Energetics and Kinetics Requirements for Organic Solar Cells to 2 Break the 20% Power Conversion Efficiency Barrier

The thermodynamic limit for the efficiency of solar cells is predominantly defined by the energy bandgap of the used semiconductor. In case of organic solar cells both energetics and kinetics of three different species play role: excitons, charge transfer states and charge separated states. In this work, we clarify the effect of the relative energetics and kinetics of these species on the recombination and generation dynamics. Making use of detailed balance, we develop an analytical framework describing how the intricate interplay between the different species influence the photocurrent generation, the recombination, and the open-circuit voltage in organic solar cells. Furthermore, we clarify the essential requirements for equilibrium between excitons, CT states and charge carriers to occur. Finally, we find that the photovoltaic parameters are not only determined by the relative energy level between the different states but also by the kinetic rate constants. These findings provide vital insights into the operation of state-of-art non-fullerene organic solar cells with low offsets.


Introduction
Recently, organic solar cells based on donor-acceptor (D-A) bulk heterojunctions (BHJs) have seen a drastic increase in the device performance, with power conversion efficiencies (PCEs) currently exceeding 17%, 1,2 with 20% in sight and even 25% predicted. 3 This has sparked a renewed interest in photovoltaic applications based on organic semiconductors. The increased efficiency has been achieved by using strongly absorbing narrow-gap non-fullerene acceptors, 3,4 resulting in better complementary light absorption by both the donor and the acceptor and significantly increased short-circuit current densities ( SC ). Simultaneously, the energetic offset between the donor and the acceptor has been decreased allowing for losses in the open-circuit voltage ( OC ) to be reduced. The charge generation yield and the photovoltage in organic solar cells is ultimately determined by physical processes taking place between excitons (S), interfacial charge-transfer (CT) states, and separated free charge carriers (CS). However, the interplay between these different species continues to be a matter of controversy.
In general, the conversion of strongly bound excitons, formed in neat donor or acceptor phases under illumination, into free charge carriers (producing electricity) is mediated by CT states, constituting bound electron-hole pairs where the electron is in the acceptor phase and the hole in the donor phase. 5 Compared to excitons, CT states are considerably less bound, dissociating into free charge carriers with relatively high quantum efficiencies, although the underpinning mechanism is still debated. [6][7][8][9][10][11][12][13] Similarly, also the recombination of free charge carriers is believed to take place via CT states. While the encounter between free electrons and holes follows a diffusion-limited Langevin-like process, albeit geometrically constricted, r14 every charge encounter is expected to result in a CT state which may subsequently dissociate back into free charge carriers or recombine. [15][16][17] This interrelation has been suggested to induce a mutual equilibrium between CT states and free charge carriers in so-called reduced Langevin systems where CT state dissociation is efficient. 18 Such an equilibrium further implies that the OC in organic solar cells is entirely defined by the energetics and the (radiative and non-radiative) recombination kinetics of CT states. This has indeed been observed in fullerene-acceptor-based BHJs, characterized by a large energy offset between excitons and CT states. [18][19][20][21] Whether a similar situation applies for non-fullerene systems displaying low energy offsets between excitons (in the acceptor) and CT states has remained unclear. Recent studies of the radiation efficiency in state-of-the-art non-fullerene solar cells found these systems to possess low non-radiative losses. 22,23 This has been attributed to the formation of an equilibrium between excitons in the acceptor and CT states, causing a repopulation of the presumably more radiative excitons. [23][24][25][26] It has also been pointed out that the small offset between excitons and CT states might compromise the charge generation yield, as the driving force for exciton dissociation is reduced. 27 On the other hand, it has been suggested that if the exciton lifetime is long enough, even in case of low charge transfer rates associated with a small exciton-CT offset, an efficient charge transfer can still occur. 23 However, comprehensive analytical treatments relating the relevant rates and energetics to key performance parameters, such as the OC , in low-offset systems are still lacking. Moreover, the conditions for when an equilibrium between excitons, CT states and free charge carriers can be established, and how this equilibrium affect the overall device performance, have thus far remained elusive.
In this work, the interplay between free charge carriers, CT states and excitons, and its relation to the device performance in low-offset organic solar cells, is investigated. Based on detailed balance considerations, we derive analytical expressions relating the charge generation yield, the charge-carrier recombination coefficient, and the open-circuit voltage to the relative energetics and kinetics between the different species. Furthermore, we clarify the conditions for when mutual equilibrium between the different states can occur. Finally, we demonstrate how the relative energetics and kinetics critically determine the overall power conversion efficiency in these solar cells, suggesting that a PCEs above 20% may be obtained at energetic offsets of 0.1 eV.

Theory
We consider a BHJ device with an active layer of thickness , where the (lowest) exciton energy of the donor is much higher than both the (lowest) exciton energy of the acceptor and the CT state energy CT , such that the back transfer from CT states to donor excitons is negligible. A schematic state diagram showing the relevant rates between the different 4 species is depicted in Figure 1. Here, triplet states are deliberately omitted to simplify the analysis. 16 Furthermore, the effect of trap-assisted charge-carrier recombination is assumed to be negligible. 28 Under steady-state illumination conditions, the kinetics between excitons (in the acceptor), CT states and separate free charge carriers are described by the following rate equations: where is the elementary charge and is the time. The density of separate charge carriers is is the quantum efficiency for excitons in the donor to dissociate into CT states; is the electron transfer rate to the acceptor, * the recombination rate for excitons in the donor, and FRET the exciton FRET rate to the acceptor. Once generated via any of the above-mentioned routes, CT states either recombine directly to the ground state with the rate constant , undergo a transfer back into excitons with the rate constant , or dissociate into free charge carriers with the rate constant . The CTstate dissociation efficiency into free charge carriers, describing the ratio of CT states that do not recombine, is obtained as where the sum ̃C T = (A) + * (D) + CT represents the effective generation rate of CT states from photons. We note that the contribution to ̃C T from direct CT state excitations ( CT ) is generally negligible under 1-sun illumination conditions.

Detailed balance
In the following we assume that excitons in the acceptor, CT states, and free electron-hole pairs are relaxed in such a way that each species are in equilibrium with themselves (i.e. can be described by a separate chemical potential). Under these conditions, the corresponding densities can be generally expressed as 30 where is the Boltzmann constant and is the temperature. Here CS represents the quasi-Fermi level splitting of separate electrons and holes, CT is the chemical potential of CT states, while is the chemical potential of the excitons in the acceptor. Furthermore CS , CT and is the associated (effective) density of available states for free charge carriers, CT states, and excitons in the acceptor, respectively. Finally, CS is the effective D-A energy gap for free electrons and holes.
At thermal equilibrium (i.e., in the dark at zero bias), defined by = CT = CS = 0, no net currents of electrons and holes are present and = surf = 0 applies. Under these conditions, detailed balance dictates all transitions to be exactly balanced by their respective inverse processes. Accordingly, we expect the CT state dissociation rate to be exactly balanced by the charge-carrier encounter rate ( CT,eq = 0 CS,eq 2 ), while the exciton-to-CT rate is balanced by CT-to-exciton back transfer rate (i.e. ℎ ,eq = CT,eq and Δ eff (D) = 0); hence, in conjunction with Eq. (5) we find: where Δ CT/CS ≡ CS − CT and noting that (1 − ) ℎ ≡ . Here, Δ CT/CS corresponds to the (effective) CT binding energy (which generally includes a possible electric field dependence). [31][32][33] Based on Equations (6) and (7), Eq. (3) can be then re-expressed as being only governed by recombination rate constants and relative energetics of the different states.
A similar detailed balance is also expected for transitions between the ground state and the CT (exciton) states at thermal equilibrium: CT,eq = CT,eq ( ,eq (A) = ,eq ), where CT,eq ( ,eq (A) ) is the thermal generation rate of CT states (excitons). Note that, for excitons, we have assumed that the exciton energy of the donor is considerably higher than that of the acceptor such that exciton generation rate in the donor at thermal equilibrium, ,eq (D) , is negligibly small, , allowing for the rate constants and to be determined. Here, ,CT and , is the quantum efficiencies for an incoming photon to excite a CT state and an exciton (either in donor or acceptor), respectively, which generally include optical interference effects as well. 35 Finally, EQE EL,CT (EQE EL,S ) is the electroluminescence quantum efficiency of CT states (excitons), whereas BB ( ) ≈ is the black-body photon spectrum of the environment (ℎ is the Planck constant and is the speed of light).
It should be emphasized that the above detailed balance analysis generally only applies when free charge carriers, CT states, and excitons in the acceptor are relaxed [Eq. (5)]. If this condition is not met under steady-state operation (hot states), then the related rate constants may differ from the ones at thermal equilibrium.

Steady-state operating conditions
Under illumination, we generally have CT ≫ CT,eq and (A) ≫ ,eq (A) . Provided that excitons in the acceptor, CT states, and free charge carriers remain relaxed under steady-state conditions, simplified expressions for the current density can be derived. Based on the above detailed balance considerations, the concomitant steady-state current density of charge carriers (Eq. (4)) is obtained as where gen = CT̃CT is the generation current density in the active layer. The second term on the right-hand side of Eq. (9) represents the bulk recombination current density between free charge carriers, with being the associated dark saturation current density.
In the dark, the (thermal) generation current density is governed by its thermal equilibrium value gen = 0 , while CS ≈ , where is the applied voltage. Assuming surface recombination to be absent, the dark current density then takes the familiar form dark = 0 [exp( ⁄ ) − 1]. Under illumination (or high injection levels), however, a deviation between the applied voltage and the quasi-Fermi level splitting of charge carriers in the bulk is generally expected for ≠ 0 in low-mobility (transport-limited) systems, such as organic solar cells, resulting in CS ≠ . 36,37 The accompanying recombination rates in the bulk, 9 taking place either via CT states at the D-A interface ( CT = CT ) or via excitons in the acceptor ( = ), are given in the Supporting Information.

Results and Discussion
To demonstrate the physical meaning and how the energetics and kinetics between excitons (in the acceptor), CT states and free charge carriers collectively determine the device performance of organic solar cells, we next conduct analytical simulations based on the developed theoretical framework. In the following, we consider the case with ideal contacts

Photocurrent generation of free charge carriers
The generation current density gen in Eq. .
We note that the charge generation yield is equal to the internal quantum efficiency, assuming a charge collection efficiency of 100% (at short-circuit and low light intensities).
In Figure 2a, the corresponding CGY is shown as a function of exciton-CT offset − CT at different CT-state binding energies Δ CT/CS . The CGY generally follows a sigmoidal shape with the offset. We note that this is consistent with previous experimental observations. 23

Bimolecular recombination between free charge carriers
To avoid charge collection losses in the Fill Factor (FF) and the SC , a minimal charge-carrier recombination current is desired. The associated effective bimolecular recombination rate of free charge carriers in the bulk is given by CS where CT ≡ 0 ⁄ = (

The open-circuit voltage and associated photovoltage losses
While increasing the energy offset between excitons and CT states results in improved charge generation yields and lower , this is however also expected to increase photovoltage (or OC ) losses. Provided that surface recombination remains negligible ( surf = 0), we expect CS = OC to apply at open-circuit ( = 0). Combining Eq. (9) and (10), we then find the following relation for the open-circuit voltage: where ̃C T = (A) + * (D) + CT . The OC critically depends on the interplay between excitons and CT states. Paradoxically, however, the OC is ultimately independent of the energy and kinetics of free charge carriers. This is a direct consequence of the fact that the charge-carrier encounter rate is in this case exactly balanced by the dissociation rate of CT states, CT = 0 CS 2 , resulting in a mutual equilibrium between free charge carriers and CT states. Indeed, further making use of Eq. (5) and (6), we find CT = CS , independent of CT .
Hence, at open-circuit, free charge carriers and CT states are always in equilibrium with each other.
We note that, despite of the existing mutual equilibrium at open-circuit, the corresponding recombination rates of CT states, CT = CT , and of free charge carriers (to the ground state), CS = CS 2 , are generally not the same. In fact, in accordance with CT = 0 CS 2 , it follows that CS = CT ( CT + ′ CT ). Hence, even in the limit ′ ≪ , we still have CS = CT CT ; this is because there will always be a fraction 1 − CT of the generated CT states that recombine directly to the ground state without ever forming free charge carriers.
Note that for ′ ≫ , in turn, most CT states formed upon charge-carrier encounter undergo back-transfer and recombine as excitons. Finally, it is important to emphasize that the equilibrium between free charge carriers and CT state is only maintained as long as CT = 0 CS 2 . When surf ≠ 0 (or ≠ 0), however, this condition no longer applies, and the equilibrium between CT states and free charge carriers is subsequently disrupted ( CT ≠ CS ).
Similarly, this condition is also violated in case of a considerable additional generation channel for charge carriers directly from hot excitons (effectively acting as a non-zero surf < 0 at open-circuit).
In this regime, the recombination of free charge carriers is predominately taking place via excitons in the acceptor. Concomitantly, the OC is limited by the recombination and energy of excitons (in the acceptor), becoming independent of the energetics and kinetics of CT states altogether.
At small offsets, a mutual equilibrium between excitons in the acceptor, CT states, and free charge carriers is established as → CT = CS . In general, the relationship between the chemical potentials of excitons in the acceptor and CT states at open-circuit is given by where = (A)̃C T ⁄ . Furthermore, in accordance with the above, we have CT = CS = OC (assuming surf = 0). The corresponding is shown by dashed lines in Fig. 3 for the case = 1. As expected, the difference between and OC (= CT ) indeed approaches zero as the offset between excitons and CT states is reduced in this case. For a given exciton energy, this regime corresponds to the maximum chemical potential and open-circuit voltage that can be extracted from the cell. We note that for the case when CT state generation is dominated by electron transfer from the donor, ≪ 1, a deviation between and CT is always present, independent of offset. However, provided that is close to unity, this deviation is generally negligible at small offsets with excitons in the acceptor and CT states effectively being in equilibrium with each other ( ≈ CT ). 15 The requirement for an (effective) equilibrium between excitons in the acceptor, CT states and free charge carriers to be established (at open-circuit) is given by − CT < Δ eq , where ] is the related critical offset (assuming surf = 0). As a result, at large enough offsets, when − CT > Δ eq , the excitons and CT states are no longer in equilibrium with each other, and a considerable chemical potential difference is formed between these states. In this regime, the associated photovoltage loss is proportional to the offset and takes the form − CT → − CT − Δ eq . Note that Δ eq depends on the ratio between the CT recombination constant and the hole transfer rate constant ℎ , and generally increases with decreasing

Implications for the optimum power conversion efficiency
Our findings are consistent with the notion that a reduction of the offset generally correlates with a concomitantly reduced photovoltage loss, resulting in an increased OC . This is, however, counterbalanced by a corresponding reduction of the charge generation yield and increase of the charge-carrier recombination coefficient, generally causing losses in both the SC and the FF. The maximum PCE is subsequently determined by a trade-off between these two competing aspects.
To explicitly relate the generation current gen (= CS ) and the charge-carrier recombination coefficient to SC and FF, a relationship between the applied voltage and the quasi-Fermi level splitting CS is needed in Eq. (9). In principle, this requires numerical drift-diffusion simulations using CS , , and the mobilities as input parameters. However, to maintain analytical tractability, yet qualitatively account for charge collection losses, we instead use an approximative analytical model developed by Neher et al. 37 Based on this socalled modified Shockey diode model, the quasi-Fermi level splitting can be simplified as ), allowing for SC and FF to be approximated. 37 Figure 4 shows the corresponding PCE under 1-sun illumination as a function of the energy offset between excitons and CT states for the system under consideration. As shown, at smaller CT binding energies Δ CT/CS (Figure 4a), a lower energy offset between excitons and CT states can be afforded, simultaneously resulting in higher PCE values. We note that since the OC is independent of Δ CT/CS , the different curves in Figure 4a share the same OC (given by green line in Figure 3). Subsequently, the PCE reduction obtained with increasing Δ CT/CS is mainly caused by an increasingly inefficient charge generation yield (reduced ), in accordance with Figure 2a. Similarly, reducing (and thus ′ ) shifts the PCE maximum towards lower − CT allowing for higher PCE values to be reached. Conversely, by decreasing (Figure 4b), the PCE maximum is instead shifted towards higher energy offsets.
Hence, the position of the maximum PCE is strongly sensitive to which parameter is reduced. The exact values of the exciton and CT state rate constants are directly related to their respective radiation efficiency (or electroluminescence quantum efficiency) and the related non-radiative (NR) OC losses within the active layer. For CT states, it has been suggested that these NR losses generally increases with decreasing CT state energy CT , in accordance with the energy-gap law. 20,21 As an increased NR loss for excitons and CT states, respectively, is expected to directly translate into an increased and , this is not only expected to result in a lower open-circuit voltage, but also in a decreased charge generation yield and increased bimolecular recombination coefficient. Hence, NR losses is expected to influence the overall device performance as well as the position of the optimum energy offset.
To demonstrate this effect, we calculate the predicted PCE versus the optical gap, noting that the optical gap is represented by the exciton energy of the acceptor in this case. is a function describing the relationship between the energy and the radiation efficiency of CT states. For simplicity, we assume a similar energy dependence for the radiation efficiency of excitons (but with CT replaced by ), i.e. EQE EL,S = EL ( ).
The predicted PCE at different energy offsets is shown in Figure 5a, assuming EL ( CT ) to be given by the minimum NR loss model (for CT states) proposed by Nelson and co-workers. 21 The corresponding maximum PCE (closed symbols) is shown as a function of the exciton-CT energy offset at different binding energies Δ CT/CS in Figure 5b. Thus, by accounting for the energy gap dependence of and , the optimum PCE is obtained at ≈ 1.5 eV (corresponding to ≈ 10 7 s -1 and ≈ 4 × 10 7 s -1 ) for an energy offset below 0.1 eV, depending on the CT binding energy. For comparison, we also included the case with EL ( CT ) given by the lower empirical limit reported by Vandewal and co-workers 20 (as indicated by open symbols), representing the NR loss of current state-of-the-art organic solar cells.
Because of the larger and in this case, a correspondingly larger PCE loss is obtained.
Nevertheless, at low binding energies, PCEs between 15-20% can be expected for offsets around 0.10-0.15 eV in this case. To enhance the PCE it is therefore crucial to reduce NR losses. An open question in this regard is whether similar NR models apply for both CT states and excitons. For example, it has been suggested that strong electronic coupling between exciton and CT states in low-offset systems cause exciton-CT-state hybridization, leading to suppressed NR recombination. 22 Answering this question is of particular importance for lowoffset systems dominated by excitons in the acceptor ( ′ ≫ ), where minimizing the nonradiative recombination of excitons may prove to be key. Independent of the values of and , however, it is clear that a small Δ CT/CS is desired to ensure maximum device performance.

Conclusions
In conclusion, we have derived an analytical framework, based on detailed balance, to clarify how the interplay between excitons, CT states, and free charge carriers influences charge generation and recombination in organic solar cells. Based on this framework, we further investigate the effect of the relative energetics and kinetics between these species on the charge generation yield, the charge-carrier recombination coefficient, and the open-circuit voltage in organic solar cell systems exhibiting small energy offsets between excitons and CT states. We find that the overall device performance is strongly dependent on the relative energetics, but critically depends on the kinetic parameters as well. Our findings highlight the importance of slow exciton recombination in low-offset systems, predicting an optimal PCE at energetic offsets of around 0.1 eV.