1 Introduction

Organic solar cells are a promising technology for electrical energy generation from sunlight due to it is low cost, fast production, flexibility, and already existing production technologies [1, 2]. Over the last two decades, significant progress in the field has been made over different aspects of the technology in order to enhance device efficiency and stability. Researchers tackled the devices in a variety of ways, including manufacturing new materials, introducing different structures, developing experimental procedures for measuring device parameters, and building theoretical models for the faster study of the device [3,4,5]. Organic solar cells can be used in different fields due to their properties, which makes them more suitable than silicon-based devices. Based on the transparent properties of organic devices, it can be used in fields like agriculture applications allowing for sunlight to pass to plants and another portion for energy generation for higher productivity and farm monitoring[6]. The flexibility advantage made it easy for organic solar cells in indoor and outdoor applications in which area and fixation of devices was a major problem. It can be printed on clothes for human sensing applications and implanted with IoT devices in remote areas [7,8,9].

In the literature, many theoretical studies have been conducted to evaluate the possible enhancements of organic devices. In [10], an investigation of the effect of an interfacial layer such as ZnO on device parameters such as resistance and short circuit current has been carried out. Several works have evaluated the effect of active layer thickness on power conversion efficiency, including [11,12,13]. According to [14], optimization of an organic solar cell based on P3HT:ICBA for indoor applications had been accomplished by addressing device structure at varying illumination intensities. In [15], a hybrid theoretical and experimental technique for quantifying distinct recombination types with carrier lifetime was presented, which combined theoretical and experimental techniques. In the last few years, more techniques are being deployed in the field for the sake of rapid enhancement and device understanding. Machine learning algorithms are used for the selection of suitable organic materials as an active layer minimizing the time of experimental and extended search periods [16]. They were considering the examples of reported work that had been provided. As a result, the need for fast theoretical investigations has become essential for the development process.

The main contribution of this work is to explore the performance of various active materials theoretically by combining the simple diode model as a macroscopic model and the microscopic parameters of the organic solar cell such as mobility, recombination, and generation rates. The optimal macroscopic parameters of different devices, such as short circuit current density (\(J_{sc}\)), open-circuit voltage (\(V_{oc}\)), and fill factor (FF) will be optimized first, and then they will be mapped to a lower level in order to optimize the microscopic parameters. The results can be taken to the device design level, allowing for more rapid and cost-effective development. In Sect. 2 a brief description of the model is introduced. Section 3, represents the methodology while Sects. 4 and 5 represents results and the conclusion.

2 Modeling

The single diode model is among the most straightforward models that could be used to analyze solar cells in general. In this model, the solar cell is modeled as a current source connected in parallel with a diode and additional shunt resistance to accommodate the losses. This model is useful for exploring the overall behavior before the fabrication process, which provides fast design iterations and investigates the device’s current-voltage characteristics. The model can establish the link between experimental measurements and theoretical results using parameters extraction [17,18,19,20]. Considering that the process of charge generation and collection in organic solar cells differs from that of silicon solar cells, the use of such models should be carried out carefully. The single diode model can be considered a composition consisting of three current components, each representing a microscopic process in the photovoltaic process. Considering the single diode model can be thought of as a composition of three current components [21].

$$\begin{aligned} J = J_{0}\left[ \exp {\frac{q(V-JR_{s})}{nk_{B}T}}-1\right] +\frac{V-JR_{s}}{R_{sh}}-J_{ph} \end{aligned}$$
(1)

where \(J_{ph}\) the photo-generated current density, n is the ideality factor of a diode, \(R_s\) represents the series resistance, \(R_{sh}\) is the shunt resistance, \(k_B\) is Boltzmann’s constant, T is the operating temperature and q is the elementary charge. \(J_0\) is the saturation current density which can be expressed as follows

$$\begin{aligned} J_{0} = q k_r n_i^\beta d \end{aligned}$$
(2)

where \(k_r\) is the recombination factor and \(n_i\) is the intrinsic carriers density [22].

Equation 1 is considered as a behavioral model of the solar cell. The behavioral model is particularly beneficial when evaluating the performance from the upper level in terms of performance parameters without going in details to the microscopic parameters when accuracy is not the primary concern compared to speed. They are widely employed by applications required to develop and implement a specific functionality within a limited time. Instead, physical models are more realistic in representing the device’s physics, but they take longer to simulate a single structure run. Physical models are simplified by behavioral models, which map the physical parameters from the microscopic to the macroscopic level. The components of the single diode model are characterized by a range of physical parameters. The main objective is to maximize the values that will support the required functionality. The photovoltaic process begins with the generation of charge carriers. It is necessary to accomplish this process depending on various parameters, including illumination, charge carrier mobility, and active layer thickness, to maximize the output current density. A model for \(J_{ph}\) has been developed in [23], which is given by

$$\begin{aligned} J_{ph} = \frac{-1+\sqrt{1 + 4Gq \left( \frac{\mu _{n}+\mu _{p}}{\epsilon _{0}\epsilon _{r}\mu _{n}\mu _{p}}\right) \left( \frac{L^2_{n}L^2_{p}}{V_{n}V_{p}}\right) }}{ \left( \frac{\mu _{n}+\mu _{p}}{\epsilon _{0}\epsilon _{r}\mu _{n}\mu _{p}} \right) \frac{2d}{V}\left( \frac{V_{n}}{L^2_{n}}+\frac{V_{p}}{L^2_{p}}\right) }, \end{aligned}$$
(3)

where \(\mu _n\) is the electron mobility, \(\mu _p\) is the hole mobility G is the generation rate, \(L_n, L_p\) is electron and hole diffusion length, d is the active layer thickness, \(\epsilon _0\) is the permittivity of free space, \(\epsilon _r\) is the relative permittivity of the active material and V is the applied voltage. While \(\frac{\mu _{n}+\mu _{p}}{\mu _{n}\mu _{p}}\) is the effective mobility.

Equation (3) has been driven based on the following assumptions

  1. (a)

    Illumination of the sample is uniform,

  2. (b)

    Electrodes are non-injecting,

  3. (c)

    Trapped carriers cannot be de-trapped by light excitons or free carriers,

  4. (d)

    There is no exciton-exciton interactions,

  5. (e)

    The excitonic injection of charge does not occur at the electrodes.

Recombination processes limit the transportation of generated charge carriers to their corresponding electrodes. In the single diode model, the diode ideality factor n and the saturation current density \(J_0\) are the parameters representing the recombination process in the solar cell. Several types of recombination can occur within the device, including direct, Shockley-Read-Hall, and Auger recombination. Depending on the reaction order (\(\beta \)), each type has a corresponding value for the ideality factor that can be calculated by

$$\begin{aligned} n=\frac{2}{\beta } \end{aligned}$$
(4)

For direct recombination \(\beta =2,\) Shockely–Read–Hall recombination \(\beta =1\) and Auger recombination \(\beta = 3\).

3 Methodology

The overall performance of the solar cell is determined by several factors, namely: short circuit current,open circuit voltage, parasitic resistances and fill factor. To obtain best performance, it is desired to have high efficiency, high fill factor, low series resistance and high shunt resistance. A solar cell is considered a multi-layer device with each layer contributing differently to the performance. These layers include the active layer, the front electrode and the back electrode. Thickness and type of material of these three layers has a direct impact on the efficiency, fill factor, and parasitic resistances [24,25,26]. To get the maximum performance of a solar cell, the efficiency and fill factor should be high, while the resistances should be low. The value of resistance arises from the processing of the device and material properties. It is controllable through different experimental ways. The device resistance has been related to mobility of charge carriers, device area and charge carriers concentration [27]. For the sake of fast solution effect of mobility is considered as the main contributing factor as its effect is common with different parameters. Along with carriers mobility ideality factor effect has been investigated as well.

Matlab is used to solve the \(I{-}V\) equation by substituting equations 4 and 2 in Eq. 1. The thickness of the active layer is assumed to be greater than the diffusion length of holes and electrons. Diffusion length is assumed to be equal for both carriers.

$$\begin{aligned} d\ge l_{n}+l_{p}\\ \frac{L_{p}}{d}=\frac{L_{n}}{d}=0.5 \end{aligned}$$

Different materials were used for this study. In Table 1, given the collection of active layers alongside with the measured parameters collected from experimental data in literature. Ranges around these values are used in this work. Our way of studying the performance is to change single parameter each time, starting by mobility then ideality factor and its related parameters and the generation rate a the end.

Table 1 Numerical parameters of different organic active materials
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Optimization of the performance is a multi-parameter iterative problem. An iterative approach is used to maximize the performance by maximizing the photo-generated current while minimizing the other two current components given by Eq. 1.

4 Results and Discussion

The effect of mobility along with ideality factor and the generation rate will be introduced in the following subsections, with a dedicated subsection for each parameter representing its effect on device performance.

4.1 Mobility

Performance of the device, identified by \(J_{sc}\) and FF is calculated based on \(J{-}V\) data in response to change in mobility. In first set of trials, hole mobility is fixed and electron mobility is changed. In second set, both are changed from trial to trial but they are equal. In first case \(\mu _{h} = 1 \times 10^{-3}\) cm2 V−1 s−1 while \(\mu _{e}\) is changed in the range (\(1 \times 10^{-4}{-}1 \times 10^{-7}\) cm2 V−1 s−1). The maximum short circuit current density was found at \(\mu _{e} = \times 10^{-7}\) cm2 V−1 s−1 at which the photo-generated current is the same as the short circuit current, indicating that the generation and collection of carriers happen at maximum efficiency. At \(\mu _{e} =\times 10^{-4}\) cm2 V−1 s−1 a mismatch between \(J_{ph}\) and \(J_{sc}\) is found. It is noticed also that the change in both current densities is not significant. The maximum difference between two trials is \(\approx\,0.007.\) Table 2 represents the values of short circuit current, photo-generation current and fill factor at the condition of non-equal mobilities.The fill factor is constant for the device at different mobilities which indicates that the unbalanced mobilities does not affect the fill factor. Figure 1 represents the J–V relation at unbalanced charge carrier mobilities.

In case of equal mobilities the change in current densities is significant from case to case. The maximum short circuit current is obtained at \(\mu _{n} = \mu _{h} = 10^{-7}\) cm2 V−1 s−1. The behavior of the device at this value is deviated from the behavior of the diode, it is closer to the behavior of a resistance. It is advised to go for different mobility to maintain the device physics. I-V and P-V characteristics of the device at different mobilities is shown in Fig. 2

Fig. 1
figure 1

J–V characteristics of solar cell with unbalanced mobilities

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Table 2 Device performance at unequal mobilities
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Fig. 2
figure 2

J–V characteristics of solar cell with balanced mobilities

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4.2 Ideality Factor

Ideality factor affects the performance as well. The value of the ideality factor determines the short circuit current. In this study, the ideality factor has been changed from \(n=1{-}2\) at active layer thickness of \(d=100{-}30\) nm. The relation between the ideality factor and the short circuit current shows that the effect of the ideality factor is significant under certain threshold and above this limit the contribution becomes less significant as the \(J_{sc}{-}n\) curve saturates at higher values of n close to \(n=2.\) For the same device parameters, different ideality factors of the device affects the \(V_{oc}\) which indicated by Fig. 3.

Fig. 3
figure 3

J–V characteristics of solar cell with balanced mobilities

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4.3 Generation Rate

Generation rate G is defined as the number of generated electrons in the device due to absorption of incident light. It is dependent on the device thickness. In this study, it has been used as a constant value over the device thickness. It contributes significantly to the device parameters like \(J_{sc}\) as higher G results in higher \(J_{sc}\). Parameters such as ideality factor is expected to inhibit it is effect due to recombination.

5 Conclusion

This work show the effect of different parameters on organic solar cells through combination of single diode model by doing several iteration on different parameters such as charge carriers mobility which results in a significantly low performance when the balance between hole and electron mobilities does not exist. Balanced mobilities shows superior performance than the unbalanced one. A theoretical maximum short circuit current can be obtained at \(\mu _{h} = \mu _{e} = 1 \times 10^{-7}\) cm2 V−1 s−1. Also, the contribution of ideality factor to the device current shows a significant contribution at lower values of n while the device saturates at values near \(n=2\). The introduced work can be used as a pre-fabrication step to get organic solar cells targeting specific applications. Tuning of different parameters can affect transparency, conductivity, and flexibility to include organic solar cells in energy harvesting applications and sensing devices.