1 Introduction

Organic solar cells (OSCs) have attracted great attention as the third generation solar cells due to their low cost, flexibility (Chowdhury and Alam 2014; Krebs et al. 2011; Srinivasan et al. 2015), and semitransparency (Bou et al. 2014). Recently, efficiency of over 10% has been reported for OSCs (Liu et al. 2014). However, the efficiency of OSCs is not enough for commercial applications. In order to improve it, efforts have been made to understand the physical processes of light-to-electricity conversion mechanisms which occur in OSCs.

The impact of various parameters such as charge carrier mobility (Ebenhoch et al. 2015), morphology (Chen et al. 2012; Hedley et al. 2013), layer thickness (Chen et al. 2016; Namkoong et al. 2013; Sievers et al. 2006), and device architecture (Albrecht et al. 2012) on device performance has been experimentally investigated. In addition to experimental efforts, modeling and simulation are powerful tools for studying the photovoltaic mechanisms in OSCs (Liang et al. 2014; Mahmoudloo and Ahmadi-Kandjani 2015). The optoelectronic model (Kirchartz et al. 2008; Koh et al. 2011; Kotlarski et al. 2008), which couples the optical with electrical models, provides insight into understanding the internal (charge transport and recombination) processes of OSCs.

In OSCs, energetic disorder should be taken into account. Disorder results in broadening of energy levels around the conduction band and valence band, known as tail states (Belmonte et al. 2010). The energetic distribution of tail states betweenthe highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of active layer can be approximated with a Gaussian or exponential density of states (DOS) (Blakesley and Neher 2011).

Since recombination is an important key loss mechanism in OSCs, to optimize the performance of these devices, an efficient description of charge recombination mechanism should be taken into account in modeling and numerical simulations. Several models have been proposed to explain recombination mechanisms in OSCs, such as direct bimolecular recombination or Langevin recombination (Koster et al. 2006) and trap-assisted recombination (Mandoc et al. 2007a). The most common techniques to determine recombination mechanisms in OSCs arecharge extraction (CE) (Shuttle et al. 2008a), transient photovoltage (TPV) (Shuttle et al. 2008b), light and temperature dependent current density–voltage measurements (Mauer et al. 2010; Mihailetchi et al. 2005), charge extraction from a linear increasing voltage (CELIV) (Armin et al. 2012; Baumann et al. 2012; Pivrikas et al. 2010), and impedance spectroscopy (Proctor et al. 2013).

Koster et al. (2005a) proposed a combination of recombination through a charge transfer (CT) state with Langevin recombination. The results of this model closely match the experimental results under illumination. In other words, this model can predict the light ideality factor but cannot reproduce the dark ideality factor. The Langevin recombination model can only predict an ideality factor of unity (Soldera et al. 2012). However, several experimental results have indicated the deviation of ideality factor from unity (Kuik et al. 2011; Wetzelaer et al. 2012), which confirms the presence of a trap-assisted recombination mechanism (Kirchartz et al. 2011; Street et al. 2010).

Kirchartz et al. (2011) proposed a model based on exponential tail states and free-to-trap recombination mechanism, which can well explain both dark and light ideality factors. Also, MacKenzie et al. (2011, 2012) reproduced both dark and light current density- voltage curve by considering a charge density (n, p) dependent mobility into the Langevin recombination prefactor k L  = k L (n, p), where R L  = k L (n, p)np.

As mobility is one of the key factors determining efficiency, this study discusses the impact of varying mobility on charge transport and recombination processes. In spite of energetic disorder, mobility can be assumed to be charge density- and field-independent (Coehoorn et al. 2005; Cottaar and Bobbert 2006).

We study bulk heterojunction OSCs based on thieno[3,4-b]thiophene/benzodithiophene: [6,6] phenyl C71-butyric acid methyl ester (PTB7:PC71BM) using an optoelectronic model with a combination of direct recombination and trap-assisted recombination mechanisms; the latter has been proposed by Kirchartz et al. (2011). Photovoltaic parameters such as open circuit voltage (Voc), short circuit current (Jsc), fill factor (FF), and power conversion efficiency (PCE) are calculated as the function of mobility. Moreover, charge carrier density and recombination rate in active layer are simulated. The simulation current density–voltage curve is in good agreement with experimental data, as determined by Rauh et al. (2012). Finally, the nature of recombination mechanism for various mobilities is determined by the slope of the open circuit voltage versus generation rate.

2 Optical and electrical model

As the thickness of layers of OSCs is in order of the wavelength of the incident light, it is important to include the optical interference effects in the calculations. Transfer matrix method (TMM) is a standard formalism introduced by Heavens (1965), and first applied to OSCs by Pettersson et al. (1999) and others (Burkhard et al. 2010; Dennler et al. 2009; Sievers et al. 2006). We use this formalism for calculating the optical process in devices.

Energy dissipation in position x at normal incidence is determined by (Pettersson et al. 1999):

$$Q(x,\lambda ) = \frac{1}{2}c\varepsilon_{0} \alpha n\left| {E(x,\lambda )} \right|^{2}$$
(1)

where c, ε0, and α are the speed of light, the permittivity of free space, and the absorption coefficient of the active layer, respectively. Also, n is the real part of complex refractive index, and |E(x, λ)|2 is the modulus square of the optical electric field intensity in position x and at wavelength λ.

Assuming the photon-to-exciton conversion efficiency to be unity, the exciton generation rate is calculated by (Albrecht et al. 2012):

$$G(x) = \int_{{\lambda_{1} }}^{{\lambda_{2} }} {\frac{\lambda }{hc}Q(x,\lambda )d\lambda }$$
(2)

where h is Planck’s constant and λ is light wavelength from 350 to 800 nm.

Exciton generation rate in the active layer is an input to the electrical model.

The excitons generated in the active layer should be dissociated into free electrons and holes. The dissociation of the bound electron–hole pair through a CT state is described by Onsager–Braun model (Braun 1984; Onsager 1938). The dissociation probability of the bound electron–hole pair which is electric field (E)-and temperature (T)- dependent is expressed by (Koster et al. 2005a):

$$p(x,E,T) = \frac{{k_{diss} (x,E,T)}}{{k_{diss} (x,E,T) + k_{f} }}$$
(3)

where x is the electron–hole pair distance, kf is the decay rate to the ground state, and kdiss is the dissociation rate given by (Koster et al. 2005a):

$$\begin{aligned} k_{diss} (x,E,T) & = \frac{{3k_{L} }}{{4\pi a^{3} }}e^{{{\raise0.7ex\hbox{${ - E_{B} }$} \!\mathord{\left/ {\vphantom {{ - E_{B} } {k_{B} T}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${k_{B} T}$}}}} {{J_{1} \left( {2\sqrt { - 2b} } \right)} \mathord{\left/ {\vphantom {{J_{1} \left( {2\sqrt { - 2b} } \right)} {\sqrt { - 2b} }}} \right. \kern-0pt} {\sqrt { - 2b} }} \\ & = \frac{{3k_{L} }}{{4\pi a^{3} }}e^{{{\raise0.7ex\hbox{${ - E_{B} }$} \!\mathord{\left/ {\vphantom {{ - E_{B} } {k_{B} T}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${k_{B} T}$}}}} \left( {1 + b + {{b^{2} } \mathord{\left/ {\vphantom {{b^{2} } 3}} \right. \kern-0pt} 3} + \cdots } \right) \\ \end{aligned}$$
(4)
$$b = \frac{{q^{3} E}}{{8\pi \varepsilon_{0} \varepsilon_{r} k_{B}^{2} T^{2} }}$$
(5)

where kL is the Langevin recombination prefactor, EB is the electron–hole binding energy, kB is Boltzmann’s constant, and J1 is the first-order Bessel function. εr is the relative permittivity, and q is the elementary charge.

Because of disorder, Eq. 3 should be integrated over a distribution of separation distances

$$P(a,E,T) = \int_{0}^{\infty } {p(x,E,T)f(a,x)dx}$$
(6)

where f(a,x) is a normalized distribution function given by

$$f(a,x) = \frac{4}{{\sqrt \pi a^{3} }}x^{2} e^{{{{ - x^{2} } \mathord{\left/ {\vphantom {{ - x^{2} } {a^{2} }}} \right. \kern-0pt} {a^{2} }}}}$$
(7)

To simulate steady-state current density–voltage curves of OSCs, drift–diffusion equations are coupled with current density continuity equations for electrons and holes and Poisson equation. These equations which are solved self-consistently via finite difference method (FDM) with boundary conditions are:

$$\frac{dE(x)}{dx} = \frac{ - q}{{\varepsilon_{0} \varepsilon_{r} }}(n(x) + n_{t} - p(x) - p_{t} )$$
(8)
$$J_{n} (x) = qn(x)\mu_{n} E(x) + qD_{n} \frac{dn(x)}{dx}$$
(9)
$$J_{p} (x) = qp(x)\mu_{p} E(x) - qD_{p} \frac{dp(x)}{dx}$$
(10)
$$\frac{{dJ_{n} }}{dx} = q\left( {G(x) - R(x)} \right)$$
(11)
$$\frac{{dJ_{p} }}{dx} = - q\left( {G(x) - R(x)} \right)$$
(12)

where nt (pt) is the trapped electron (hole) density in tail states, and n (p) is the free electron (hole) density.

Jn (Jp), μn (μp) and Dn (Dp) are the electron (hole) current densities, electron (hole) mobility, and electron (hole) diffusion coefficients, respectively. G and R denote exciton generation rate and total recombination rate in the active layer, respectively.

The distribution of density of tail statesis expressed by exponential model (Blakesley and Neher 2011):

$$g_{n} = \frac{{N_{tc} }}{{k_{B} T_{c0} }}\exp \left( {\frac{{E - E_{c} }}{{k_{B} T_{c0} }}} \right)$$
(13)
$$g_{p} = \frac{{N_{tv} }}{{k_{B} T_{v0} }}\exp \left( {\frac{{E_{v} - E}}{{k_{B} T_{v0} }}} \right)$$
(14)

where Ntc (Ntv) and Tc0 (Tv0) are trapped-state densities and the characteristic temperature of the exponential model in conduction-band (valence-band) tail states, respectively. Ec (Ev) denotes the conduction (valence) energy level.

2.1 Recombination model

To consider the presence of band tail states, the recombination via exponential tail states proposed by Kirchartz et al. (2011) is used. This is given by

$$\begin{aligned} R_{tail} (x) = & \int_{{E_{v} }}^{{E_{c} }} {\frac{{N_{tc} }}{{k_{B} T_{c0} }}\exp \left( {\frac{{E - E_{c} }}{{k_{B} T_{c0} }}} \right)\frac{{c_{p}^{ - } c_{n}^{0} \left( {n(x)p(x) - n_{\text{int}}^{2} } \right)}}{{c_{n}^{0} \left( {n(x) + N_{c} \exp \left( {\frac{{E - E_{c} }}{{k_{B} T}}} \right)} \right) + c_{p}^{ - } \left( {p(x) + N_{v} \exp \left( {\frac{{E_{v} - E}}{{k_{B} T}}} \right)} \right)}}dE} \\ & \quad + \int_{{E_{v} }}^{{E_{c} }} {\frac{{N_{tv} }}{{k_{B} T_{v0} }}\exp \left( {\frac{{E_{v} - E}}{{k_{B} T_{v0} }}} \right)\frac{{c_{p}^{0} c_{n}^{ + } \left( {n(x)p(x) - n_{\text{int}}^{2} } \right)}}{{c_{n}^{ + } \left( {n(x) + N_{c} \exp \left( {\frac{{E - E_{c} }}{{k_{B} T}}} \right)} \right) + c_{p}^{0} \left( {p(x) + N_{v} \exp \left( {\frac{{E_{v} - E}}{{k_{B} T}}} \right)} \right)}}dE} \\ \end{aligned}$$
(15)

where c − p and c 0 n are capture rate coefficients for the conduction-band tail, and c 0 p and c + n are capture rate coefficients for the valence-band tail. Moreover, Nc (Nv) and nint indicate the effective density of states of conduction (valence) band and the intrinsic density of electrons and holes, respectively.

In addition to the recombination via exponential tail states, recombination between free charge carriers is calculated by the Langevin recombination rate (Langevin 1903):

$$R_{L} (x) = k_{L} \left( {n(x)p(x) - n_{\text{int}}^{2} } \right)$$
(16)
$$k_{L} = \frac{q}{{\varepsilon_{0} \varepsilon_{r} }}\left( {\mu_{n} + \mu_{p} } \right)$$
(17)

And geminate recombination via CT states introduced by Koster et al. (2005a) is described by:

$$R_{CT} = \left( {1 - P} \right)R_{L}$$
(18)

The total recombination rate in the active layer is the sum of recombination via exponential tail states, the Langevin recombination, and recombination via CT states R = R tail  + R L  + R CT , replacing in Eqs. (11) and (12).

2.2 Assumptions and input of optoelectronic model

The device is considered with an ITO/PEDOT:PSS/PTB7:PC71BM/Ca/Al structure based on the work of Rauh et al. (2012). The optical constants of Indium Tin Oxide (ITO), Aluminum (Al), poly (3,4ethylenedioxythiophene): poly (styrene sulfonate) (PEDOT: PSS), and Calcium (Ca) were taken from Petoukhoff et al. (2014), and the data for PTB7:PC71BM were obtained from Hedley et al. (2013). For simplicity, mobility is assumed to be the same for electron and hole. Moreover, drift–diffusion model is applied approximating an effective medium theory, in which the bulk heterojunction blend is considered as a homogeneous simiconductor with an effective band gap (Eg) which is the energy difference betweentheLUMO level of the acceptor and the HOMO level of the donor. It is assumed HOMO and LUMO are aligned by electrodes work function, thus no potential barrier is present in the cells.

3 Results and discussions

The absorption spectrum in active layer obtained by TMM is depicted in Fig. 1. The major absorption in the active layer is obtained in the range of 430–700 nm. Light absorption at longer wavelengths is due to PTB7 conjugated polymers (Hedley et al. 2013).

Fig. 1
figure 1

Normalized absorbance in 105 nm-thick active layer

Full size image

Figure 2 shows the distribution of exciton generation rate in the active layer from anode to cathode. As shown here, maximum photo-generated carriers were centered in the middle of active layer.

Fig. 2
figure 2

Distribution of exciton generation rate in active layer

Full size image

Figure 3 depicts the simulated current density–voltage of the device with the thickness of 105 nm under AM1.5G spectra at 100 mW/cm2. In the present study, the required material parameters which were used for the best fit are summarized in Table 1. For comparison, the experimental results obtained from Rauh et al. (2012) are presented in Fig. 3. The simulated curves slightly deviate from experimental results. This behaviour can be explained by the slight difference of real parameter values and parameter values used in our simulation. However, this figure shows that the simulated and experimental results fit very well at dark and illuminated current. This fitting reveals the validation of the model and the parameters used in the simulation.

Fig. 3
figure 3

Current density–voltage curves; dotted-line and solid line curves represent simulated results (present work) and experimental results (Rauh et al. 2012), respectively

Full size image
Table 1 Parameters used for best fit in this simulation
Full size table

Below, charge carrier mobility is chosen in the range of 10−7–1 (cm2/Vs), which is the mobility range of organic semiconductors, and other parameters are kept constant.

The Voc value as a function of mobility is shown in Fig. 4. As can be seen here, the Voc has a maximum value at a mobility of 10−5 (cm2/Vs). Starting from a mobility of 10−5 to 1 (cm2/Vs), the carrier transport velocity increases, resulting in an efficient extraction, which will cause a reduction of free charge carriers in the device (Fig. 5), leading to small splitting of quasi-fermi levels of electrons and holes (Fig. 6). As Voc is related to the difference between the quasi-fermi levels of electrons and holes (Qi and Wang 2012), its value decreases from 0.72 to 0.55 V. At a low mobility [10−7–10−5 (cm2/Vs)], despite the increasing mobility, carrier extraction is not efficient, leading to an increase in free electron and hole densities in the active layer (Fig. 5). Consequently, the quasi-fermi levels of electrons and holes become far apart (Fig. 6), leading to the increase of Voc.

Fig. 4
figure 4

Voc as a function of mobility

Full size image
Fig. 5
figure 5

Distribution of free electron and hole densities in active layer under open circuit conditions for different mobilities; solid line and dashed line represent electron and hole densities, respectively

Full size image
Fig. 6
figure 6

Quasi-fermi levels in active layer under open circuit conditions for various mobilities; solid line and dashed line represent quasi-fermi level of electron and hole, respectively

Full size image

The results derived from Fig. 4 are consistent with the literature (Deibel et al. 2008; Mandoc et al. 2007b), which has reported that a high mobility results in the fast extraction of charge carrier which, in turn, reduces Voc.

Equations 3, 4 and 6 reveal that the dissociation probability of the electron–hole pair through a CT state is also dependent on electron and hole mobilities.

Dissociation probability at maximum power point is shown in Fig. 7. For mobilities lower than 10−4 (cm2/Vs), the dissociation probability starts to decrease steeply by decreasing the mobility. For mobilities higher than 10−4 (cm2/Vs), one does not expect that higher moblities would lead to an improvement of dissociation probability because it approaches 1. As a result, at low mobilities, the electron–hole pairs do not dissociate into free electron and hole carriers, but decay to their ground state.

Fig. 7
figure 7

Dissociation probability at maximum power point versus mobility

Full size image

Recombination rate versus charge carrier mobility at maximum power point is plotted in Fig. 8.

Fig. 8
figure 8

Langevin (RL), tail states (Rtail), CT states (RCT), and total (Rtotal) recombination rates at maximum power point versus mobility

Full size image

Simulation results show that Langevin recombination rate is an increased function of charge carrier mobility, while bothrecombination rate via tail states and CT states have a maximum value at μ = 10−5 (cm2/Vs). In addition, Fig. 8 demonstrates the upper limit of Langevin recombination and total recombination; for μ ≥ 10−4 (cm2/Vs) both Langevin and total recombination are mobility independent. The behaviour of Langevin recombination agrees with (Wagenpfahl et al. 2010) which reported the upper limit of Langevin recombination.

Simulation results, shown in Fig. 9, indicate that free electron and hole density increases by increasing the charge carrier mobility from 10−7 to 10−5 (cm2/Vs) and decreases for a charge carrier mobility higher than 10−5 (cm2/Vs). By comparing Figs. 9 and 8, it is found that the Langevin recombination is governed by charge carrier mobility, while the changes of free charge densities are dominant over the recombination via tail states. As shown in Fig. 7, for a mobility higher than 10−5 (cm2/Vs), the dissociation probability grows steeply, so (1 − P) decreases (Eq. 18), resulting in reduction of recombination via CT states and for μ = 10−1 & 1 (cm2/Vs), RCT approaches zero (not shown in Fig. 8). Based on simulation results illustrated in Fig. 8, for μ > 10−5 (cm2/Vs), Langevin recombination governs in the devices, whereas for μ < 10−5 (cm2/Vs), trap-assisted recombination is dominant.

Fig. 9
figure 9

Distribution of free electron and hole densities in active layer at maximum power point for different mobilities; solid line and dashed line represent electron and hole densities, respectively

Full size image

The effects of losses at short circuit current can be shown by the internal quantum efficiency (IQE) curve, which is expressed by \(IQE(\% ) = \frac{{J_{sc} }}{{qG_{total} }} \times 100\), where Gtotal (cm−2/s) is the total generation rate of exciton inside the active layer. IQE as a function of mobility is plotted in Fig. 10. From μ = 10−5 cm2/Vs, IQE strongly increases with increasing the mobility, and for μ > 10−3 (cm2/Vs), more than 95% of absorbed photogenerated carriers participate in the photocurrent generation. Moreover, the inset illustrates Jsc values for various mobilities. The Jsc curve has the same behaviour as IQE curve.

Fig. 10
figure 10

IQE as a function of mobility. The inset shows Jsc versus mobility

Full size image

Figure 11a, b shows FF and PCE as a function of mobility, respectively. FF first increases with mobility, then is saturated at μ = 10−2 (cm2/Vs). The enhancement of FF is in agreement with (Ebenhoch et al. 2015; Tress et al. 2012), which reported an improvement in FF with increasing the mobility. Figure 11b reveals that PCE has a maximum value at μ = 10−2 (cm2/Vs); therefore, a higher mobility will not lead to enhancement of the efficiency of cells. PCE is defined by \(PCE\,(\% ) = \frac{{V_{oc} \times J_{sc} \times FF}}{{P_{in} }} \times 100\), where Pin is the incident solar power which equal 100 mW/cm2 in this study. The drop in PCE for μ〉10−2 (cm2/Vs) is attributed mainly to the reduction of Voc. The prediction of an optimum value of mobility in this study is in accordance with the literature (Deibel et al. 2008; Mandoc et al. 2007b; Wagenpfahl et al. 2010) which has stated that PCE has a maximum point versus mobility.

Fig. 11
figure 11

a FF, b PCE as function of mobility

Full size image

The slope of Voc as a function of logarithm of light intensity (I) determines the type of the recombination mechanism. Considering a bimolecular recombination, Voc dependence on generation rate derived by Koster et al. (2005b) is defined as

$$V_{oc} = \frac{{E_{g} }}{q} - \frac{{k_{B} T}}{q}\ln \left( {\frac{{\left( {1 - P} \right)k_{L} N_{c} N_{v} }}{PG}} \right)$$
(19)

Generation rate (G) is proportional to the light intensity; thus, Eq. (19) represents V oc  ∝ ln (I). For a bimolecular recombination, the slope of the curve is equal to \(\frac{{k_{B} T}}{q}\), while in the presence of trap-assisted recombination, this is higher than \(\frac{{k_{B} T}}{q}\) (Kuik et al. 2011; Mandoc et al. 2007c).

In this step, we assume the generation rate is a constant value in the range of 1019–1024 cm−3/s. Voc versus logarithm of generation rate for various mobilities is calculated and plotted in Fig. 12. The slopes of lines derived from Fig. 12 are summarized in Table 2.

Fig. 12
figure 12

Voc as a function of the logarithm of the generation rate

Full size image
Table 2 The slopes of lines derived from Fig. 12
Full size table

The slope of lines increases from 1.04 to 1.76 kBT/q when the mobility decreases from 1 to 10−7 (cm2/Vs), which corresponds to the dominance of trap-assisted recombination with decreasing of mobility, because at a low mobility, stronger occupation of traps is expected as charge carriers do not efficiently escape tail states. These results are in agreement with results reported by Ebenhoch et al. (2015) for PTB7:PC71BM OSCs. They found that bimolecular recombination increases upon increasing the mobility, whereas trap-assisted recombination was the dominant recombination at a low mobility.

4 Conclusions

In this study, an optoelectronic model was employed to simulate OSCs based on PTB7:PC71BM active layer. Langevin recombination, recombination via exponential tail states, and recombination via CT states were considered to study the effect of charge carrier mobility oncharge transport and recombination rate. It was found that Voc was mainly dependent on mobility and had a maximum value at μ = 10−5 (cm2/Vs). Simulation results demonstrated that the maximum of PCE occurred at μ = 10−2 (cm2/Vs), and the prediction of existence of an optimum value for mobility was found to be in agreement with the literature. The slope of Voc versus generation rate showed that at high mobilities, Langevin recombination governed in the devices, and at low mobilities, recombination via tail states dominated the loss mechanism in the devices.