Relationship between fill factor and light intensity in solar cells based on organic disordered semiconductors: The role of tail states

The origin of the relationship between fill factor and light intensity (FF-I) in organic disordered-semiconductor based solar cells is studied. An analytical model describing the balance between transport and recombination of charge carriers, parameterized with a factor, Γ!, is introduced to understand the FF-I relation where higher values of Γ! correlate to larger FF. Comparing the effects of direct and tail state mediated recombination on the FF-I plot, we find that for low mobility systems direct recombination with constant transport mobility can only deliver a negative dependence of Γ!,#$% on light intensity. By contrast, tail state mediated recombination with trapping and de-trapping processes can produce a positive Γ!,& vs. Sun dependency. The analytical model is validated by numerical drift-diffusion simulations. To further validate our model, two material systems that show opposite FF-I behaviour are studied: PTB7-Th:PC71BM devices show a negative FF-I relation while PTB7-Th:O-IDTBR devices show a positive correlation. Optoelectronic measurements show that the O-IDTBR device presents a higher ideality factor, stronger trapping and de-trapping behaviour, and a higher density of trap states, relative to the PC71BM device, supporting the theoretical model. This work provides a comprehensive understanding of the correlation between FF and light intensity for disordered semiconductor based solar cells.


I. INTRODUCTION
Organic semiconductors [1][2][3][4][5] are widely studied material systems for photovoltaic applications, due to their ease of processing, chemical tunability, low cost, flexibility, and low weight. However, as the materials are intrinsically disordered, they often have lower mobilities and increased density of trap states relative to more ordered semiconductors [6]. Consequently, when used as active materials for thin-film photovoltaics, the competition between charge carrier extraction and charge recombination is a key concern affecting the magnitude of the photogenerated current density at operating point. These losses often result in a reduction in the current density-voltage (J-V) curve fill factor (FF) of devices. The FF is determined by the ratio between maximum power generated, which is the product of the current density ( ! ) and voltage ( ! ) at maximum power point (MPP), and the product of short circuit current density The performance of a photovoltaic device is therefore related to FF through = '( )( '( [10,11,14]. Only a limited number have investigated FF [15] owing to the difficulty in describing its physical origins and accounting for the many factors that contribute to it. In most of those studies based on organic solar cells, FF has been shown to decrease with increasing light intensity. [16][17][18][19][20][21] A small number of studies, however, have shown the reverse, namely that FF increases with light intensity for intensities below one sun. [15,22] The first type of behaviour has been rationalized by either a super-linear increase in the bimolecular recombination rate with charge density and hence light intensity, [21] or series resistive effects. [23] In the second type of behaviour, where FF increases with increasing light intensity, the reasons are less clear. Researchers have proposed that the leakage current due to low shunt resistance in organic solar cells (OSCs) [15,23] controls the FF under low light intensity, resulting in a reduction of FF at very low light intensities (less than 10 -5 Sun). [23] [15] However, the leakage current cannot easily be differentiated from the dark saturation current [24,25] making it difficult to extract the key information solely from the shunt resistance values measured using the dark current. At present no complete model exists to explain these two types of behaviour.
The FF of low mobility semiconductor-based solar cells has been correlated to the competition between charge recombination and charge extraction [26,27] with the earliest study dating back to 1932 by Hecht. [28] Bartesaghi et al. [29] adapted this concept and applied it to organic solar cells successfully. The concept was later used to derive analytical expressions for the J-V curve of a low mobility diode [30] and was extended [31] to take recombination mechanisms, space-charge effects, and contacts into account. All of these models have been successfully applied to OSCs under standard solar illumination (1 Sun). At lower light intensities however, the carrier-density dependence of transport and recombination processes becomes more important and is expected to affect the FF-I relation. Considering the disordered nature of OSCs, caused by the distribution of conjugation lengths, disorder in conformation, and crystallinity, etc., the extended density of electronic states (DOS) and the associated dispersive charge transport and recombination processes must be included in any analysis that aims to explain the behavior of the FF over orders of magnitude in light intensity.
In this paper, we derive an analytical model to describe the correlation between FF and light intensity in organic disordered semiconductor-based solar cells. We consider separately the effects of direct and trap-mediated recombination on the FF-I plot. Our analytical models are verified using a more complex one-dimensional numerical drift-diffusion simulation based on the General Purpose Photovoltaic Device Model (gpvdm) software. [32,33] Our results suggest that, for low mobility systems, devices that are limited by direct recombination always show negative dependence of FF on light intensity, while devices with tail states can show the opposite behavior. In order to test the proposed model we study two different types of organic blend device that showed different FF-I relationships, one based on a poly [4,8- the non-fullerene acceptor (5Z,5′Z)-5,5′-(( (4,4,9,9-tetraoctyl-4,9- [35,36] instead of the fullerene PC71BM. The theoretical analysis is supported by experimental estimation of the trap states in the two device types. The PTB7-Th:O-IDTBR device is shown to have a higher ideality factor, stronger trapping and de-trapping behaviour, and higher trap density than the PC71BM device, leading to a positive FF-I relation as opposed to the negative relation shown in the PC71BM device.

II. THEORY AND MODEL
The FF of a classical inorganic solar cell depends on the two resistive elements of the standard equivalent circuit of a solar cell, namely the series and parallel resistance ( ' and * ). [37] In addition, the voltage dependence of the recombination current %+( matters. This voltage dependence is approximately exponential, scaling with where , / is the thermal voltage and $# is the ideality factor that provides information about the dominant recombination mechanism. [38,39] Several studies have discussed how resistive effects and recombination mechanisms change the relationship between fill factor and light intensity using diode equation, which we will call the FF-I relationship [40,41]. In the absence of resistive effects and recombination through trap states, the FF should depend on light intensity in a similar way to )( , i.e. increased light intensity results in a higher )( and FF. [37,40] Series resistance losses, however, increase with increasing current density and may cause an associated decrease of the FF with higher light intensities. The addition of trap recombination or shunt resistances may result in an increase of the FF with light intensity. [40] However, these insights are based on the standard equivalent circuit model of a solar cell under illumination, and do not account for disorder or inefficient charge collection. In the case of disordered organic or inorganic absorber materials, low mobilities are generally undesirable. Hence, to compensate for the effects of low mobilities, the device design of disordered organic or inorganic materials is typically chosen such that the absorber layer is fully or nearly-fully depleted. For low mobility-lifetime products, the wide field-bearing depletion zone helps to achieve efficient charge extraction, relative to a partially depleted design [42]. The electric field in a fully depleted organic solar cells is approximately given by ( ,$ − )/ , where ,$ is the built-in potential, is the applied voltage and is the active layer thickness. Because the electric field affects the probability of charge collection [26,27], the recombination current can be voltage and illumination dependent, as opposed to the standard equivalent circuit description of a solar cell, and consequently, the superposition principle [43] can no longer be applied. Instead, a range of different effects may influence the light intensity dependence of the fill factor. These have been described variously as light intensity-and voltage-dependent photocurrents, [26,27] recombination currents, [21] internal series resistances, [44] or even ideality factors, [45] all of which may modify the device current-voltage curve.
Previous modelling studies [29][30][31] had great success in understanding the limitations on FF under 1 Sun. For example, a study by Koster and co-workers [29] introduced a factor, q, representing the recombination-to-extraction rate at short circuit as a way to quantify collection efficiency and indicate FF. This approach is referred to as the Koster Model in this paper. However, previous analyses have not considered the impact of charge carrier densitydependent carrier mobility [46,47] nor have they considered the situation at maximum power point. Thus, an adapted analysis is required to properly model disordered systems with significant densities of tail states.
In the model described herein, we compare cases with and without carrier densitydependent charge transport. We begin by considering two types of recombination present in real-world solar cells (rather than assuming only Langevin-type-second-order recombination [29]): 1) Second-order, direct, free electron-to-free hole recombination with no trapmediated recombination and constant mobility.
2) First order, trap-mediated recombination and transport as they are often seen as the dominating loss mechanism in OSCs. [33,[48][49][50][51] We model a device directly at maximum power point (MPP), under the assumption that ! is proportional to )( : This has been validated numerically using gpvdm [32,33] (see Fig.   S8 in the Supplemental Material [52]) and is often the case for real-world OSCs. [53] It follows that ! can be expressed as a constant fraction ( ) of )( ( < 1) such that: We consider uniform absorption profile, and charge transport to be drift-dominated at MPP. We also assume that quasi-Fermi levels are spatially invariant at MPP. We parameterize the model using the transport-to-recombination factor at MPP: ! . In the model, we define ! as the ratio between the drift transport ( #% ) and recombination ( %+( ) rate constants (both in the unit of -. ) such that ! = #% / %+( . For interest, we compare our approach with that taken by Koster and co-workers [29] in Table S1 in the Supplemental Material [52].  [52].

A. A model for devices dominated by direct, second order recombination without tail-states
Direct recombination occurs between a free electron and a free hole, and can be radiative, [55] as illustrated in Fig With an ideality factor $#,#$% of 1, the free charge carrier density at MPP ( /,! ) can then be expressed as We can then describe a pseudo-first order recombination rate 'constant' At MPP, we assume that carrier transport is drift-dominated, which should be valid provided that ! < ,$ and a large enough electric field is maintained. [42] We use the drift rate coefficient (s -1 ), #%,#$% ( ! ) as a proxy for the extraction rate coefficient at MPP to describe the average rate for carriers to drift to the respective contacts: [29] #%, Here is the constant transport mobility (we assume balanced electron and hole mobilities), is the layer thickness, and $7&,! is the internal electrostatic potential drop across the absorber layer at MPP. The internal voltage $7&,! is given by The transport-to-recombination factor for direct recombination !,#$% is then Since )(,#$% is proportional to the log of the light intensity ( ) in Eq. (10), i.e. )(,#$% ∝ ( ), !,#$% should decrease with light intensity as long as > . 5 is assured (common for practical devices), indicating that direct recombination could only deliver a negative dependence between !,#$% and , and hence a negative dependence of FF on light intensity.
This relationship shows that if the device is limited by direct recombination, FF tend to higher values at lower light intensities, as is often reported for high efficiency devices. [18,34] We expect the same result using the Koster model, [29] since in that model the factor θ is proportional to the generation rate (equivalent to light intensity), and the mobility is constant. Therefore, the model devices limited by direct recombination and constant mobility cannot produce a positive FF-I correlation.

B. A model for devices dominated by tail state-mediated recombination
Tail state models have often been used to understand the unusual behavior in OSCs [33,48,49] and, as discussed above, are essential to a comprehensive model of devices operating at low light intensity. As we show in this section, only a model including trapmediated recombination and trap-mediated transport can reproduce the positive FF dependence on light intensity described in the introduction.
This approach is motivated by two key observations in the field of OPVs, i.e. 1) Most devices present ideality factors greater than 1 [56,57]; 2) Langevin-type second-order bimolecular recombination mechanism, that is defined by = ; , seldom holds, [58,59] with ; = < = ! = " O 7 + * P, and > the vacuum permittivity, and % the relative permittivity of the blend, and 7 and * are the electron and hole mobility, respectively. These observations lead to the following assumptions: 1) The DOS of organic semiconductors is distributed in energy, and follows an exponential-type distribution function; 2) Charge transport is correlated to charge carrier density through trapping and detrapping processes, as opposed to the carrier-density independent mobility approximation that is commonly used [29].
By substituting Eq. (13) into Eq. (11) we obtain: Here, the open circuit voltage ( )(,& ) is given by (see Supplemental Material for details [52]) The ideality factor $#,& is defined by assuming the same characteristic energy for both conduction and valance band (see Supplemental Material for the derivation [52]): The free charge density at OC ( /,)( ) can be directly related to the light intensity ( ) based on Hence, the free charge carrier density at MPP ( /,! ) can be expressed as Following the same method as Section II.A, we assume that the carrier drift rate coefficient #%,& ( ! ) parameterizes the charge transport rate at MPP for electrons [42]. However, the multiple trapping model requires that we substitute the mobility term for an effective mobility +// as shown in Eq. (20).

III. MODEL RESULTS: ANALYTICAL VERSUS NUMERICAL
In this section, analytical model results of versus ! are compared with numerical drift diffusion simulations performed using gpvdm [32,33]. Comparisons are carried out firstly at 1 Sun illumination, then over a range of different illumination intensities.

A. Comparison of the analytical and numerical models at 1 Sun illumination
We first performed calculations using the proposed analytical model (Eq. (10) and Eq.
(23)) under 1 Sun using a large parameter space to obtain a comprehensive picture of the correlation between FF and !,#$% , and between FF and !,& . Figure  obtained from J-V curves calculated using gpvdm [32,33,62] in Fig. 2. We note that gpvdm has been validated against experimental data in the past. [33,63,64] Despite that for extreme low values of !,#$% FF goes down with !,#$% , within commonly observed FF values in the range from 50% to 70%, we find that FF increases with ! for both direct and trap-mediated recombination. This is a similar trend to that first observed by Koster et al. [29] This agreement supports the validity of our analytical model under 1 Sun illumination.  Table S2 in the Supplemental Material [52]. Note that this analytical model relies on low mobility semiconductors. In the case of crystalline silicon solar cell, the transport is fast, the FF increases with increasing light intensity based on diode equation analysis [40]. We also find that in our drift-diffusion simulations, high mobility devices follow the ideal diode equations, while low mobility devices follow our analytical model, as shown in Fig. S2. A simple way to understand the effect of low mobility is to introduce a high transport (series) resistance in a diode model, and more discussion can be found in Section VI in the Supplemental Material [52]. Therefore, the commonly observed negative FF-I relation in organic solar cells can be explained by the low-mobility induced transport resistance. While our analytical model cannot explain devices with ideal transport, it is useful for understanding devices based on low mobility materials ( < 10 -1 cm 2 V -1 s -1 ).
The analytical results were also compared to FF values extracted from one-dimensional drift diffusion simulations of J-V curves over a range of light intensities and direct recombination coefficients ([10 -19 , 10 -11 m -3 s -1 ]) using the same base parameter set (Table S3 of the Supplemental Material [52]) in gpvdm [32,33,62] (see Fig. 3b)

Tail state-mediated recombination
The effect of tail states on the !,& . relation calculated using the expression given in Eq. (23) is shown in Fig. 4 (a-f) The input parameters used in the analytical model are listed in Table I.
We also performed one-dimensional drift-diffusion simulations with an exponential distribution of trap states using gpvdm [32,33,62] and the FFs were calculated at different light intensities. A comparison of the results based on the same parameters as given in Table S3 are shown in Fig. 4 (g-l). Figure S7 shows the effect of capture cross sections on the FF-I relation; Values of the cross-section for capture of free to trapped electrons (holes) were chosen to ensure fast charge capture (trapping) rate, and hence traps to be active. The capture cross section of the conduction band tail is chosen to be at least three orders of magnitude higher for capture of electrons (trapping) relative to the capture cross sections for holes (recombination).
For the valence band tail, this ratio is inverted with hole capture being more efficient than electron capture. Thereby, the conduction (valence) band tail is heavily populated with trapped electrons (holes) without the recombination rate being overwhelmingly high. The capture cross sections of holes (electrons) in the conduction (valence) band tail are, however, still high enough to ensure that recombination occurs primarily via tail states. The high ratio between trapping and recombination cross sections ensures that charge density is able to build up in the tail before recombining, leading to light intensity dependent effects such as the light intensity dependent mobility that is often seen in organic semiconductors [46,47].
With a low value of & (30 ), +// at MPP does not show a notable variation with light intensity (Fig. 4 (g)). Simultaneously, the FF shows a continuous increase with reduced light intensity with a shallow gradient due to negligible recombination ( Fig. 4 (j)). In this instance, the effect of traps density is negligible, provided the effective trap density ( & +@* ) is less than 10 24 m -3 eV -1 . With higher values of & (0.06 or 0.10 eV) but low & +@* (10 18 m -3 eV -1 ), +// at MPP remains unchanged with varied light intensity ( Fig. 4 (h,i)), and FF shows a similar trend to the & = 30 meV case ( Fig. 4 (k,l)). However, when & +@* is increased, +// at MPP starts to decrease notably with reduced light intensity ( Fig. 4 (h,i)) and the slope of FF-I plot switches from negative to positive with light intensity (Fig. 4 (k,l)). Sun making the ratio / /( & + / ) much larger than at lower light intensities.
The effects of the effective mobility ( +// ) on the FF-I relation can also be explained in terms of the relative recombination rate. Inefficient charge transport will result in a higher recombination rate at a given light intensity and voltage. It follows that, at low light intensity, the recombination rate relative to the generation rate at voltages less than )( , ( ( )/ ) is expected to be higher than that under 1 Sun illumination.  Fig. 4 (l). With low & +// (Fig. 5 (a)), ( )/ at ! is lower at low light intensity (0.01 Sun) relative to 1 Sun, indicating lower recombination, consistent with higher FF. However, with high & +@* (Fig. 5 (b)), at 0.01 Sun ( )/ is significantly higher than at 1 Sun across the range of scanned voltages, which suggests higher recombination rates and lower FF at = 0.01 Sun. The analysis in terms of relative recombination rate is consistent with that based on the effective mobility.
These results based on numerical simulations using gpvdm are consistent with the observations from our analytical model. We have also ruled out the possibility that interfacial contact barriers at the electrodes could produce a positive FF-I dependence (see Section X in Supplemental Material for further details [52]). We conclude that the influence of tail states can theoretically account for different FF-I relationships. These results combined with Fig. 3 also show that our analytical models can be useful for both 1 Sun and light-intensity dependent analysis for FF.   Note: DOS = Density of states. ,$ follows effective band gap 3 , since we consider ideal Ohmic contact with no contact barriers. The choice of 3 or ,$ is made based on the recent development on novel non-fullerene acceptors [70][71][72][73][74], which often presents a 3 of 1.6 eV.
We also note here that the FF-I relation is maintained regardless of the value of ,$ (see Figure  S10 in the SI).

IV. EXPERIMENTAL RESULTS
Having investigated the relationship between FF and light intensity theoretically, we now proceed to demonstrate the modelled FF-I behavior using practical organic solar cell devices.

A. Experimental FF-I relation
To investigate the FF-I relation of practical organic materials based solar cells, inverted architecture ( Fig. 6(a)) OSCs based on PTB7-Th [34] as the donor, and either blended with the fullerene acceptor PC71BM or the non-fullerene acceptor O-IDTBR [35] were fabricated (see Experimental Section in the Supplemental Material [52] for more details regarding device fabrication). The energy level alignment of the studied materials [35,[75][76][77][78][79][80][81] and contacts [82] is presented in Fig. S11 in the Supplemental Material [52]. As the effect of leakage current has often been used to explain the FF reduction at low light intensity, [15,23] we first compared the dark current density with light current density at different illumination intensities, as shown in Fig. 6(c) and (d). Although the reverse dark current of PC71BM based device is around one order of magnitude lower than that of the O-IDTBR device, we find that the dark current for both devices is greater than one order of magnitude lower than the current density under the lowest light intensity (3 mW cm -2 ). In addition, the reduction of FF for O-IDTBR devices becomes apparent at 1 Sun, where the current density is at least two order of magnitude higher than the dark current density suggesting the origin is unlikely to be the leakage current in this case. However, the fact that the O-IDTBR device presents higher reverse dark current than the PC71BM device is an

B. Quantifying trap states
We have shown that both the analytical and numerical (drift-diffusion) modelling results indicate that direct recombination cannot produce a positive dependence of FF on light intensity in low mobility semiconductor-based solar cells. Conversely, the existence of a significant density of exponential-type trap states can affect the FF-I relation in such a way that FF is reduced at lower light intensities. In this section we show that the different FF-I dependencies of the two devices studied can be directly related to their different trap state densities.

Ideality factors
Measurements of device ideality factors have frequently been used to indicate the degree of recombination via trap states in OSCs [57]. As derived in Eq. (17), higher ideality factors correspond to a greater proportion of recombination via trap states.
We extracted the ideality factor ( $#,J ) of the measured devices using )( versus light intensity plot (Suns-)( ), as shown in Fig. 7. The ideality factors calculated from the slope of the curve fits were 1.00 ± 0.10 and 1.60 ± 0.20, for the PC71BM and O-IDTBR devices, respectively. We also estimated the dark ideality factor $#,# using dark J-V curves, showing the same trend as $#,J (see Methods section and Fig. S12 in Supplemental Material for further details [52]).
The O-IDTBR device presents notably higher ideality factors (closer to 2) than the PC71BM device (close to 1), indicating that trap mediated recombination is likely to play a bigger role in the O-IDTBR devices than in the PC71BM devices.

Low frequency capacitance
Ideality factor measurements indicate that the O-IDTBR device likely presents more trap-mediated recombination than the PC71BM device. In this section we directly measure the trap state density of devices using low frequency (10 kHz) capacitance measurements. [51,[83][84][85] Since the measurement frequency approaches the time scale of trapping and de-trapping, trapped carriers can respond to the alternating internal electric field. It has previously been argued that an increase in capacitance at higher applied DC voltages can be attributed to trapstates mediating the charge distribution and transport, [51,[83][84][85] suggesting that an extended density of trap states is the origin of the low frequency capacitance enhancement.
We have adapted this concept to understand the influence of illumination intensity on the low frequency capacitance within the multiple trapping and de-trapping model. In the low frequency regime studied, the effects of deep traps cannot be detected while shallow traps can be. Under low illumination, the trap states are not fully occupied, and carriers mostly fill the deep states. As such the de-trapping rate is low owing to its exponential dependence on trap A # B ) . Deeply trapped carriers therefore do not strongly influence carrier dynamics. With increased light intensity, however, the shallow states start to be filled. In this situation, the rate of de-trapping becomes significant such as to influence transport and result in an increase in the measured capacitance. [83] Hence, the enhancement of capacitance at high light intensity is an indication of trap-mediated charge dynamics. The effect of trapping is expected to be more pronounced with low internal field, where charge extraction is slow. Measuring the capacitance response at low frequency under a range of applied voltages is therefore a useful method to detect the shallow trap states.
Here we apply these concepts to our OPV devices and perform capacitance-voltage (C-V) measurements under 10 kHz frequency at different applied DC voltages starting from -2 V to 2 V. Figure 8(a) shows the capacitance measured using a 50 mV AC voltage set at 10 kHz for the PC71BM and the O-IDTBR based devices measured at various light intensities (from 1 Sun to the dark). On applying a negative bias of -2 V, the capacitance converges to the geometric capacitance. In this regime, the strong electric field under large negative applied bias efficiently removes carriers such that charge recombination, transport or redistribution caused by trapping and de-trapping is small and can hardly interfere the dynamics of charge carriers.
With increased voltage, the electrostatic potential difference between the contacts drops, leading to reduced drift currents. From this point, trap states start to play an active role in mediating charge carrier transport processes resulting in the increased capacitance seen in Fig.   8  According to Ref. [83], under low frequency and high light intensities, the capacitance caused by trapping and de-trapping processes gives information of the lower limit of the trap density corresponding to the shallow traps. Here, deeper trap states can only be probed at low frequency on the same order as their de-trapping rates. Assuming the additional capacitance at close to ,$ is caused by trapped carrier being released to the mobility edges, the trap-charge density of accessible trap states can be estimated using Eq. (24). [83,86,87] where is the dielectric constant, is the capacitance, is the device area, and is the applied voltage.
We obtained the trap density by fitting 1 5 . plots at 1 Sun over the voltage range close to ,$ , as shown in Fig. 8(b). The slope of 1 5 . in the PC71BM device is significantly higher than that of the O-IDTBR device (comparing positive values), indicating that the trap density is much lower in the PC71BM device than in the O-IDTBR device. Using Eq. (24), we obtain values of 2.5×10 22 m -3 , and 2.0×10 23 m -3 for the PC71BM and the O-IDTBR device at 10 kHz, respectively. The exact values of the total trap density in the two devices are difficult to determine since it's hard to obtain clean signals at extremely low frequencies due to experimental system noise, and the trap densities estimated above can only be related to the accessible traps at 10 kHz frequency and under 1 Sun illumination. However, the higher trap density extracted from the capacitance-voltage analysis is strong evidence that the O-IDTBR device has a higher total trap state density than the PC71BM device.
To verify our conclusions from the capacitance measurement, we performed C-V simulations at 10 kHz with a 50 mV AC voltage under different light intensities (the same as experimental C-V measurements) using gpvdm [32]. These C-V simulations are fully timeresolved and no additional assumptions have been made such as linearization of the equations.
Since we do not know the precise parameters e.g. trap profile and density in the real devices, we do not perform a fitting routine to the experimental C-V curves, but rather aim for qualitative agreement. The simulations were carried out at & = 0.10 with the same parameter set as list in Table S3. The built-in voltage was set to 1 V and zero field mobility was set to 1×10 -3 cm 2 V -1 s -1 . We compared low (10 18 m -3 eV -1 ) and high (10 24 m -3 eV -1 ) effective trap densities.
As shown in Fig. 8(c), although the magnitude of calculated C-V characteristics are much higher than that of the measured C-V, the simulated device with high trap density shows a much larger capacitance enhancement than the device with low trap density when we increase the light intensity from dark to 1 Sun. This result is consistent with the prior theory [83], namely that traps have a strong influence on the capacitance signal under a low frequency AC voltage.
We also calculated 1 5 . characteristics at 1 Sun as shown in Fig. 8(d). A clear slope difference is observed between low and high trap density device, which is also qualitatively consistent with the experimental results. The good agreement between experiments and simulations strongly supports our conclusions that the O-IDTBR device has a higher trap state density than the PC71BM device. In summary, the O-IDTBR device presents a higher ideality factor, stronger trapping and de-trapping behaviour, and higher trap density than the PC71BM device. Consistent with our analytical model, the reduction of FF at low light intensity can be correlated to the existence of a significant density of tail states mediating carrier transport.

V. CONCLUSIONS
We proposed an analytical model parameterized by the transport-to-recombination factor ! to help to understand the correlation between fill factor and light intensity in organic