## 1. Introduction

Organic solar cells (OSCs) based on bulk heterojunction (BHJ) structure are promising candidates for the generation of clean, affordable electricity owing to their low fabrication cost compared to inorganic solar cells [

1,

2]. Recent research efforts into the development of new donor-acceptor blends have led to a significant improvement in the power conversion efficiency (PCE) of OSCs, above 11% [

3,

4,

5,

6], which is gradually moving OSCs closer to commercialization. However, poor material properties such as low charge carrier mobilities, low dielectric constants, and poor band offsets have led to the low PCE of OSCs relative to their silicon counterparts [

1]. Hence, optimization of these parameters is essential for improving the performance of OSCs. Considerable attention has recently been focused on exploring strategies for the further optimization of the PCE, guided by a thorough understanding of the fundamental mechanisms that govern the photovoltaic performance of OSCs [

7]. The following four processes distinguish BHJ OSCs from their inorganic counterparts: (i) photon absorption and exciton formation; (ii) exciton diffusion to the donor-acceptor (D-A) interface; (iii) exciton dissociation into free charge carriers at the D-A interface; and (iv) charge transport and collection at their respective electrodes [

8,

9]. These processes should be efficient enough to achieve an enhanced PCE of BHJ OSCs.

One of the important parameters of BHJ OSCs is the open-circuit voltage

${V}_{OC}$, which is the voltage at which the net current flowing through the device is zero. It is established that in BHJ OSCs,

${V}_{OC}$ depends on the energetic difference between the lowest unoccupied molecular orbital (LUMO) of the acceptor material

$({E}_{\mathrm{LUMO}}^{\mathrm{A}})$ and the highest occupied molecular orbital (HOMO) of the donor material

$({E}_{\mathrm{HOMO}}^{\mathrm{D}})$, called the donor-acceptor effective energy gap

$({E}_{DA})$ [

10,

11,

12,

13,

14]. Until now, the most explored approach to increasing

${V}_{OC}$ of BHJ OSCs has been to increase the energy of the LUMO of the fullerene acceptor material or decrease the energy of the HOMO of the donor material, or both [

10,

11,

12,

13,

14,

15]. Scharber and co-workers [

10] empirically defined

${V}_{OC}$ as:

where

$q$ is the elementary charge and

$\Delta {E}_{\mathrm{loss}}$ is an empirical value denoting the energy losses occurring in transporting the charge carriers to the electrodes [

16,

17,

18].

The validity of Equation (1) has been established for a number of donor-acceptor (fullerene) blends with a voltage loss

$\Delta {E}_{\mathrm{loss}}$ ranging from 0.3−0.6 eV [

19,

20]. The question that may be asked is: what are the possible causes of the voltage loss

$\Delta {E}_{\mathrm{loss}}$ and how can it be reduced? The origin of

$\Delta {E}_{\mathrm{loss}}$ is still being debated, with several possible mechanisms such as bimolecular recombination, coulombic interactions, energetic disorder, etc., [

12,

16,

17,

18,

21,

22]. An important loss mechanism in BHJ OSCs is the bimolecular recombination of free charge carriers with its rate

$R$ given by [

23,

24]:

where

$\gamma ={\gamma}_{pre}\frac{q}{{\epsilon}_{o}{\epsilon}_{r}}({\mu}_{e}+{\mu}_{h})$ is the bimolecular recombination coefficient (m

^{3}s

^{−1}):

${\gamma}_{pre}$ is the dimensionless reduction prefactor,

${\epsilon}_{0}$ is the permittivity of free space,

${\epsilon}_{r}$ is the dielectric constant, and

${\mu}_{e}({\mu}_{h})$ is the electron (hole) mobility (m

^{2}/Vs) in the active organic layer.

By minimizing the nonradiative bimolecular recombination in BHJ OSCs [

25], one can minimize the voltage loss and eventually increase

${V}_{OC}$. Recent studies [

26,

27] have shown that

${V}_{OC}$ can be determined more accurately by measuring the bimolecular recombination rate. These studies have also considered the influence of light intensity on

${V}_{OC}$ by varying the carrier concentration due to photoexcitation. Lange et al. [

28] studied the dependence of

${V}_{OC}$ on the charge carrier concentration in polymer:fullerene blends using bias amplified charge extraction (BACE) measurements. In addition, as stated above, the energy loss also occurs due to the energetic disorder, which reduces the

${V}_{OC}$. A few recent studies have shown the effect of energetic disorder on

${V}_{OC}$ using a Gaussian density of states (DOS) [

29,

30,

31]. Garcia et al. [

29] developed a model to predict the

${V}_{OC}$ values by introducing DOS distributions of the HOMO in donor and the LUMO in acceptor materials. In their study, the energetic disorder is modeled by the Gaussian DOS with the mean energy equal to

${E}_{\mathrm{LUMO}}^{\mathrm{A}}$ and standard deviation

${\sigma}_{e}$ for the acceptor material and corresponding

${E}_{\mathrm{HOMO}}^{\mathrm{D}}$ and

${\sigma}_{h}$ for the donor material. In other studies [

16,

25,

32,

33,

34,

35,

36], the role of energetics with respect to charge carrier dynamics and coulombic interactions have been analyzed. Credgington and Durrant [

32] studied the dependence of

${V}_{OC}$ on the generation and recombination dynamics; they studied the dependence of

${V}_{OC}$ on the charge carrier concentration and carrier recombination lifetime. Vandewal et al. [

25,

33,

34,

35] derived a relationship between

${V}_{OC}$ and the energy of the charge transfer (CT) exciton state

${E}_{CT}$ by incorporating the energetic disorders. However, other works [

16,

36] have derived different expressions relating

${V}_{OC}$ and

${E}_{CT}$ in OSCs which exclude the influence of energetic disorder.

Despite numerous attempts to study

${V}_{OC}$ and the associated energetic loss mechanisms in BHJ OSCs, there appears to be few attempts being made to understand whether it depends on the concentration of photoexcited charge carriers in the donor and acceptor materials. This may be due to the fact that as the two materials, donor and acceptor, are blended together in a BHJ structure, they cannot be separately excited experimentally. However, in order to understand the contributions of photoexcitation of donor and acceptor materials to the performance of a BHJ OSC, one may seek to assess the performance considering the two cases separately: (1) when absorption and photoexcitation occur only in the donor and (2) when these occur only in the acceptor. Such a study can only be carried out theoretically, as presented here. When an exciton is excited in the donor in a BHJ OSC, its electron can be transferred from the donor LUMO to the acceptor LUMO at the D-A interface because the acceptor LUMO lies at a lower energy (see

Figure 1a). In this process, an exciton becomes a CT exciton as its electron is transferred to the acceptor and its hole remains in the donor, as shown in

Figure 1c. Likewise, when an exciton is excited in the acceptor, its hole can be transferred from the acceptor HOMO to the donor HOMO, being at a lower hole energy (see

Figure 1b), which also creates a CT exciton with an electron in the acceptor LUMO and a hole in the donor HOMO, as shown in

Figure 1c.

This has prompted several researchers to investigate the above two processes in more detail [

37,

38,

39,

40,

41,

42,

43,

44,

45,

46]. Recently, two different external quantum efficiencies (EQE) due to different donor and acceptor absorption bands have been reported [

37]. The difference in the absorption bands can be attributed to the difference in the energy levels involved. The times of transfer of electrons from the donor LUMO to the acceptor LUMO and that of holes from the HOMO of the acceptor to the HOMO of the donor have been measured [

38,

39] and have been found to be different; the hole transfer was found to be faster than the electron transfer. This may be expected to lead to generating different concentrations

$n$ and

$p$ of electrons and holes. Different charge generations by exciting the donor and acceptor individually have also been reported in some donor-acceptor combinations [

40], and different internal quantum efficiencies (IQE) have also been observed [

41]. Some studies have demonstrated that BHJ OSCs based on thienothiophene-substituted diketopyrrolopyrrole (DPP) polymers (PDPP2TT-T) possess an EQE of 0.63–0.78 within the wavelength range of acceptor-fullerene absorption (400–650 nm), whereas a lower EQE of 0.35–0.50 is achieved within the range of donor-polymer absorption (650–900 nm) [

41,

42]. Such a difference may not be attributed to a difference in absorption of photons in the different materials, but may be due to the difference in charge generation. In another study, Baulin et al. [

43] compared the dynamics of CT states generated through electron-transfer and hole-transfer exciton-dissociation pathways in three BHJ OSCs, and they found that the dynamics of CT state recombination are very similar for the two charge-generation processes, implying that the nature of the generated CT states is independent of the pathways of CT formation. They did, however, observe that the generation of CT states and free charge carriers greatly depends on the actual LUMO energy offset

$\Delta {E}_{\mathrm{LUMO}}={E}_{\mathrm{LUMO}}^{\mathrm{D}}-{E}_{\mathrm{LUMO}}^{\mathrm{A}}$ (electron transfer) and the HOMO energy offset

$\Delta {E}_{\mathrm{HOMO}}={E}_{\mathrm{HOMO}}^{\mathrm{D}}-{E}_{\mathrm{HOMO}}^{\mathrm{A}}$ (hole transfer) of the blended system [

44]. These energy offsets are independent of the pathway of the formation of CT excitons. Singh et al. [

45,

46] recently studied the contribution of exciton generation in the donor and acceptor materials and its dissociation to the performance of BHJ OSCs. They found that the absorption and dissociation rates as well as exciton diffusion lengths are comparable whether excitons are generated in the donor or acceptor.

In this paper, we investigate in detail the difference in the open circuit voltage from the photoexcited donor and photoexcited acceptor separately in the following seven different polymer:fullerene blends: poly[2-methoxy-5-(3′,7′-dimethyloctyloxy)-1,4-phenylenevinylene] (MDMO-PPV), regioregular poly(3-hexylthiophene) (P3HT), poly[N-9′-heptadecanyl-2,7-carbazole-alt-5,5-(4′,7′-di-2-thienyl-2′,1′,3′-benzothiadiazole)] (PCDTBT), poly[2-(3,7-dimethoxyoctyloxy)-5-methoxy-1,4-phenylene vinylene] (OC_{1}C_{10}-PPV), poly[4,8-bis[(2-ethylhexyl)oxy]benzo[1,2-b:4,5-b′]dithiophene-2,6-diyl][3-fluoro-2-[(2-ethylhexyl)carbonyl]thieno[3,4-b]thiophenediyl] (PTB7), mono-DPP, and bis-DPP blended with different derivatives of methano-fullerene[6,6]-phenyl C_{61},-butyric acid methyl ester (PCBM). We derived two different expressions for ${V}_{OC}$ in BHJ OSCs. The first expression, which is a result of photoexcitation of the donor, depends primarily on the energetic difference between the hole quasi-Fermi level $({E}_{F}^{h})$ and the HOMO of the donor material $({E}_{F}^{h}-{E}_{\mathrm{HOMO}}^{\mathrm{D}})$ and the electron concentration $n$. The second expression, which is a result of photoexcitation of the acceptor, depends primarily on the energetic difference between the LUMO and the electron quasi-Fermi level $({E}_{F}^{e})$ of the acceptor material $({E}_{\mathrm{LUMO}}^{\mathrm{A}}-{E}_{F}^{e})$ and the hole concentration $p$. Thus, the two open circuit voltages calculated in each selected D-A blend are presented here and their influences on the performance of BHJ OSCs are discussed. Following this, the simultaneous photoexcitation of both the donor and acceptor materials is also considered and the corresponding ${V}_{OC}$, which is different from the above two, is derived, and the calculated ${V}_{OC}\mathrm{s}$ are compared with their corresponding measured values and a reasonably good agreement is found between them.

## 3. Results

For calculating the

${V}_{OC}$ using Equations (10), (15), and (18), we require some material-dependent input parameters. For this, as stated above, we chose seven donor-acceptor blends with known input parameters available in the literature, as listed in

Table 1.

The values of

$\gamma $ for PCDTBT:PC

_{71}BM [

58], mono-DPP:PC

_{71}BM [

61], and bis-DPP:PC

_{71}BM [

61] are taken from the literature, and those of the remaining four blends, MDMO-PPV:PC

_{61}BM P3HT:PC

_{60}BM, OC

_{1}C

_{10}-PPV:PC

_{61}BM, and PTB7:PC

_{71}BM, are calculated using the bimolecular recombination coefficient equation described in the text. In addition, we need the values of

$({E}_{F}^{h}-{E}_{\mathrm{HOMO}}^{\mathrm{D}})$ and

$({E}_{\mathrm{LUMO}}^{\mathrm{A}}-{E}_{F}^{e})$, as well as the carrier concentrations

$n$ and

$p$ in each blend to calculate

${V}_{OC}$ from Equations (10), (15), and (18), which are also listed in

Table 2 and

Table 3. However, it may be noted that both

$n$ and

$p$ are only available for PTB7:PC

_{71}BM [

62] as given in

Table 2 and

Table 3. In mono-DPP:PC

_{71}BM and bis-DPP:PC

_{71}BM, only the electron concentration

$n$ is known in the literature [

63]. Consequently, to find the hole concentration

$p$ in these two blends, we used Equation (7) and calculated

$p$; these

$n$ and

$p$ thus obtained are listed in

Table 2 and

Table 3. We first used the

$n$ and

$p$ values determined above for PTB7:PC

_{71}BM, mono-DPP:PC

_{71}BM, and bis-DPP:PC

_{71}BM in Equations (4) and (5), and calculated their energetic distances

$({E}_{\mathrm{LUMO}}^{\mathrm{A}}-{E}_{F}^{e})$ and

$({E}_{F}^{h}-{E}_{\mathrm{HOMO}}^{\mathrm{D}})$, respectively, at 300 K by assuming the effective density of states

${N}_{C}={N}_{V}$ = 1 × 10

^{25} m

^{−3} in each blend.

For the other four blends considered here, MDMO-PPV:PC

_{61}BM, P3HT:PC

_{60}BM, PCDTBT:PC

_{71}BM, and OC

_{1}C

_{10}-PPV:PC

_{61}BM, the

$n$ and

$p$ concentrations are not known. As the

$n$ and

$p$ concentrations in all the three materials are of the order of 10

^{22} m

^{−3} (see

Table 2 and

Table 3), we used the average values of

$n$, which is 1.8 × 10

^{22} m

^{−3}, and

$p$, which is 0.95 × 10

^{22} m

^{−3}, for PTB7:PC

_{71}BM, mono-DPP:PC

_{71}BM, and bis-DPP:PC

_{71}BM, and calculated the energetic distances

$({E}_{F}^{h}-{E}_{\mathrm{HOMO}}^{\mathrm{D}})$ and

$({E}_{\mathrm{LUMO}}^{\mathrm{A}}-{E}_{F}^{e})$ for the other four blends (see

Table 2 and

Table 3). That is to say, using Equations (4) and (5) with

${N}_{C}={N}_{V}$ = 2.5 × 10

^{25} m

^{−3} known for MDMO-PPV:PC

_{61}BM [

17,

57],

${N}_{C}={N}_{V}$ = 2 × 10

^{26} m

^{−3} for P3HT:PC

_{60}BM [

1],

${N}_{C}={N}_{V}$ = 5 × 10

^{25} m

^{−3} for PCDTBT:PC

_{71}BM [

52], and

${N}_{C}={N}_{V}$ = 2.5 × 10

^{25} m

^{−3} for OC

_{1}C

_{I0}-PPV:PC

_{61}BM [

1,

59], in addition to the corresponding average charge carrier concentrations

$n$ = 1.8 × 10

^{22} m

^{−3} and

$p$ = 0.95 × 10

^{22} m

^{−3}, we calculated

$({E}_{\mathrm{LUMO}}^{\mathrm{A}}-{E}_{F}^{e})$ and

$({E}_{F}^{h}-{E}_{\mathrm{HOMO}}^{\mathrm{D}})$, respectively, in the other four blends.

Using these parameters in Equations (10) and (15), the

${V}_{OC}$ is calculated in all seven blends for the photoexcitation of the donor and acceptor as given in

Table 2 and

Table 3, respectively. The calculated values of

${V}_{OC}$ from Equation (10) when the donor is excited in most selected materials are somewhat comparable with those calculated from Equation (15) when the acceptor is excited, as also shown in

Figure 3. In addition, considering the case of simultaneous photoexcitation of donor and acceptor materials, the open circuit voltage in each blend is calculated from Equation (18) as given in

Table 4. All the three

${V}_{OC}\mathrm{s}$ obtained from the asynchronous and simultaneous photoexcitations of donor and acceptor are found to be comparable and agree well with the corresponding measured values [

51,

54,

59,

61,

64,

65].

The open circuit voltages

${V}_{OC}\mathrm{s}$ calculated from Equations (10) and (15) are plotted as a function of the effective band gap

${E}_{DA}/q$ as shown in

Figure 3, which illustrates that

${V}_{OC}$ calculated from Equation (10) is slightly higher than that from Equation (15).

Finally, as stated previously, in a BHJ OSC the donor and acceptor materials are blended together. Consequently, both the donor and acceptor are excited simultaneously, resulting in one open circuit voltage as derived in Equation (18), which is used to calculate the

${V}_{OC}^{DA}$ for simultaneous photoexcitations of both the donor and acceptor in each of the selected blends, as listed in

Table 4. The measured values of open circuit voltage

${V}_{OC}^{\mathrm{meas}.}$ are also given in

Table 4 for comparison.

## 4. Discussions

A comprehensive study of the open circuit voltage in BHJ OSCs has been carried out. Three different expressions for

${V}_{OC}$ are derived; the first in Equation (10) is obtained when the donor material in a BHJ OSC is excited, the second in Equation (15) is obtained when the acceptor material is excited, and the third in Equation (18) is obtained when both the donor and acceptor materials are excited simultaneously. As stated previously, when the photoexcitation occurs in the donor, an electron is transferred from the LUMO of the donor to the LUMO of the acceptor, and conversely, when photoexcitation occurs in the acceptor, a hole is transferred from the HOMO of the acceptor to the HOMO of the donor [

48]. The photoexcitation of the donor and acceptor individually produces comparable

${V}_{OC}$ values from Equations (10) and (15), as shown in

Figure 3. Moreover, according to

Figure 3, the linear correlation between

${V}_{OC}$ and

${E}_{DA}/q$ agrees well with previous studies [

20,

30,

32,

33,

64]. The third open circuit voltage

${V}_{OC}^{DA}$ calculated in Equation (18) from the simultaneous excitations of both the donor and acceptor, as given in

Table 4, is also comparable with the other two

${V}_{OC}\mathrm{s}$. All three

${V}_{OC}\mathrm{s}$ thus derived produce comparable results and agree reasonably well with the experimental values (see

Table 2,

Table 3 and

Table 4). However, the calculated

${V}_{OC}$ values in

Table 2,

Table 3 and

Table 4 are obtained from slightly different energetic parameters; as a result, we obtained three different sets of

${V}_{OC}\mathrm{s}$ for the photoexcitation of the donor, photoexcitation of the acceptor, and simultaneous photoexcitation of both. However, experimentally, there is only one set of measured values of

${V}_{OC}$ to compare the theoretical results with. As a result, the calculated range of

${V}_{OC}$ for the materials considered here is 0.50–0.96 V, which is slightly different from the range of experimental values (0.52–0.85 V). Thus, as stated above, this discrepancy may be attributed to the different energies

$({E}_{F}^{h}-{E}_{\mathrm{HOMO}}^{\mathrm{D}})/q$ used in the calculation of

${V}_{OC}$ from Equations (10), (15), and (18).

In addition, the two

${V}_{OC}\mathrm{s}$ in Equations (10) and (15) depend linearly on

$\mathrm{ln}n$ and

$\mathrm{ln}p$, respectively, and this agrees well with previously measured results [

28,

63]. This also holds for

${V}_{OC}$ derived from Equation (18), which depends on

$\mathrm{ln}(\sqrt{np)}$. By setting

$n=p$ in Equations (10), (15), and (18), we found that all three

${V}_{OC}\mathrm{s}$ increase linearly with

$\mathrm{ln}n$ and have the same slope equal to

$\frac{{k}_{B}T}{q}/\mathrm{log}(\mathrm{exp}(1))\approx 2.3\times 2{k}_{B}T/q\approx 119\text{}\mathrm{mV}$. This implies that the materials with higher photo-generated charge carrier concentration will have higher

${V}_{OC}$ in comparison with materials with lower carrier concentration.

It is also worth noting that the dependence of

${V}_{OC}$ on the incident light intensity has been studied [

17,

24]. Substituting

$np=\frac{G}{\gamma}\propto {I}_{abs}={I}_{o}[1-\mathrm{exp}(-\alpha t)]$ into Equation (18), where

${I}_{abs}$ is the absorbed light intensity,

${I}_{0}$ is the incident light intensity,

$\alpha $ is the absorption coefficient, and

$t$ is the thickness of the active layer, we get:

The

${V}_{OC}$ in Equation (20) depends linearly on

$\mathrm{ln}{I}_{0}$ with a slope of

$\frac{{k}_{B}T}{q}$, which agrees perfectly well with the plotted

${V}_{OC}$ as a function of

$\mathrm{ln}{I}_{0}$ in [

17,

24].

Orlowski et al. [

66] used the energetics of quasi-Fermi levels to study

${V}_{OC}$ in a heterojunction solar cell. They found that for a particular donor (p-type)-acceptor (n-type) heterojunction, the electron concentration

$(n)$ in the conduction band (LUMO) of the acceptor and the hole concentration

$(p)$ in the valence band (HOMO) of the donor, coupled with their respective quasi-Fermi levels, can create open circuit voltages like two independent cells. This agrees very well with our concept of deriving two different

${V}_{OC}\mathrm{s}$ in Equations (10) and (15), although both the expressions produce similar results.

Furthermore, we find that the

${V}_{OC}^{D}$ lies 0.34–0.49 V below the

${E}_{DA}/q$ in column 5 of

Table 2 when the donor is excited, comparable to 0.34–0.48 V below the

${E}_{DA}/q$ in column 5 of

Table 3 when the acceptor is excited. This implies that both the donor and acceptor photoexcitations can contribute to the achievable open circuit voltage as well as the photovoltaic performance of BHJ OSCs; this agrees with our earlier results [

45]. The voltage offsets

$(\raisebox{1ex}{${E}_{DA}$}\!\left/ \!\raisebox{-1ex}{$q$}\right.-{V}_{OC}^{j})$ (where

$j=D,\text{\hspace{0.17em}}A\text{\hspace{0.17em}}or\text{\hspace{0.17em}}DA$) of the various blends listed in

Table 2,

Table 3 and

Table 4, ranging from 0.34–0.50 V, agree well with the experimental values measured at room temperature in the range of 0.25–0.48 V for small molecule OSCs [

67], 0.32 V for polymer:fullerene solar cells of indacenoedithiophene (IDT) polymer [

68], 0.34–0.44 V for a range of donor-acceptor blends [

69], 0.38 V for OC

_{1}C

_{10}-PPV:PCBM solar cells [

12], and 0.30–0.60 V for BHJ OSCs [

20]. Other low voltage offsets

$(\raisebox{1ex}{${E}_{DA}$}\!\left/ \!\raisebox{-1ex}{$q$}\right.-{V}_{OC}^{j})$ that have been reported recently are 0.25 V for an evaporated bilayer OSC [

70], 0.23 V and 0.26 V for diketopyrrolopyrrole-thieno[2,3-f]benzofuran (DTD):PC

_{60}BM and DTD:naphthalene diimide acceptor-polymer (N2200) systems, respectively [

71], and 0.34–0.40 V for BHJ OSCs [

72], with which our calculated values also somewhat agree in the range.

Overall, it may be deduced from our results that separate and simultaneous photoexcitations of the donor and acceptor produce comparable

${V}_{OC}\mathrm{s}$, and the highest loss in

${V}_{OC}$ is found to be contributed by the energetics of the donor and acceptor materials. Therefore, to achieve a high

${V}_{OC}$, it is necessary to choose higher efficiency polymer-donor and acceptor materials, as well as control the donor and acceptor material energetics, which may lead to improvements in the PCEs of BHJ OSCs [

32].