Open circuit voltage and efficiency in ternary organic photovoltaic blends

Abstract

Organic bulk heterojunction solar cells based on ternary blends of two donor absorbers and one acceptor are investigated by experiments and modeling. The commonly observed continuous tunability of the open circuit voltage V_OC with the donor1 : donor2 ratio can quantitatively be explained as quasi-Fermi level splitting due to photocreated charges filling a joint density of states that is broadened by Gaussian disorder. On this basis, a predictive model for the power conversion efficiency that accounts for the composition-dependent absorption and the shape of the current–voltage characteristic curve is developed. When all other parameters, most notably the fill factor, are constant, we find that for state-of-the-art absorbers, having a broad and strong absorption spectrum, ternary blends offer no advantage over binary ones. For absorbers with a more narrow absorption spectrum ternary blends of donors with complementary absorption spectra, offer modest improvements over binary ones. In contrast, when, upon blending, transport and/or recombination kinetics are improved, leading to an increased fill factor, ternaries may offer significant advantages over binaries.

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Broader context

Solar cells based on blends of organic semiconductors almost routinely achieve power conversion efficiencies (PCEs) in the 10% range. One of the strategies that has been proposed to push this limit further upward is the use of so-called ternary blends in which two absorbing donor materials instead of one are blended with the electron acceptor. Systematic improvement of PCE by ternary blending has been reported. Despite the experimental success, the current understanding of both the PCE enhancement and the underlying puzzling tunability of the open circuit voltage V_OC is limited and essentially qualitative. In this contribution, we develop and validate a quantitative and predictive framework for both the V_OC and PCE of ternary organic solar cells, which allows us to propose design rules. In contrast to current wisdom, we demonstrate that PCE enhancement in ternary blends hardly results from optimizing the overlap with the solar spectrum but predominantly from synergetic improvement of the charge transport and recombination kinetics. The model can be easily extended to include more details and will allow other researchers to upfront evaluate the photovoltaic potential of ‘any’ ternary material combination.

Introduction

Despite the very promising evolution of bulk heterojunction organic photovoltaics (BHJ-OPVs), leading to single junction efficiencies currently exceeding 10% PCE,^1,2 incomplete absorption of the solar spectrum remains a limiting factor for the power conversion efficiency (PCE). Ternary BHJs, comprising two donors with complementary absorption spectra, promise to increase the generation of charge carriers as a result of increased sunlight absorption.³ However, the introduction of a second, more red-absorbing donor with a shallower HOMO is intuitively expected to pin the open circuit voltage (V_OC) to the lower of the two values corresponding to the binary blends, thus limiting the PCE. Surprisingly enough, experiments show that absolute pinning does not take place. Instead, the open-circuit voltage in ternary OPVs is shown to be continuously tunable with donor composition,^4–7 allowing improvement of the PCE when combined with properly, i.e. complementary chosen donor absorption spectra. Likewise, addition of a second acceptor with a different LUMO level leads to a similar tunability of V_OC.^8,9

The reason for the continuously tunable V_OC remains somewhat in the dark. Recently, Yang et al. proposed a ‘parallel junction’ model to phenomenologically describe this behavior.¹⁰ On the same basis, Savoie et al. proposed an empirical expression for V_OC as a current-weighted average of the two sub-systems and used this to estimate the maximal theoretical PCE (13%) for varying donor bandgaps.¹¹ Apart from lacking a formal basis, there are two problems with the parallel junction picture. First, it relies on a specific morphology in the active layer that allows different regions to act as independent binary cells that are electrically decoupled. Even if a fortunate morphology would lead to the presence of two (hole-) percolating networks that are electrically isolated from each other, the presence of metal contacts does unavoidably enforce equal quasi-Fermi levels in the subcells. The subcells cannot be treated independently. This problem becomes evident while considering the equivalent circuit of two parallel photodiodes. When biased beyond the lowest V_OC, i.e. in the regime where the high-V_OC subcell is supposed to boost V_OC, the low-V_OC subcell is in fact biased in its forward direction and will short the system. Alternatively, Street proposed a model for the formation of organic alloys in ternary blends based on two fullerene acceptors and a single donor.¹² For this model to apply the charge wavefunction must be delocalized over many sites; for example for a 1 : 9 acceptor ratio, delocalization over more than 10 acceptor units is required to ‘feel’ on average at least one instance of each acceptor type. Although this may be conceivable for well-packed fullerenes, this intuitively seems to be rather unlikely for mixtures of heterogeneous polymers. Nevertheless, Khlyabich et al. recently argued for such an alloying effect in polymers, although no explicit calculations of V_OC were presented.⁴

Hence for a quantitative assessment of the (im)possibilities of ternary bulk heterojunction solar cells there is need for a somewhat more rigorous model. Here, we implement a simple physical model based on state filling in a disordered Gaussian density of states. By first assuming a constant occupation the experimentally observed gradual dependence of V_OC on the donor1 : donor2 (D1 : D2) ratio is explained. The assumption of constant occupation for all different polymer fractions is then expanded to include absorption and transport allowing a quantitative analysis of the achievable short circuit current (j_SC) and power conversion efficiency in ternary blends. In agreement with experiments, it is predicted that for materials with a wide and strong absorption spectrum no improvement can be expected going from binary to ternary devices when all other parameters, like the fill factor, are unaffected by the blending. On the other hand, a moderate improvement is expected for ternary over binary blends for more narrow-spectrum absorbers. In contrast, we demonstrate that when the transport and/or recombination dynamics are simultaneously improved, ternary blends may offer significant advantages over binaries.

In the present work, we focus on donor1:donor2:acceptor BHJ OPVs but the concept should be equally applicable to donor:acceptor1:acceptor2 devices. It is unimportant if the constituent donor(s) and acceptor(s) are polymers, oligomers or small molecules. Moreover, for transparency we shall refer to the central energy of the disorder-broadened HOMO and LUMO levels as the HOMO or LUMO energy, which differs from what is probed by common electrochemical methods like cyclic voltammetry. Also the optical bandgap, when taken to be the absorption onset, is substantially smaller than the HOMO–LUMO gap due to (a) broadening and (b) excitonic effects. Because of the large uncertainties associated with both the absorption onset and the exciton binding energy we refrain from attempts to convert HOMO–LUMO gaps to optical gaps. This has no effect on the conclusions of this work.

Results

Open circuit voltage of ternary blends

In this section, we will demonstrate why V_OC does not pin to the HOMO of the shallower donor (V_OC2 in Fig. 1). In order to study the connection between the donor polymer composition and V_OC, different ternary BHJ OPVs were fabricated by mixing the acceptor PC₇₁BM with pairs of the donor polymers TQp6, TQm6 and TQm12 in compositional steps of 10%. The polymers were synthesized as indicated in ref. 13 and their molecular structures and full names can be found in Fig. S1 in the ESI.† These polymers were chosen due to their similar structures leading to good miscibility. The atomic force microscopy images were consistent with good miscibility. We note that poor miscibility of the two donors need not lead to different behaviors. This is due to the presence of metal contacts that, as explained above, cause an equilibration of the Fermi levels throughout the device. However, for donor–donor demixing not to cause problems, it is essential that both donors (continue to) mix well with the acceptor, which leads to a system consisting of spatially separated mostly binary regions. Also the phase separation between the donor1- and donor2-rich phases should not cause additional morphological problems like overly rough or discontinuous films. The relative HOMO level energies of our donor polymers give rise to ternary systems with either a relatively small or a relatively large difference in V_OC between the binary extremes. For all devices, the total donor : acceptor ratio was kept constant at 1 : 1. For the given D1 : D2 ratio, the individual D : A ratio is not optimized as in this work transparency of the results is favored over highest performance. Further details about device fabrication are given in the Experimental section. The open circuit voltages under 1 sun illumination vs. composition are shown in Fig. 2.

Fig. 1 Schematic band diagram depicting the offset of LUMO and HOMO levels for a donor pair D1 and D2 and an acceptor A. Limiting binary V_OCs are indicated as being proportional to the difference between the LUMO of the acceptor and the HOMO of the donors. The V_OC of the D1 : D2 : A ternary blend is continuously tunable between V_OC1 and V_OC2.

Fig. 2 Experimental (symbols) and simulation (lines) data of V_OCvs. composition for TQm6:TQm12:PC₇₁BM (black) and TQp6:TQm12:PC₇₁BM (blue) DDA ternary blends. The x-axis is the ratio of the concentration of the first polymer to the total polymer concentration. The estimated standard deviation in V_OC as determined from several devices on multiple substrates did not exceed 10 meV (plotted error bar) for any of the V_OC measurements. The parameter values used for the simulations are listed in Table S1 in the ESI.† The inset shows the same data normalized to the values at x = 0 and x = 1.

The open circuit voltage of ternary systems based on TQm6:TQm12 (black symbols in Fig. 2) and TQp6:TQm12 (blue symbols) shows smooth and near-monotonous composition dependence. The two outlying V_OC data points of the TQm6:TQm12 at 0.5 and 0.6 polymer ratios were investigated by AFM but no systematic (deviations in) phase separation was observed (Fig. S2 in ESI†); neither did any of the other performance indicators indicate specific problems for these two devices. Irrespective of the particular functional shape, all ternary systems are consistent with previous reports that V_OC is continuously tunable via the D1 : D2 ratio. Moreover, they strongly suggest an increasing bowing of the trend line with increasing difference between the V_OCs of the binary extremes. This is better visible in the inset to Fig. 2 where the normalized V_OC shifts are plotted.

In order to rationalize these observations, we apply a simple state filling model where the disordered Gaussian density of states for the two donor polymers and the acceptor are defined according to

E_HOMO and E_LUMO are the donor and acceptor energy levels and centers of the Gaussian DOS. The parameter σ defines the broadening of the DOS. The independent variable E is the energy. In the present simulation, the relevant energy levels are the HOMO levels of the two donors and the LUMO level of the acceptor. The effective hole DOS, G_eff is calculated according to

G_eff = f_D1·G_D1 + (1 − f_D1)·G_D2 (2)

where f_D1 is the composition ratio of D1 to the total donor concentration and G_D1 and G_D2 are the Gaussian DOS of the two polymers (eqn (1)). For electrons, that reside on the single acceptor, G_eff = G. The key assumption at this point is that the electron and hole populations in the entire BHJ are in equilibrium, i.e. there are no lateral variations in the quasi Fermi levels E_F. Hence, the electron quasi Fermi level E_F,el can be calculated from the condition where n/N₀ is the electron occupation with the total site density N₀ and the Fermi–Dirac distribution f_FD(E − E_F,el) The temperature is set at 300 K and k_B is Boltzmann's constant. For holes similar expressions are used to obtain E_F,hole from p/P₀. V_OC is then obtained as the difference between the electron and hole quasi Fermi levels E_F,el and E_F,hole:

V_OC = E_F,el − E_F,hole (5)

The results of the aforementioned physical model are depicted in Fig. 2 as solid lines. In these simulations the (hole) occupation is an effective parameter, accounting for light intensity and absorption, transport, recombination etc., that is kept constant for all compositions. The main trends of the experiments, i.e. a continuous transition between the binary extremes at x = 0 and x = 1 with a bending that increases with the increasing ΔHOMO, are well reproduced. This behavior is explained in Fig. 3 where the effect of the donor HOMO level offset ΔHOMO on the composition dependence of the quasi Fermi level for holes, and hence on V_OC is demonstrated. The left panel of the figure shows a pure donor1 (D1):acceptor (A) composition resulting in a deep Fermi level for the holes. When a small fraction of a second donor (D2) with a less deep HOMO level is added, two extreme scenarios can be discerned. When the HOMO of the second donor is far from that of D1, it will effectively pin the hole Fermi level since (a) the occupation is always much less than the fraction of D2 and (b) the DOS of D1 is negligible at the Fermi energy: all holes sit on D2 sites. When the HOMO of the second donor is close to that of D1, condition (b) above is no longer met, and the Fermi level is determined by the relative weights of the D1 and D2 DOS at the Fermi energy, giving rise to a more gradual shift of E_F,hole with the D1 : D2 ratio.

Fig. 3 Schematic representation of the calculated V_OC as the difference between the electrons and holes quasi-Fermi level for a given occupation. Left: binary D1 : A; middle, right: D1 : D2 = 0.7 : 0.3 showing small and large HOMO level differences, respectively. The DOS and DOOS of the donors and acceptor are represented by light and dark gray areas, respectively. The DOS contributions of D1 and D2 are shown by dark gray lines.

As a further confirmation we used the same model to successfully describe experimental V_OC data for three optimized D1:D2:A ternary systems reported in the literature.^4,5 The fitting results and used parameter values are shown in the ESI,† as Fig. S5–S7 and Tables S2–S4, respectively. These fitting results confirm the conclusion from Fig. 2 and 3 above, that larger differences in V_OCs between the binary extremes lead to a more pronounced curvature of the V_OCvs. composition curve. Importantly, it does so without invoking any structural or alloying effects.^4,12

Having established a simple, but physically consistent model to describe the composition dependence of V_OC in ternary blends, the question arises if the ‘tunability’ of V_OC indeed allows ternary OPVs to outperform binaries. Up to now, simulations have been based on the assumption that occupation is constant for all compositions, a fact that does not allow addressing the key (supposed) advantage of ternaries, a better overlap with the solar spectrum, leading to higher j_SC and PCE. In the next section, we will therefore extend our model to include the composition dependent absorption.

Power conversion efficiency of ternary blends

The need of quantifying the occupation of the density of states, i.e. knowing the number of charge carriers in the device, was solved by implementing an optical model that takes into account the net absorption of the constituent absorbers, here assumed to be the donors. The respective absorption lengths L_D1,2 as functions of the wavelength are given bywhere L_D10 and L_D20 are the absorption lengths (or attenuation lengths) of the pure materials and the donor1 fraction f_D1,2 = D1/([D1] + [D2]). G_D1,2(λ) is the absorption profile that is assumed to be Gaussian – for organic semiconductors this choice is more realistic than a step-like absorption onset as used in e.g. the Shockley–Queisser analysis:¹⁴with E_g1,2 and W_D1,2 the maximum and full width at half maximum (FWHM) – both in nm – of the absorption. The absorption maximum corresponds to the HOMO–LUMO gap as described in the previous section and Fig. 1. E_g1,2 and W_D1,2 are chosen to fit the real absorption spectrum of the donors under study (Fig. S8 in ESI†). Although we will not pursue this here, we note in passing that eqn (7) can readily be modified to account for additional absorption maxima or the absorption of sensitizers.^3,10

The effective absorption length L_eff for the blend is given by

For our semi-transparent devices of device thickness L_device the absorption is approximated as Using the absorption spectrum eqn (9) the flux of photogenerated holes and electrons is calculated by integrating the product of the solar spectrum and the absorption spectrum aswhere, IQE is the internal quantum efficiency that was set at 0.85 defining a moderate device performance. AM1.5 is the solar spectrum and E_ph is the photon energy in eV needed for dimensional reasons. Again, we note that eqn (10) can be readily extended to include more advanced absorption models that take into account e.g. reflection losses of cavity effects. For the present purpose, focusing on relative differences and semitransparent cells, there is little need to do so.

In order to correlate the above with experiments as straightforward and transparent as possible, we adopt a phenomenological transport model. This avoids explicitly addressing the complicated charge transport in the blend, including morphological effects, (reduced) bimolecular and trap-assisted recombination, density and field dependent mobilities, etc. The free carrier concentration is approximated by multiplying the carrier flux ṅ or ṗ with the respective lifetime τ_el or τ_hole:

n = ṅ·τ_el (11a)

p = ṗ·τ_hole (11b)

Although the lifetimes of electrons and holes can be expected to be different, we found that using different values led to an under-determinedness in our model. While running simulations in which the electron and hole lifetimes were varied between 1 and 100 μs, both concertedly and independently, we found only minor quantitative differences. The conclusions from the following sections are therefore independent of charge carrier lifetimes. Hence, equal τ_el and τ_hole are taken and set at 10 μs. This assumption corresponds to the optimal situation of balanced charge transport, which also minimizes space charge effects as it leads to n = p.

V _OC is calculated from the occupation and quasi Fermi levels as described earlier in eqn (3)–(5). Finally, the short circuit current j_SC and the power conversion efficiency PCE are calculated as

j_SC = q·ṗ·L_device (12a)

P_out = FF(⋯)·V_OC·j_SC (12b)

PCE = P_out/P_in (12c)

where, FF(⋯) is the fill factor that in principle is a complicated and essentially unknown function of many parameters. P_in is the input solar power ∼1 kW m⁻² defined as the integral of the AM1.5 spectrum over the wavelength. As we primarily are interested in the difference in PCE as a function of HOMO and LUMO energies, the fill factor will be kept constant. The dependency of the PCE on the FF will be explicitly addressed in the final section of this paper.

As the first test, by implementing the optical model using the parameter values in Table S1 in the ESI,† the calculated V_OC, j_SC and PCE closely fit the trends in the experimental data of the TQm6:TQm12:PC₇₁BM system, as seen in Fig. 4. The simulation results show the expected increasing current with decreasing open circuit voltage due to an increased absorption. The fill factor for this specific blend was measured to be fairly constant at 0.45 and this value was used in the simulations. As mentioned above, the three donor polymers were selected to give, in combination with PCBM, two ternary systems with weak and strong composition dependence of V_OC, respectively, see Fig. 2. At the same time, the rather similar absorption spectra lead to the absence of clear trends in j_SC and PCE for the systems where TQp6 is mixed with TQm6 and TQm12 polymers, that can, however, still be surprisingly accurately reproduced given the simplicity of the model presented above (see ESI,† Fig. S9, S10 and Tables S6, S7).

Fig. 4 Simulation (lines) and experimental (symbols) data for V_OC (a), j_SC (b) and PCE (c) vs. composition for the TQm6:TQm12:PC₇₁BM ternary OPV system.

Selecting optimal donor materials and compositions

In this final section, the performance limits of ternary OPVs will be assessed. The goal is to establish an estimate for the maximum improvement ternary devices may have over the corresponding binary devices. For the following calculations, we keep all parameters constant and equal to experimentally realistic values except for the LUMO or the HOMO levels of the donors. As a consequence, these calculations focus on ternary systems with complementary absorption spectra, in contrast to the materials used in the experiments discussed above. Since we are interested in the maximum achievable PCE, for each specific pair of donors, with given HOMO and LUMO levels, the ratio of the concentration of the first donor to the total donor concentration D1/(D1 + D2) that gives rise to the highest PCE is calculated. An example of a typical PCE vs. D1/(D1 + D2) curve is given in the ESI,† Fig. S11.

Case 1 – LUMO variation. This case is inspired by the work of Yang et al.,¹⁵ who investigated the performance of ternary OPVs for a range of polymer donors with nearly identical HOMO levels and varying LUMO levels. This system is unlikely to lead to an optimal ternary device in absolute sense due to two reasons. First, by keeping the HOMO levels fixed it cannot benefit from voltage gains associated with adding a more ‘blue’ absorber, cf.Fig. 1 and 2. Second, varying the LUMO levels implies that mostly non-optimal LUMO offsets between donor(s) and acceptor, causing energy losses upon charge transfer, will be evaluated. The results of this type of ternary system are nevertheless very instructive.

In these simulations, the LUMO levels for both donors 1 and 2 are varied between −2.5 eV and −4.1 eV, while their HOMO level is fixed at −5.4 eV. The acceptor LUMO level for PC₇₁BM is set at −4.1 eV. Wherever possible we used parameters that correspond to ref. 15 (see ESI,† Table S8). The results of the simulations are shown in Fig. 5, with the left panel showing the optimal D1 fraction for the given LUMO level combination and the right panel the corresponding PCE, i.e. at the optimal D1 fraction.

Fig. 5 (a) Optimal fraction of donor1 and (b) associated PCE for all the different LUMO level combinations; the HOMO level is fixed at −5.4 eV. The diagonal dashed line indicates equivalent D1 and D2, i.e. a binary system; as expected graphs a and b are anti-symmetric and symmetric, respectively, w.r.t. to this line. The highest overall PCE of 8.75% is obtained for ternary blends indicated by the dashed ellipses.

In agreement with ref. 15, for most D1–D2 combinations the optimal composition is a ternary one, i.e. in Fig. 5a bright yellow (pure D1) and blue (pure D2) regions are relatively rare and do not come near the overall optimal PCE of 8.75% that is found for a ternary system with a D1 : D2 ≈ 1 : 1. The latter is rather close to the value of 8.5% that is reported by Yang et al. for (PTB7 : PBDTT-SeDPP = 1 : 1) : PC₇₁BM. The donor LUMO levels for which the optimal PCE is found in the simulations, ∼−3.25 eV and ∼−3.85 eV are very consistent with the −3.31 eV and −3.70 eV of PTB7 and PBDTT-SeDPP, respectively.

The reason why, in the case of a fixed and common donor HOMO level, ternary systems outperform binary ones is the following. The constant HOMO energy makes V_OC essentially constant, so the only thing to be optimized is the overlap with the solar spectrum. For a given finite absorption width two donors are, in virtually all cases, better at that than a single one. For the used parameters, the results indicate that ternary devices are the better choice over binary ones and that they can increase the PCE by ∼33% as the best binary device has a PCE of ∼6.5%, whereas best ternary device has 8.75%. For comparison, Yang et al. found binary PCEs between 6.2% and 7.2%, again consistent with our model. It should, however, be stressed that the superiority of ternary OPVs is not a general result; especially when not the donor HOMO but the donor LUMO is kept fixed. In this case, higher performances can be achieved leading to a dramatically changed picture.

Case 2 – HOMO variation. In this case, the LUMO level of the donors is fixed at −3.8 eV creating a (near) optimal offset of 0.3 eV from the PC₇₁BM LUMO which we fixed at −4.1 eV. In contrast to the previous case with a fixed HOMO/variable LUMO, changing the HOMO affects both V_OC and the overlap with the solar spectrum, i.e. j_SC: a deep HOMO level increases V_OC, but also the absorption bandgap which eventually reduces the current. On the other hand, a more ‘red’ absorber maximizes the overlap with the solar spectrum and j_SC, but the required shallow HOMO level will result in a smaller V_OC.

An important technicality that needs to be addressed here is the sensitivity of the model results to the choice of parameters. First, all results presented in this work are robust to parameter variations in the sense that the qualitative conclusions remain unaffected. However, the quantitative results depend on parameter choices and especially on the shape of the absorption profile. For the calculation of Fig. 5 above quantitative agreement with ref. 15 in Fig. 5 is best when a nm-axis is used. The reason for that is likely the fact that the actual absorption profiles are rather symmetric on a nm-axis. For the present calculations (variable HOMO), it turns out that evaluating eqn (7) on a nm-axis leads to the results that are in conflict with experimental observations. Therefore in the following section an eV-axis will be used; otherwise all parameters are kept as in Fig. 5.

The results of Fig. 6 indicate a highest achievable PCE of ∼13%. Surprisingly, this is found for binary devices – in Fig. 6b the superiority of binary devices is shown as a yellow cross. Mapping this cross on a panel shows that the horizontal and vertical branches correspond to pure D1 and pure D2, respectively. The two small ternary regions, indicated by dashed ellipses, show that for certain polymer combinations optimized ternary systems can have an improved PCE over the corresponding binaries. At the same time, these specific ternaries do not give an optimal overall PCE.

Fig. 6 (a) Optimal fraction of donor1 and (b) associated PCE for all different HOMO level combinations for an absorption width (FWHM = 0.8 eV); the LUMO level is fixed at −3.8 eV; a constant fill factor of 0.65 was used. The diagonal dashed line indicates equivalent D1 and D2, i.e. a binary system. Small ternary regions are indicated by dashed ellipses.

There are multiple reasons for the abrupt jumps in the optimal composition in Fig. 6a. The jump around the ‘binary line’ (dashed diagonal) reflects the fact that the overall-optimal device is binary: for D1,2 combinations where neither donor is optimal, still the optimal D1 : D2 ratio is the one where one has as much as possible from the donor that comes closest to the optimal donor, i.e. again a binary system. In other words, disadvantages in terms of reduced current or V_OC outweigh any advantages of having a ternary system. The other jumps are associated with the presence of two maxima in the PCE vs. composition curve as explained at Fig. S11 in the ESI.†

It is important that the maximum PCE is now predicted to be substantially larger than in the previous variable-LUMO case. The reason is that in the present case energy losses associated with electron transfer to the acceptor are much reduced. This is equivalent to the fact that increasing the bandgap in this (variable HOMO) case does, but in the other (variable LUMO) case does not lead to an increased V_OC, while having the same effect on the short circuit current.

Comparing to the work of Scharber et al. on donor selection rules,¹⁶ where they predict a PCE of 10% for binary devices, the higher value of 13% that is found here can mostly be attributed to the used higher IQE of 85% (instead of 65%). A close fitting of the present model to the one of Scharber et al. is included in Fig. S12 (ESI†) using the parameters of Table S9 (ESI†).

A remarkable observation in Fig. 6 is the position of the optimal HOMO level at −5.78 eV, corresponding to a bandgap of ∼2.0 eV. The difference with experimental findings where optimal HOMO energies around −5.3 eV are typically found^1,2,15 is partially due to the difference in the way the bandgap and HOMO and LUMO energies are defined. Here, central energies are used; the parameters in Fig. 6 give in fact rise to an absorption onset at ∼1.5 eV (∼830 nm), in good agreement with the PTB7 absorption spectrum in ref. 15 that we took as the reference for the absorption width. At the same time, the present results indicate that substantial improvement in PCE can be obtained for a PTB7-like material with the same bandgap but downward-shifted HOMO and LUMO levels – PTB7 has a LUMO at ∼−3.3 eV, giving rise to a large charge transfer energy loss. This suggestion is corroborated by the results of He et al. who showed that PTB7-Th, having slightly deeper HOMO and LUMO levels, indeed gives improved performance.²

Using parameters that correspond to current state-of-the-art donor materials, ternary compositions show no improvement in PCE over binaries for the vast majority of energy level positions (Fig. 6). The picture becomes different if a less optimal absorber is used. Setting the absorption FWHM from 0.8 eV in Fig. 6 to 0.5 eV leads to a situation where the overall optimal PCE is produced by a ternary blend, see Fig. 7. Note that this FWHM corresponds to ref. 10 where, at the same time, rather optimistic absorption lengths of 10–20 nm were used in combination with a 200 nm device thickness. In Fig. 7b a clear cross shape corresponding to near-optimal binaries is visible. As a general design rule, we find that narrow but strong absorbers give rise to systems where ternary compositions offer improved performance over binary ones; broadening the absorption (cf.Fig. 6 and 7) or weakening the absorption length shifts the balance to binary-dominated ones. This is further illustrated in Fig. S13 of the ESI.† The rationalization of this rule consists of two parts. First, for strong absorbers the penalty for diluting the absorber that is closest to the optimal donor by adding a small fraction of another donor, is not so large as most light will still get absorbed. Second, for narrow absorbers the gain of adding another absorption band is relatively large, especially if the added absorber is a strong one.

Fig. 7 (a) Optimal fraction of donor1 and (b) associated PCE for all different HOMO level combinations for a narrower absorption width (FWHM = 0.5 eV); the LUMO level is fixed at −3.8 eV; a constant fill factor of 0.65 was used. The diagonal dashed line indicates equivalent D1 and D2, i.e. a binary system. Ternary regions indicated by dashed ellipses give an overall optimal PCE of ∼8.2%.

An instructive intermediate situation arises when a material with a wide absorption band, as in Fig. 6, is mixed with the one with a narrow absorption, as in Fig. 7. Intuitively one might expect the narrow band to complement the wide one. It does, but at the price of diluting the wide absorption material. The net result is current and PCE loss with respect to the overall optimal binary device. Alternatively, one can rationalize this result on the basis of Fig. 6 by realizing that adding a narrow absorber must in all cases be inferior to adding a wider one.

The above findings lead to the conclusion that in the case of realistic good absorbers with close to optimal energy levels the enhanced overlap with the solar spectrum cannot be expected to lead to significant improvement in performance of ternary blends of the D1:D2:A type. However, this conclusion depends critically on the assumption that all other parameters are independent of composition. In practice, especially the fill factor can depend significantly on composition, as e.g. in ref. 6, 7 and 17, leading to optimal ternary compositions. Where this is not the case, e.g. in ref. 5 and 18, optimal PCEs are found for close to binary compositions, in agreement with the conclusions from Fig. 6 and 7.

In order to illustrate the critical role the composition-dependent fill factor plays in achieving ternary systems that outperform their binary counterparts, the simulations of Fig. 6 were re-run with a fill factor that is a function of the D1 fraction f_D1:

FF(f_D1) = af_D1² + (FF_D1 − FF_D2 − a)f_D1 + FF_D2 (13)

with FF_D1 and FF_D2 the fill factors at the binary extremes corresponding to donors 1 and 2 and a a bowing constant. A positive (negative) bowing parameter a reflects degraded (improved) charge transport and recombination kinetics with respect to a linear interpolation between the binary extremes and can, for example result from morphological effects like changes in molecular packing and phase separation.

In the simulations of Fig. 8, we used typical values of FF_D1 = 0.65, FF_D2 = 0.5 and a = −0.45, corresponding to a maximum FF of 0.7 at f_D1 = 0.66, as shown in Fig. S14 in the ESI.† The calculation results now show that for virtually all HOMO level combinations, including the overall-optimal one, a ternary compound is preferred. Comparison of these results with that in Fig. 6 allows two important conclusions. First, in the case of ternary blending using state-of-the-art absorbers, minor or no improvements due to a better overlap with the solar spectrum can be expected (Fig. 6 and 7), but ‘secondary effects’ that lead to an improved fill factor can give rise to significant improvements (Fig. 8). This is in agreement with the experimentally observed correlation between the optimal fill factor and PCE.^6,7,17 Second, in Fig. 8 the overall optimal PCE is still located on the binary line, where both donors have the same HOMO and LUMO. This reiterates the superiority of a single optimal bandgap provided the absorption is sufficiently broad. Hence, the ideal ternary system consists of two heterogeneous donors, both having this optimal bandgap, and showing a synergistic behavior in terms of FF, i.e. a negative a in eqn (13). Experimentally, this may be hard to achieve and one may be forced away from the overall PCE optimum.

Fig. 8 (a) Optimal fraction of donor1 and (b) associated PCE for all different HOMO level combinations. The calculation parameters are same as Fig. 6 but with a composition-dependent fill factor given by eqn (13) with FF_D1 = 0.65, FF_D2 = 0.5 and a = −0.45, corresponding to a maximum FF of 0.7 at f_D1 = 0.66. The dashed ellipse indicates the best overall PCE of ∼14.1% at a D1 fraction of 0.66.

As for our own experimental system, that was selected to investigate V_OC, the virtual absence of spectral differences between the donors and the composition-independent fill factor preclude ternary compositions from having any advantages over binaries.

We shall finally draw an analogy between the present results and the 3-phase morphology encountered in binary OPV systems in which pure and aggregated donor and acceptor phases coexist with an amorphous mixed phase.¹⁹ Ignoring complications associated with the fact that charge transfer to the acceptor may be affected for better or worse, the mixed and pure donor phases can be considered as donor1 and donor2 phases, with slightly shifted energy levels due to the aggregation. From the discussion of Fig. 6 and 7 it can be said that such a 3-phase morphology is sub-optimal in the case of a donor with a broad absorption spectrum.

Experimental

Bulk heterojunction ternary (D1 : D2 : A with (D1 + D2) : A = 1 : 1 weight ratios) OPVs where made from polymer-fullerene solutions of 25 g L⁻¹ concentration in 1,2-dichlorobenzene (ODCB) according to the following process. A 40 nm thin film of poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) was spin coated on pre-cleaned ITO/glass substrates in air. After baking the active layer was spin coated in a glove box. A 0.6/90 nm LiF/Al top contact was evaporated on the active layer under high vacuum conditions. The device areas and thicknesses were measured to be 0.044–0.048 cm² and 90–110 nm, respectively. The jV-curves were measured under simulated AM 1.5 illumination. For all material combinations (TQm6:TQm12, TQp6:TQm6, and TQp6:TQm12) device performance indicators (j_SC, V_OC, and PCE) were measured and averaged on typically 3 devices based on 2 different samples (6 devices in total), leading to relative errors that never exceeded 5%.

Conclusions

We have investigated the device physics of organic bulk heterojunction solar cells based on ternary blends consisting of two absorbing donors (D1 and D2) and a single acceptor, focusing on the relationship between the D1 : D2 ratio and the open circuit voltage V_OC and on the maximally attainable power conversion efficiency (PCE). We experimentally demonstrated that the shape of the V_OCvs. D1-fraction curve depends on the difference between the V_OC of the binary extremes. Larger V_OC differences give rise to a stronger curvature, i.e. a faster drop of V_OC to the lowest binary value. This behavior is quantitatively reproduced by a simple physical model that is based on the notion that photocreated charges fill the joint density of states. By incorporating a simple absorption model that accounts for the composition dependent overlap with the solar spectrum, we are able to make quantitative predictions of the PCE of ternary systems. The PCE model compares favorably to available literature data and indicates that for state-of-the-art absorbers, having a wide and strong absorption spectrum, ternary blends offer no advantages over optimized binaries when all other parameters, most notably the fill factor, are constant. When, in contrast, a composition-dependent fill factor is used, reflecting improved transport and/or recombination kinetics for ternary compositions, the ternaries gain a clear advantage over binaries. Hence, our results suggest that the advantage of ternary OPV systems should not be sought in an improved overlap with the solar spectrum but in synergistic effects related to charge transport and recombination, leading to a higher fill factor. Compared to blending electro-optically inert materials like poly(styrene), which have been shown to improve the transport characteristics of small molecule OPV systems,²⁰ the crucial advantage of blending in a second absorbing and transporting material is that the absorption can remain (mostly) uncompromised, avoiding the need for thicker layers.

Acknowledgements

EW acknowledges the Swedish Research Council for financial support.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ee03095a

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