Introduction

High absorption coefficients of organic molecules make them attractive for photovoltaic applications1. However, the high absorption coefficients come with the downside of exciton binding energies that are much higher than in inorganic materials. Splitting the exciton therefore requires a type II heterojunction between an electron-donating and an electron-accepting molecule. While the process of exciton dissociation and charge generation in organic solar cells has frequently worked remarkably well2,3,4,5,6 in the past, this was often—at least partly—due to the acceptor molecule being a fullerene. However, the predominance of fullerenes has been an obstacle for high performance. While fullerenes do absorb light and may contribute substantially to the photocurrent7, they do not absorb light as efficiently as the typically used donor molecules8. Thus, the percentage of fullerene molecules in the film had to be limited for reasons of high absorption even if a higher fullerene concentration would have been helpful for better charge transport9,10. Till recently, photovoltaic devices using non-fullerene-based acceptor molecules (NFAs) have not shown high performances11,12. This has changed in the last few years and a variety of new non-fullerene acceptor molecules are now available and often achieve higher efficiencies than their fullerene-based counterparts13,14,15,16,17,18,19,20,21. This is due to non-fullerene acceptors absorbing light more efficiently15 and due to open-circuit voltage losses being reduced relative to the case of devices based on fullerene acceptors18,21,22,23,24. The variety of available non-fullerene acceptors also provides great flexibility in the choice of energy gaps and positions of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) levels. This high flexibility does not exist when using fullerenes, which mainly absorb in the ultraviolet (UV) range of the spectrum and whose energy levels are typically changed by using fullerene multiadducts which rarely show good photovoltaic performance25.

When trying to optimize the absorption spectra and energy levels of donor and acceptor molecules, different design criteria apply26 to the situation of donor molecules combined with less strongly absorbing fullerenes or with highly absorbing small-molecule acceptors. With two molecules contributing roughly equally to the total absorptance, the question arises what the ideal band gaps or absorption onsets for the molecules should be. The answer to this question is affected by a series of parameters such as the voltage loss between absorption onset and open-circuit voltage, the spectral width of the absorption of the molecules and the product of peak absorption coefficient and layer thickness.

The aim of this article is to develop design criteria for when to choose either complementary or congruent (overlapping) absorption spectra of donor and acceptor molecules. By taking these design criteria into account, the new flexibility and the promising absorption properties of the NFAs can be utilized and thus contribute to an optimization of the solar cell performance. To develop these criteria, we investigate and systematically differentiate the influence of the various parameters mentioned above.

Results

Model

First, we take a closer look at the absorption spectra of organic semiconductors and formulate a suitable parameterization of the absorption coefficient α(E). The parameterization of α(E) should reflect the important features of the experimental behavior. In addition, this parameterization provides a basis for the further determination of the absorption coefficient αeff(E) of the donor–acceptor blend, as well as the absorptance a(E) of the active layer and finally the solar cell efficiency ηeff.

Unlike in the case of inorganic semiconductors, organic semiconductors have a high absorption coefficient only in a limited spectral region with a typical width of about ~0.5 eV1. As an example, Fig. 1a shows the measured absorption coefficients of the non-fullerene acceptors FBR27 and IDTBR28, the donor polymer PTB7-Th29,30, and their blends (Supplementary Fig. 1 shows the molecular structures). A particular feature being typical for organic semiconductors is that the shape of the absorption peak is asymmetric with the rising edge at lower energies being steeper than the decay at higher energies1.

Fig. 1
figure 1

Comparison between the experimental and simulated absorption coefficients of organic molecules for OPV. (a) Absorption coefficients of the non-fullerene acceptors FBR27 (red) and IDTBR28 (cyan), the donor polymer PTB7-Th29,30 (blue), and their blends (ratio 1:2) (light blue, purple) determined from UV–Vis measurements. (b) Exemplary curves of simulated absorption coefficients α1(E) and α2(E) representing the organic donor and acceptor material. The coefficient αmax and the full-width half-max (FWHM) describe the strength and shape of the absorption peak. In addition, the energies Eip,1 and Eip,2 indicate the inflection points of the corresponding absorption edges whose variation causes a shift in peak position. The effective absorption coefficient αeff(E) of the blend material, used in our simulations, results from the superposition of the respective absorption coefficients α1(E) and α2(E). The corresponding parameters of the simulation are listed in Supplementary Table 1, and Eip,1 = 1.7 eV and Eip,2 = 2 eV

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We simulate this absorption behavior by using a parameterization, in which the absorption coefficient of the organic material is formed by the product of two distribution functions (error functions). This approach allows us to have parameters to adjust the absolute height of the absorption coefficient (α0), the width of the rising and falling edge (σup, σdown), and the distance (∆E) between the inflection points of rising and falling edge. The curves of the absorption coefficients α1(E) and α2(E) illustrated in Fig. 1b are examples of this parameterization. In order to have a consistent definition for the absorption edge, we choose the inflection points of the absorption coefficient as the characteristic energy of the absorption onset or band gap of the material31. The supplementary information (Eqs. S1–S5, Supplementary Table 1, and Supplementary Figs. 2 and 3) provides a detailed overview of the used equations and parameters whose variation adjusts the peak shape and height.

Contrary to the traditional assumption that a significant energy offset between the donor and acceptor levels is necessary to provide a driving force for exciton dissociation25,32,33, recent developments in material science for organic photovoltaics (OPVs) have shown that some non-fullerene organic solar cells (OSCs) with type II heterojunctions exhibit fast and efficient charge separation despite having a negligible energy offset18,34. In these cases, no spectral signature is observed in absorption or luminescence spectra that can be directly attributed to the charge-transfer state, suggesting that the energy ECT of the charge-transfer state is very close to that of either the donor or acceptor molecule. For the purpose of this article, we therefore choose this best-case scenario, where the luminescence of the donor–acceptor blend is not red-shifted relative to the emission of either donor or acceptor. Thus, the effective absorption coefficient αeff(E) of the blend results from the superposition of the donor and acceptor behavior described by the coefficients α1(E) and α2(E). As a further simplification, we also assume that the absorption coefficients of the donor and acceptor materials, which have the same peak shape but differ in their spectral position, contribute equally to the total absorption of the blend. A shift of the energetic position, adjustable by varying the respective inflection-point energy Eip, leads to different effective absorption coefficients αeff(E). In this way, different absorption characteristics of the blend material can be produced, ranging from congruent (Eip,1 = Eip,2) to complementary (Eip,2 ≫ Eip,1). These effective absorption spectra differ in their spectral distribution, shape, and height, as well as their position of the absorption onset. The purple spectrum in Fig. 1b is an example of an effective absorption coefficient αeff(E) generated in this way. It results from a superposition of the single spectra α1(E) and α2(E) also presented in Fig. 1b as red and cyan lines.

A systematic energy shift of the two absorption peaks is used to answer the initial question, how the absorption onsets of the two organic materials should be arranged in order to achieve the highest possible efficiency ηmax. These simulations enable us to develop design criteria for organic solar cells with well-absorbing non-fullerene acceptors, in which donor and acceptor contribute equally to absorption.

Note that determining the efficiency η requires us to calculate the absorptance aeff(E) of the photoactive layer from the effective absorption coefficient, the absorber layer thickness d, and an optical model. In the first part of the paper, we intend to keep the discussion generic (and not layer-stack specific) so we use a simple Lambert–Beer-type model without interferences. In the last section (“Simulations with optical interferences and collection losses”), we assume a specific layer stack which allows us to include interferences in our optical model. The simple Lambert–Beer model for the absorptance used in the first part of the paper assumes complete reflection at the rear contact of the solar cell and neglects reflection at the front or parasitic absorption. Thereby, the absorptance aeff(E) increases monotonously with increasing layer thickness d of the absorber and is given by

$$a_{{\mathrm{eff}}}\left( E \right) = 1 - {\mathrm{exp}}\left( { - 2\alpha _{{\mathrm{eff}}}\left( E \right)d} \right).$$
(1)

In addition, we assume in the first part of the paper that charge collection is efficient, that is, that the external quantum efficiency Qe(E) is equal to the absorptance. Thus, we calculate the short-circuit current Jsc density via

$$J_{{\mathrm{sc}}} = q\mathop {\int }\limits_0^\infty a_{{\mathrm{eff}}}\left( E \right)\phi _{{\mathrm{sun}}}\left( E \right){\mathrm{d}}E,$$
(2)

where q is the elementary charge and ϕsun(E) is the AM 1.5G solar spectrum. The optimum Jsc therefore only depends on the product αeff(E)d, which is however an energy-dependent quantity. For a given shape (α0 = 2.2 × 105 cm−1, σup = 0.08 eV, σdown = 0.6 eV, ΔE = 0.4 eV) the absorption coefficient is entirely characterized by its maximum value. Thus, we use the product αmaxd in the following as one of the key parameters controlling the value of the optimum inflection points Eip,1and Eip,2.

In order to be able to perform the calculation of the efficiency η, an expression for the open-circuit voltage Voc is required. Non-radiative recombination losses are a significant loss mechanism in organic solar cells reducing the achievable open-circuit voltage Voc and thereby the efficiency η. The mechanisms causing these losses and being responsible for their high extent in organic solar cells are not fully understood yet. A common and frequently used approach is Langevin’s theory of bimolecular and diffusion-limited recombination35, which however often fails to provide an accurate description of experimental behavior36,37,38. Langevin recombination assumes losses independent of the energy gap between the excited and the ground state. However, according to Benduhn et al.39, there is an empirical relation between the non-radiative recombination rate and the energy gap between the excited charge-transfer and the ground state. This relation results if we assume that the diffusion of carriers towards each other (and the donor–acceptor interface) is not the slowest step in recombination. Instead, we have to assume that the dissipation of energy by excitation of molecular vibrations limits the time constant of non-radiative recombination. The non-radiative recombination rate shows an approximately exponential dependence on the energy gap in semiconductors40,41, large organic molecules42, and complexes43. This loss mechanism would then be intrinsic, caused by molecular vibrations of the C–C bonds in organic semiconductors39.

The magnitude of non-radiative voltage losses ΔVnr depends on the internal reorganization energy λ and the energy ECT of the charge-transfer state, which is the energy difference between the ground state and the CT excited state. In addition, it may also depend on the molecular packing and the alignment of donor and acceptor molecules at the donor–acceptor interface44,45. Increasing this charge-transfer state energy reduces non-radiative losses, because these transitions become less likely for larger energy gaps, since more phonons are involved. The non-radiative recombination rate knr is approximately proportional to the so-called Franck–Condon factor Spe−S/p! and is thereby linked to the number p of vibrational modes necessary for the transition46,47. This number, in turn, is given by ECT/ħω, where ħω is the mean energy of a phonon involved in the transition. S is the so-called Huang–Rhys factor, which is associated with the strength of the electron–phonon coupling and which is in the order of 1 in organic semiconductors48,49. The Franck–Condon factor has the mathematical form of a Poisson distribution and has a peak at S = p and decays strongly for the practically relevant case where p ≫ S (typical values of p are on the order 10). The dependence of knr on ECT translates into a dependence of the non-radiative voltage loss ΔVnr on ECT. For this article, we use the empirical relation

$$\Delta V_{\mathrm{nr}} = 0.547\, {\mathrm{V}} - 0.184 \, {\mathrm{VeV}}^{-1} ( {E_{{\mathrm{CT}}}} - \lambda )$$
(3)

introduced in ref.39 to account for energy-gap-dependent non-radiative voltage losses.

The energy ECT of the charge-transfer state corresponds to the energy at the intersection between normalized absorptance aeff(E) and normalized electroluminescence ϕEL(E) of the blend material50. Fig. 2 shows an example of this definition for both Fig. 2a experimentally determined and Fig. 2b simulated spectra. For a red-shifted absorptance and EL emission, also a smaller energy for the charge-transfer state is obtained. Hence, in a blend material consisting of two organic semiconductors, the one with the smaller energy gap decisively determines the amount of non-radiative losses. Furthermore, the internal reorganization energy λ affects the amount of non-radiative losses. As shown in Fig. 2, the distance between the charge-transfer state energy ECT and the maximum of the EL spectrum defines the value of reorganization energy λ, which increases for example with a spectral broadening of the absorption onset.

Fig. 2
figure 2

Definition of the charge-transfer state energy ECT and the internal reorganization energy λ. Comparison of (a) experimental and (b) simulated normalized electroluminescence spectra ϕEL(E) and the external quantum efficiency Qe(E) and absorptance aeff(E), respectively. The energy at the intersection point of the normalized absorptance aeff(E) and the EL emission ϕEL(E) of the blend material defines the energy ECT of charge-transfer state, which is used to calculate the non-radiative voltage losses according to ref39. In addition, the internal reorganization energy λ, defined by the distance between charge-transfer state energy ECT and the maximum of the EL spectrum, is also incorporated into the calculation of the non-radiative losses ∆Vnr. A broadening of the absorption onset further results in a red shift of the EL spectrum, as a result of an increasing energy λ. A red shift of absorptance and EL emission is accompanied by a lower charge-transfer energy ECT, leading to higher non-radiative voltage losses

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If the recombination losses, the absorption coefficient of the blend, and an optical model are defined, we are able to calculate the open-circuit voltage via51,52

$$V_{{\mathrm{oc}}} = \frac{{k_{\mathrm{B}}T}}{q}{\mathrm{ln}}\left( {\frac{{J_{{\mathrm{sc}}}}}{{J_{0,{\mathrm{rad}}}}}} \right) - \Delta V_{{\mathrm{nr}}}\ {\mathrm{for}}\ J_{{\mathrm{sc}}} \gg J_{0,{\mathrm{rad}}}$$
(4)

with the Boltzmann constant kB, the temperature T, the saturation-current density

$$J_{0,{\mathrm{rad}}} = q \,{\int \nolimits_0^\infty} a_{{\mathrm{eff}}}\left( E \right)\phi _{{\mathrm{bb}},\, 300{\mathrm{K}}}\left( E \right){\mathrm{d}}E$$
(5)

in the radiative limit, the short-circuit current density Jsc according to Eq. (2), and ϕbb,300 K(E) the black body spectrum at 300 K. The external light-emitting diode (LED) quantum efficiency QLED is linked to the non-radiative voltage losses by52,53

$$Q_{{\mathrm{LED}}} = {\mathrm{exp}}\left( { - \frac{{q\Delta V_{{\mathrm{nr}}}}}{{k_{\mathrm{B}}T}}} \right).$$
(6)

The current–voltage characteristic of the solar cell is then given by

$$J = \frac{{J_{0,{\mathrm{rad}}}}}{{Q_{{\mathrm{LED}}}}} \cdot \left( {{\mathrm{exp}}\left( {\frac{{qV}}{{k_{\mathrm{B}}T}}} \right) - 1} \right) - J_{{\mathrm{sc}}}.$$
(7)

The electrical power density P = –JV results from the multiplication of total current density J and voltage V. Finally, the efficiency η of the solar cell is calculated by weighting the maximum electrical power density with the incident power density Psun of the sun according to the AM 1.5G spectrum, so η = max(P)/Psun. In the ideal case—which neglects the series resistance from the contacts and assumes an ideal diode—the fill factor (FF) only depends on Voc54 and has values around 90%. We use this ideal FF behavior54 in the first two parts of the paper (subsections “Simulations with ideal collection and absorption properties,” “Simulations with parameterized absorption coefficients”) and thus assume ideal transport properties where all generated charge carriers can be collected. For real n–i–p-type thin film solar cells, the FF strongly depends on the thickness d of the active layer and the electrical transport properties of the absorber materials itself55,56,57,58,59. Therefore, in the subsequent simulations in the last part of this paper (“Simulations with optical interferences and collection losses”), electronic losses in the active layer are included. For organic solar cells with NFA-based blends, the FF is often between 60 and 75%15,18,19,60,61,62,63 even for devices with thin active layers of about 100 nm, while certain fullerene-containing blends exceed 70% FF at thicknesses above 200 nm64,65,66,67. High FFs are enabled by efficient charge carrier collection which is governed by the ratio between how long charge carriers are free before they recombine versus how long it takes them to reach the contacts where they are extracted57,58,59. Therefore, the thickness of the blend layer does not only influence its absorptance aeff but also the FF55, because it determines the average distance that charge carriers need to travel before being extracted at the contacts. Previously55, we have expressed the competition between transport and recombination by the electronic quality factor

$$Q = \frac{{\mu ^2}}{{k^{0.8}}},$$
(8)

where μ is the mobility and k is the coefficient of direct recombination. For undoped devices, the FF is affected by thickness and built-in voltage, which are independent of the absorber material, and the photo-generation rate of charge carriers. After eliminating the influence of these parameters on the FF, the FF only depends on the electronic quality factor Q which is a material-specific quantity. As a consequence, the choice of a certain donor–acceptor combination will not only determine the spectral shape of absorption but also affect the blend’s Q—and therefore the FF of the resulting solar cell.

To reach a realistic description for typical organic solar cell devices in the subsection “Simulations with optical interferences and collection losses,” we will—in addition to considering charge-collection issues—drop the simple Lambert–Beer model for absorption expressed in Eq. (1) by taking interference into account which introduces absorptance maxima at certain thicknesses. We optically simulate a typical n–i–p cell stack composed of glass/ITO (150 nm)/ZnO (40 nm)/PTB7-Th:NFA (1:2)/MoO3 (10 nm)/Ag (100 nm). The thickness d of the active layer is varied. Optical simulations are based on the Transfer Matrix Method68. The resulting generation profiles under AM 1.5G illumination are used as input to numerical drift-diffusion simulations of the charge carrier dynamics with the Advanced Semiconductor Analysis (ASA) software (https://www.tudelft.nl/en/eemcs/the-faculty/departments/electrical-sustainable-energy/photovoltaic-materials-and-devices/software-platform/asa-software/), which generates current–voltage data. The electronic simulation parameters are listed in Supplementary Table 2.

Our main goal in the subsection “Simulations with optical interferences and collection losses” is to illustrate, for a given absorption spectrum, the influence of interference and electronic quality on the thickness dependence of the efficiency. We compare the two cases of congruent and complementary absorption and discuss the implications for the design criteria of organic solar cells from non-fullerene acceptors.

Simulations with ideal collection and absorption properties

In order to illustrate the fundamental impact of non-radiative losses on the efficiency of a solar cell, the first simulations consider a solar cell with ideal collection and absorption properties, characterized by a step-like absorptance

$$a_{\mathrm{step}}\left( E \right) =\left\{\begin{array}{cc} {0,} & E \, < \, E_{\mathrm{g}} \\ {1}, & E \ge E_{\mathrm{g}}\end{array}\right.$$
(9)

Fig. 3 shows the contour plot of the resulting efficiency η as a function of band-gap energy Eg and the amount of non-radiative recombination losses expressed by the external LED quantum efficiency QLED. The external LED quantum efficiency QLED varies between 1 and 10−8, which approximately equates to voltage losses in the range from 0 to 480 mV (≈60 mV per decade). First of all, the contour plot shows that the magnitude of non-radiative losses determines the efficiency limit of the solar cell, since these losses reduce the maximum open-circuit voltage. If non-radiative losses do not occur (QLED = 1), the efficiency η reaches the Shockley–Queisser limit69. But with increasing losses, the efficiency of the solar cell decreases. Besides, non-radiative losses affect the ideal band-gap energy Eg at which the maximum efficiency is achieved for the respective amount of losses. The white line in Fig. 3 highlights this dependence and illustrates that the ideal band-gap shifts with increasing losses to higher energies. The discrete steps of the ideal band-gap curve are not caused by the discretization of the numerical simulation, but by the spectral profile of the solar spectrum (AM 1.5G), which shows abrupt dips due to molecular absorption (e.g., of O2, O3, H2O, and CO2) in the atmosphere.

Fig. 3
figure 3

Simulation with ideal collection and absorption properties. Contour plot of the efficiency η of a solar cell with ideal collection properties and ideal, step-like absorptance astep(E), as a function of its band-gap energy Eg and the external LED quantum efficiency QLED, which is linked by the reciprocity relation with the non-radiative recombination losses. The white line indicates the ideal band gap as a function of QLED showing that the higher the losses, the larger this ideal band gap and the lower the maximum efficiency of the solar cell that can be achieved. The black lines mark equipotential lines at efficiencies of 20, 25, and 30%

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Simulations with parameterized absorption coefficients

In this section, we investigate what the ideal band gaps should be in case the active layer consists of a donor–acceptor blend with two molecules that equally contribute to the total absorptance aeff(E). Within the scope of this study, we use the model and the equations introduced in the subsection “Model” to determine the solar cell efficiency η. For this purpose, we shift the two absorption coefficients α1(E) and α2(E) representing the donor and acceptor of the blend material by varying their inflection-point energies Eip,1 and Eip,2 (Fig. 1b). The corresponding simulation parameters are listed in Supplementary Table 1. For each step of the numerical simulation the effective absorption coefficient αeff(E) is formed. For each of these effective absorption coefficients, we subsequently calculate the efficiency η for a set of different layer thicknesses d. As outlined in the subsection “Model,” we consider this variation as a function of the key parameter αmaxd. The contour plots in Fig. 4a–c represent the dependence of the efficiency η on the inflection-point energies Eip,1 and Eip,2 for three different values of αmaxd. The efficiency values on the main diagonals correspond to a congruent configuration of the two absorption coefficients (Eip,1 = Eip,2). In contrast, the off-diagonal combinations with different inflection-point energies belong to a complementary absorption behavior of donor and acceptor. The comparison of these three contour plots in Fig. 4a–c illustrates that the parameter αmaxd has a decisive influence on which absorption behavior—congruent or complementary—achieves the highest efficiency for a given αmaxd.

Fig. 4
figure 4

Simulations of an organic solar cell with parameterized absorption coefficients and ideal collection. (a–c) Examples of the simulated efficiency η of an organic solar cell for three different value of the key parameter αmaxd as a function of the inflection-point energies Eip,1 and Eip,2 of the absorption coefficients α1(E) and α2(E) describing the donor and acceptor material in the blend. For small values of αmaxd the highest efficiency results for equal energies of Eip,1 and Eip,2, located on the diagonal of the contour plot, which means that a congruent absorption behavior of donor and acceptor is more efficient in comparison to a complementary one. The opposite applies for large values of αmaxd, as shown in (c), since the highest efficiencies occur for off-diagonal, that is, different values of Eip,1 and Eip,2. (d) Ratio of the absolute efficiency maximum ηmax and the maximum efficiency ηmax,dia on the main diagonal as a function of αmaxd. The course highlights the transition from the absolute maximum efficiency for a congruent arrangement of α1(E) and α2(E) toward the maximum for complementary absorption behavior

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For small values of αmaxd, the maximum is on the main diagonal, implying that a congruent arrangement of the absorption coefficients leads to the highest efficiencies (Fig. 4a). In this case, the choice of two complementary absorbing materials is even less favorable.

The efficiency of the solar cell continuously increases with increasing αmaxd for all combinations of Eip,1 and Eip,2, but more strongly for the complementary absorption. Consequently, at a value of αmaxd ≈ 1.6, the complementary arrangement of the absorption coefficients finally reaches a higher efficiency (Fig. 4b). From then on, an efficiency gain is achieved by selecting two absorption coefficients α1(E) and α2(E) with different inflection-point energies for the active blend. Fig. 4d summarizes these trends and shows the ratio of the absolute maximum efficiency ηmax and the maximum efficiency ηmax,dia on the main diagonal as a function of αmaxd.

For a better understanding of the dependence of the maximum efficiency on αmaxd, one should take a look at the shape of the solar spectrum ϕsun (Fig. 5a). Since this spectrum does not provide a constant photon flux over energy, the short-circuit current density Jsc—proportional to the area under the respective curves in Fig. 5a—is strongly affected by the energetic position of the absorptance a(E). Due to the weighting with the solar spectrum, it is even possible to ultimately produce higher short-circuit current densities with a lower overall absorptance. A corresponding example in Fig. 5a, b highlights this fact, where the active layer with a congruent absorptance (Fig. 5b) harvests more photons than the one with the complementary even if its overall absorptance is lower. This example shows that a loss in absorptance at higher photon energies is less relevant because of the reduced photon flux in the solar spectrum for energies above 2.5 eV.

Fig. 5
figure 5

Comparison between different absorption characteristics. Comparison between (a) the product of the AM 1.5G solar spectrum ϕsun and the absorptance a of a congruent (blue) and a complementary (red) absorbing blend layer. The corresponding absorptance a is shown in (b). This example shows that even if the overall absorptance of a congruent arrangement of α1(E) and α2(E) is lower, its short-circuit current density may be higher in the end. The corresponding values of the parameters are Eip,1 = 1.75 eV, Eip,2,con = 1.75 eV, Eip,2,comp = 2.4 eV, and d = 68 nm, the others are the standard values listed in Supplementary Table 1

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Fig. 6a shows the ideal absorptance aeff,ideal of the blend, for which the highest efficiency (Fig. 6b) can be obtained, as a function of αmaxd. For small values of αmaxd, it is more efficient that both organic materials absorb at low energies to exploit the higher photon flux in the low-energy range of the solar spectrum. However, if αmaxd increases and thus the absorptance in the low-energy region becomes high, it is at some point more beneficial to arrange the absorption coefficients of the donor and the acceptor in a complementary manner to cover a wide energy range. The larger the absorption coefficients of the molecules (larger αmax), the smaller the thicknesses needed to reach this transition point. Additionally, in this figure the transition from the maximum efficiency for a congruent arrangement at small values of αmaxd to one for a complementary arrangement at large αmaxd is directly apparent. The white lines mark the corresponding inflection-point energies Eip,1 and Eip,2, which are shown again in Fig. 6c as cyan lines for the sake of clarity. The additional curves in Fig. 6c of the resulting ideal band gaps, when non-radiative losses were kept constant in the simulations, further emphasize that the general trend also holds for constant values of QLED. However, the ideal band gaps, i.e., inflection-point energies Eip,1 and Eip,2 shift with higher non-radiative losses to higher energy.

Fig. 6
figure 6

Beneficial blend parameters resulting from simulations. (a) Absorptance aeff,ideal(E) of the blend layer, which leads to the highest efficiency ηmax at the respective value of αmaxd. The white lines indicate the corresponding inflection-point energies Eip,1 and Eip,2 of the absorption coefficients of the donor and acceptor. (b) The maximum efficiency ηmax, both for energy-gap-dependent, non-radiative losses (cyan) according to Eqs. (3) and (6) and for constant non-radiative losses (red, purple). (c) The corresponding ideal inflection-point energies Eip,ideal

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Simulations with optical interferences and collection losses

So far, the efficiency calculations are based on two simplifications: (i) absorption follows Lambert–Beer and (ii) electronic losses are neglected meaning that all photo-generated charge carriers are extracted and contribute to the current—independent of applied voltage. When the first assumption is dropped by including optical interference in the layer stack, maxima appear in the thickness dependence of absorptance and short-circuit current density (Supplementary Fig. 4a, b). These maxima transfer to the efficiency-thickness relation as can be seen in Fig. 7a, b, which depict the two discussed cases of complementary (PTB7-Th:FBR) and congruent (PTB7-Th:IDTBR) absorption. When the second assumption is dropped by accounting for finite mobilities and recombination coefficients, the FF becomes thickness dependent (Supplementary Fig. 4c, d). The different behavior is parametrized by the electronic quality factor Q in Eq. (8), which is varied along the different curves in Fig. 7a, b. Higher Qs generally result in higher efficiencies, which holds for both studied blend systems. The efficiency increase with higher mobilities and lower recombination coefficients—and thus larger Qs—is also depicted in Fig. 7c, d. Figure 7a, b show that for both cases of complementary and congruent absorption, an active layer thickness around the second (or higher) interference maximum leads to an improved efficiency only for a sufficiently large Q. Interference thus discretize the optimum thickness at the first, second, or third absorption maximum. These jumps in optimum thickness are observed clearly in Fig. 7e, f where the optimum thickness is plotted for a large range of mobilities and recombination coefficients.

Fig. 7
figure 7

Simulation with optical interferences and collection losses. Device performance obtained by drift-diffusion simulations based on experimental optical data of a system with (a, c, e) complementary (PTB7-Th:FBR) and (b, d, f) congruent absorption (PTB7-Th:IDTBR). Optical simulations account for interference which leads to efficiency maxima at certain active layer thicknesses in (a) and (b). Different values of the electronic quality factor Q lead to qualitatively different thickness dependence. High Qs lead to higher gains in maximum efficiency for (a) complementary and (b) congruent absorption. The general increase in efficiency, evaluated at the optimum thickness, with higher Qs—meaning higher mobility and lower recombination coefficient—is further illustrated in (c) and (d). The attainable efficiency varies stronger with Q in the (c) complementary than in the (d) congruent case. For a large range of Qs in (e, f), the depicted optimum active layer thickness only takes discrete values around the interference maxima visible in (a) and (b). A transition to a higher optimum thickness requires lower Q values for (e) complementary than (f) congruent absorption

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While complementary and congruent absorption behave qualitatively similar in many aspects, there are certain notable differences between the two cases. Most importantly, the increase in efficiency from the first to the second maximum—enabled by sufficiently large Qs—is less for the congruent case of IDTBR as can be seen by comparing Fig. 7a, b. The closer spacing of lines with equal efficiency in the case of complementary absorption in Fig. 7c compared to congruent absorption in Fig. 7d express the same effect. The thickness dependence of the efficiency is determined by the gain in absorptance and Jsc versus the loss in FF with increasing thickness. For congruent absorption large part of the incident light is already absorbed in a thin active layer and the achievable gain in absorptance for a thicker layer is low compared to the case of complementary absorption. In the latter case, the absorptance is low over a larger spectral range for a thin device and by increasing the thickness, the absolute value of absorptance increases significantly. Since the drop in FF with thickness and electronic quality is the same for both systems, the attainable gain in absorptance decides if a thicker layer is favorable in terms of device efficiency. For the same reasons not only the attainable gain in efficiency differs between a complementary and a congruent absorbing system but also the value of Q that is necessary to reach an efficiency improvement. For the simulations shown in Fig. 7a, b, a jump to the second interference maximum requires Q ≳ 103 cm1.6 V−2 s−1.2 for the complementary absorbing PTB7-Th:FBR, while the congruent absorbing PTB7-Th:IDTBR requires Q ≳ 104 cm1.6 V−2 s−1.2. Also, for PTB7-Th:IDTBR in Fig. 7f the transition to larger optimum thicknesses is shifted to higher values of k and μ, implying a higher Q, compared to PTB7-Th:FBR in Fig. 7e. Thus, for a blend with complementary absorption the threshold for Q, above which thicker active layers are more efficient, is generally lower.

Discussion

Donor and non-fullerene acceptor molecules absorb only well in a limited spectral energy region of about ~0.5 eV. By combining different donor and non-fullerene acceptor molecules, which contribute roughly equal amounts to the total absorptance, different absorption characteristics of the blend material can be realized, ranging from congruent (Eip,1 = Eip,2) to complementary (Eip,2 ≫ Eip,1). For which of these different combinations the highest efficiency is obtainable depends to a large extent on how the incident photon flux ϕsun provided by the sun is exploited. This photon flux per electron volt of the solar spectrum is much higher in the low-energy range and decreases with increasing energy. Thus, the energetic position of the total absorption a(E) strongly influences the maximum achievable short-circuit current density and finally the solar cell efficiency η.

We have shown that for small layer thicknesses, it is more beneficial to arrange the absorption coefficients of the donor and the acceptor in a congruent manner in order to make the best possible use of the high photon flux ϕsun in the low-energy range. Covering a broad energy range by combining two molecules that absorb complementarily is only more efficient if the absorptance of the active layer in the low-energy range is still high which requires molecules that absorb very strongly or active layers with thicknesses around the second interference maximum or higher. Such thick layers are only feasible if charge-carrier collection at the maximum power point is sufficiently efficient.

Data availability

The data that support the findings of this study are available from the corresponding author on request.

Code availability

The code and data sets generated and analyzed during this study are available from the corresponding author on reasonable request.