usually accompanied by an optical model based on the TMM which is used to determine the singlet exciton photogeneration rate profile in the active layer of the multilayer OPV structure [27]. The DDM models with three different charge carrier photogeneration profiles are used in our investigation to analyze experimental data and to achieve a good reproduction of all PCS and J-V measurements.

3.2.1.1 Model 1: TMM_DDM

The TMM_DDM model used in our simulations consists of Poisson’s equation and continuity equations for electrons and holes [28]. The electron and hole transport are drift-diffusion based on the assumption of constant electron mobility and Poole-Frankel field-dependent hole mobility [29]. The charge carrier recombination is taken to be bimolecular of Langevin type [28]. The TMM was used to determine the optical field profile in the OPVs active layer. It is considered that free charge carriers are formed by the absorption of photons directly and instantaneously with a quantum efficiency equal to 1 [29]. The Dirichlet boundary conditions were applied [6]. The TMM_DDM system of equations was discretized by the Sharfetter-Gummel approach and numerically solved by Newton’s method [30].

3.2.1.2 Model 2: A_DDM

The same model was used excluding interference effects and assuming that the charge carrier photogeneration rate in the OPVs active layer is constant. The photogeneration rate profile G(λ) was calculated based on the absorption spectrum A(λ) measured for each P3HT:PCBM film as:

 

Fig. 2. (a) Absorptance spectra for ITO/PEDOT:PSS/P3HT:PCBM/Al devices with 80, 90, 130, 190, 230, and 300 nm-thick active layer, (b) optical power spectrum of the monochromator and schematics of the ITO/PEDOT:PSS/P3HT:PCBM/Al device, and (c) charge carrier photogeneration rate for ITO/PEDOT:PSS/P3HT:PCBM/Al devices with 80, 90, 130, 190, 230, and 300 nm-thick active layer.

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3.2.1.3 Model 3: PFA_DDM

When an electric field dependent IQE described by the P-F expression [15]:

3.2.2 PCS in photodetector operation mode: optical interference considerations

It is widely accepted that OPVs behave like low-finesse optical cavities in which optical interference plays an important role [31]. The TMM_DDM model which relies on this assumption was, therefore, used to calculate PCS of considered OPVs in the photodetector working regime. The n(λ) and k(λ) obtained with FILMeasure for P3HT:PCBM thin films for six different thicknesses (80, 90, 130, 190, 230, and 300 nm) were used in the TMM calculations. The measured and TMM_DDM simulated PCS of ITO/PEDOT:PSS/P3HT:PCBM/Al devices, under 0 and -4 V reverse bias, for six different P3HT:PCBM film thicknesses, are compared in Fig. 3.

 

Fig. 3. Comparison of measured and TMM_DDM simulated PCS for ITO/PEDOT:PSS/P3HT:PCBM/Al devices with (a) 80, (b) 90, (c) 130, (d) 190, (e) 230, and (f) 300 nm-thick active layer.

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First, we comment on measured PCS. It can be seen from Fig. 3 that OPVs with 80, 90, and 130 nm-thick active layers have PCS whose peaks are positioned at the same wavelength of 550 nm as it is the peak of the absorption-based photogeneration rate G(λ) (Fig. 2(c)). This corresponds to a typical symbatic photocurrent response [21,32]. In thicker devices, the red shift of PCS is apparent (Figs. 3 (d)-(f)). It arises from longer wavelength photons which are absorbed deep in the bulk but far from electrode contacts. These photons produce charge carriers that are not affected by the surface recombination loss mechanism and notably participate in the photocurrent [21,32]. Since they have to pass a long way to be extracted to the external circuit, these carriers are more visible under high reverse voltages than in the zero-bias case. This is exactly what can be seen in our experimentally obtained PCS (Fig. 3). The red shift is more pronounced under -4 V reverse bias than under zero bias.

If we now pay attention to the TMM_DDM simulated PCS presented in Fig. 3 it can be noticed that they poorly match the measured PCS (peak wavelength or photocurrent peak value). The PCS calculated by the TMM_DDM model shows considerable change in peak position and spectral shape with changing d typical for the case of optical interference. On the contrary, the measured PCS exhibits a slight red shift with increasing d while the spectral shape stays nearly the same. Can any other process taking place in OPVs, which is reported in the literature to be ALT dependent [8–13], explain the discrepancy between theory and experiment shown in Fig. 3? It was proposed that the photocurrent is anti-correlated to the optical field intensity determined by the TMM in the vicinity of the PEDOT:PSS/P3HT:PCBM interface. If so, it is reasonable to expect the oscillation in the PCS peak position with the ALTs, which is not observed in the measured PCS. Further, the proposed thickness-dependent IQE, CCCE, recombination rate and charge carrier mobility can only have an impact on the PCS peak intensity; they cannot cause the spectral shift of the simulated PCS necessary to match the experimental data. A very important conclusion follows from the discussion above: the optical interference effects are not pronounced in our P3HT:PCBM based OPVs and thus they are not responsible for their thickness-dependent behavior.

In further discussion, we will limit ourselves to the ITO/PEDOT:PSS/P3HT:PCBM/Al devices with 80, 90, and 130 nm-thick P3HT:PCBM layer in order to avoid modeling the PCS red shift present in thicker OPV devices. To model the PCS red shift, the surface recombination at the PEDOT:PSS/P3HT:PCBM interface should be taken into account.

3.2.3 Comprehensive OPV model

In addition to the PCS calculations conducted for ITO/PEDOT:PSS/P3HT:PCBM/Al devices working in the photodetector regime under monochromatic visible light, the TMM_DDM was used to simulate the J-V characteristics of the same devices under solar simulator light in the solar cell operation mode. The light and dark J-V curves obtained by the TMM_DDM are compared with the measured ones in Fig. 4. According to Fig. 4, the TMM_DDM doesn’t reproduce the measured J-V characteristics well. The disagreement between the theory and the experiment is more pronounced for thinner devices and it mostly refers to the discrepancy in measured and theoretically predicted J-V curve slopes in the first quadrant.

 

Fig. 4. Comparison of measured and TMM_DDM simulated light J-V characteristics for ITO/PEDOT:PSS/P3HT:PCBM/Al devices with (a) 80, (b) 90, and (c) 130 nm-thick active layer. The corresponding measured and TMM_DDM simulated dark J-V curves are compared in the Insets.

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Once we have concluded that the interference doesn’t play a significant role in our OPVs, it was assumed that the absorption profile in the active layer is described by the Beer-Lambert law. Furthermore, for the considered thinner devices, it was reasonable to assume a constant absorption profile (independent of the depth in the active thin film) and, thus, a constant profile of the photogeneration rate. Therefore, the photogeneration rate was expressed by Eq. (1), and the A_DDM was used for PCS and J-V characteristics simulations. In Fig. 5 the A_DDM simulated PCS and J-V curves for devices of different d are depicted together with the measured ones.

 

Fig. 5. Comparison of measured and A_DDM simulated PCS (a)-(c) and light J-V characteristics (d)-(f) for ITO/PEDOT:PSS/P3HT:PCBM/Al devices with 80, 90, and 130 nm-thick active layer. The corresponding measured and A_DDM simulated dark J-V curves are compared in the Insets (d)-(f.).

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As it can be seen in Fig. 5(a), the experimentally obtained PCS peak positions were reproduced well by the A_DDM, but the intensity mismatch remained. By applying the A_DDM, the agreement between the calculated and measured light J-V curves was improved especially in the fourth quadrant which can be deduced from Figs. 4 and 5(d)-(f). In the first quadrant, the measured and simulated J-V curves both, light and dark, still showed substantially different slopes.

To completely match the theoretically predicted and experimentally obtained PCS of ITO/PEDOT:PSS/P3HT:PCBM/Al devices operating in the photodetector mode, an electric field dependent IQE θ(λ) was introduced. The θ(λ) was calculated by dividing the peak intensities of measured and A_DDM simulated PCS for different reverse bias voltages in the range from 0 V to -4 V. The accomplished θ(λ) dependencies for devices of different d very well obeyed the P-F effect given by Eq. (2) as it can be seen in Figs. 6(a)-(c). This confirms the validity of our theoretical approach since it is known that the electric field-assisted charge carrier photogeneration in OTFs indeed follows the P-F effect [15,33]. It should be emphasized that the a, b, and C parameters obtained for the devices of different d were different. Their thickness dependences are shown in Fig. 6(d) and (e). The a and C parameters change in a non-monotonic fashion with d, while b is increased when the thickness increases. The b(d) trend is in accord with Ref. [17]. In order to verify the obtained results, a, b, and C parameters were calculated in the same way for six samples of each ALT and the average values are presented in Table 3. The mean values of the P-F parameters show qualitatively the same thickness-dependent behavior as the parameters of the selected (most efficient in solar cell regime) devices. The parameter a represents the maximum value of IQE that can be achieved in a device with the appropriate ALT.

 

Fig. 6. Calculated θ(E) fitted by P-F expression for ITO/PEDOT:PSS/P3HT:PCBM/Al devices with (a) 80, (b) 90, and (c) 130 nm-thick active layer. The thickness dependencies of the (d) parameters a and C, (e) parameter b, and (f) µ_p0 and µ_n determined by the least square fitting of the PFA_DDM calculated J-V curves to the experimental data.

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Table 3. Average values of P-F parameters, zero field hole mobility and electron mobility

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The introduction of θ(λ) in the A_DDM, i.e. the application of the PFA_DDM led to a nearly perfect overlap of experimental and simulated PCS, but the deviation of theoretical from measured J-V characteristics was not resolved. By analyzing the impact of different PFA_DDM parameters on the calculated J-V curve, it was established that the electron mobility µ_n governs the J-V curve slope in the first quadrant and that the zero-field hole mobility µ_p0 directly affects the short circuit current density. The experimentally obtained J-V curves could only be reproduced by the PFA_DDM if µ_n and µ_p0 were allowed to be thickness dependent. The µ_n and µ_p0 as functions of d determined by the least square fitting of the PFA_DDM calculated J-V curves to the experimental ones are depicted in Fig. 6(d). The electron mobility predicted by our calculations showed a growing trend with ALT which agrees with the results published in the work of Foster et al. for annealed P3HT:PCBM based OPV devices with ALTs around 100 nm [34]. On the other hand, the zero-field hole mobility, as it can be seen from Fig. 6(f), first decreased with the change of ALT from 80 nm to 90 nm, and then it improved as d was further increased. In Ref. [34] the ALT dependent hole mobility has a declining character, but having in mind the work of Bavel et al. [19], which shows that the degree of P3HT crystallization changes non-monotonically with P3HT:PCBM film thickness, it is reasonable to expect a non-monotonic change of the hole mobility as well. The average µ_p0 and µ_n show qualitatively the same change with d as it can be deduced from Table 3. Also, the standard deviations shown in Table 3 indicate that the results for the samples of the same ALT were acceptably uniform except for µ_p0. The significant standard deviations obtained for µ_p0 are directly the consequence of considerable deviations in the J_sc which can be seen from Table 1.

When all thickness-dependent parameters are applied to the PFA_DDM, the measured and simulated PCS and J-V characteristics showed very good agreement, as can be observed in Fig. 7.

 

Fig. 7. Comparison of measured and PFA_DDM simulated PCS (a)-(c) and light J-V characteristics (d)-(f) for ITO/PEDOT:PSS/P3HT:PCBM/Al devices with 80, 90, and 130 nm-thick active layer. The corresponding measured and PFA_DDM simulated dark J-V curves are compared in the Insets (d)-(f.).

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At this point, another important conclusion can be drawn. The presented optical characterization of P3HT:PCBM thin films of different thicknesses and DDM analysis of P3HT:PCBM based OPV devices with different ALTs implies that: the P3HT:PCBM films of different thicknesses are acting as made from the distinct materials characterized by different parameter values. The parameter values change non-monotonically with d. The authors believe that the optical and electrical parameter values of the P3HT:PCBM thin films of different thicknesses determined in this paper are directly related to the morphology of each film, thus, recording the morphology is necessary to give them a physical context. This should be the subject of further research. It seems that the film morphology is correlated with the film thickness. This is supported by the literature as it was shown in the work of Bavel et al. that OTFs of different d, fabricated and annealed under the same conditions, have different degrees and forms of crystallization [19]. The crystallization degree and the distribution of crystallized domains change with the OTF thickness in a non-monotonic fashion [19]. The unresolved correlation between morphology and OTF thickness should be thoroughly investigated in the future.

4. Conclusion

In summary, the thickness-dependent behavior of P3HT:PCBM based OPVs was studied through experimental investigation and theoretical modeling. The optical parameters of P3HT:PCBM thin films of eight different thicknesses were determined from the measured transmittance spectra. It was found that, even after taking the optical interference into account, n and k still exhibit considerable variations with film thickness. The ITO/PEDOT:PSS/P3HT:PCBM/Al devices of six different P3HT:PCBM film thicknesses were fabricated and tested in photodetector and solar cell operation modes. The PCS under monochromatic visible light irradiation were recorded in the photodetector working regime. The J-V characteristics of devices operating as solar cells under solar simulator light were measured. The operation of the device in both examined regimes was modeled by the DDM. When the photogeneration rate profile in the active layer was calculated using the TMM, the theoretical predictions deviated significantly from the experimentally obtained PCS and J-V characteristics. The model using constant photogeneration rate across the ALT which was determined from a measured absorptance for each P3HT:PCBM film of different d gave much better results. The agreement with the experimental PCS was improved for thinner devices showing the photocurrent response symbatic with the absorption coefficient. To completely reproduce the OPVs PCS and J-V characteristics, it was necessary to assume the P-F electric field dependent IQE and thickness dependent material parameters.

The two observations presented in this paper could be of great importance for further OPVs efficiency improvement and optimization. First, the multiple optical interference in OPVs may not be as important as previously thought. Second, there are serious indications set out in this paper that the OTF’s morphology is strongly correlated with the film thickness, especially when thermal annealing is applied, which results in a non-monotonic thickness dependence of optical and electrical OTF parameters. A comprehensive study of the impact of thin-film thickness on the OTF’s crystallinity should be conducted in the future.