Nonequilibrium site distribution governs charge-transfer electroluminescence at disordered organic heterointerfaces

Significance

Semiconducting polymers and small molecules have promising applications in organic optoelectronic devices, such as solar cells, which typically consist of a disordered mixture of donor and acceptor materials. The performance of these devices is determined by the properties of the donor/acceptor interface. While the donor/acceptor interface is often studied by operating the solar cell device as a light-emitting diode, resulting in charge-transfer electroluminescence, the currently prevailing description of electroluminescence is not quantitative and is often inconsistent with experiments. We present an experimentally verified and quantitative model of charge-transfer electroluminescence at donor/acceptor interfaces that reconciles the inconsistencies present in the literature. Our model simultaneously and quantitatively accounts for energetic disorder and molecular vibrations governing charge transport and luminescence in organic semiconductors.

Abstract

The interface between electron-donating (D) and electron-accepting (A) materials in organic photovoltaic (OPV) devices is commonly probed by charge-transfer (CT) electroluminescence (EL) measurements to estimate the CT energy, which critically relates to device open-circuit voltage. It is generally assumed that during CT-EL injected charges recombine at close-to-equilibrium energies in their respective density of states (DOS). Here, we explicitly quantify that CT-EL instead originates from higher-energy DOS site distributions significantly above DOS equilibrium energies. To demonstrate this, we have developed a quantitative and experimentally calibrated model for CT-EL at organic D/A heterointerfaces, which simultaneously accounts for the charge transport physics in an energetically disordered DOS and the Franck–Condon broadening. The 0–0 CT-EL transition lineshape is numerically calculated using measured energetic disorder values as input to 3-dimensional kinetic Monte Carlo simulations. We account for vibrational CT-EL overtones by selectively measuring the dominant vibrational phonon-mode energy governing CT luminescence at the D/A interface using fluorescence line-narrowing spectroscopy. Our model numerically reproduces the measured CT-EL spectra and their bias dependence and reveals the higher-lying manifold of DOS sites responsible for CT-EL. Lowest-energy CT states are situated ∼180 to 570 meV below the 0–0 CT-EL transition, enabling photogenerated carrier thermalization to these low-lying DOS sites when the OPV device is operated as a solar cell rather than as a light-emitting diode. Nonequilibrium site distribution rationalizes the experimentally observed weak current-density dependence of CT-EL and poses fundamental questions on reciprocity relations relating light emission to photovoltaic action and regarding minimal attainable photovoltaic energy conversion losses in OPV devices.

Organic semiconductors show great promise for use in low-cost optoelectronic devices, such as photodetectors (1) and solar cells (2–4). In these devices the photoactive layer typically consists of a disordered bulk heterojunction (BHJ) mixture of an electron donor (D) and an electron acceptor (A). The properties of the D/A interface determine the overall performance of the optoelectronic device (5) since charge carrier generation and recombination events occur at this interface. For example, the D/A interface governs the open-circuit voltage (V_OC) and the power conversion efficiency of organic photovoltaic (OPV) devices (6, 7) which, despite their appealing features, remain outperformed by other photovoltaic technologies (8). Thorough understanding of the processes taking place at the D/A interface is critical for the development of next generations of organic optoelectronics.

The D/A interface can be conveniently probed by operating the OPV device as a light-emitting diode (LED), where the charge carriers are injected from the electrodes and, after traversing the photoactive layer, recombine at the D/A interface, giving rise to charge-transfer electroluminescence (CT-EL) (9) (Fig. 1A). CT-EL measurements are particularly well suited for the study of organic heterointerfaces since equipment requirements are rather minimal and the measurements can be carried out relatively quickly. Nevertheless, despite the ubiquitous use of CT-EL in the field, the currently prevailing description of CT-EL is not quantitative and is often inconsistent with experiments as well as organic semiconductor physics, as detailed below.

Fig. 1.

CT-EL at disordered organic heterointerfaces. (A) CT-EL spectra are typically only weakly current-density J dependent, as shown for MDMO-PPV:PC₆₁BM (1:4). (B) Schematic showing the higher-energy DOS sites sampled by the recombining carriers during the CT-EL experiment. CT-EL from adjacent lowest-energy DOS sites (dashed circles) is unlikely since such CT states are extremely rare. (C) Spatial DOS site distribution over a fraction (20 × 20 × 20 nm³) of a simulated volume (90 × 90 × 146 nm³) obtained by 3D kMC for MDMO-PPV:PC₆₁BM (1:4). Sites (light gray) with close-to-equilibrium energies (σ²/kT ± σ below DOS center) are highlighted by red and blue open circles (donor HOMO and acceptor LUMO, respectively). (D) Higher-energy sites relevant for charge transport and recombination (DOS site energies higher than σ²/kT + σ below DOS center; see the corresponding DOS schematic).

The CT-EL spectrum contains valuable information regarding the energy of the recombining carriers in their respective disorder-broadened density of states (DOS), as schematically illustrated in Figs. 1B and 2. The corresponding CT-EL peak position is commonly used as reference to estimate the CT energy (E_CT) (7, 9), which correlates with important OPV device parameters, such as device V_OC (6, 7). The CT energy is also used as a reference point for evaluating V_OC and photovoltaic energy conversion losses (10). CT-EL is commonly assumed to originate from charge carriers recombining from the “relaxed and lowest energy” CT state (11), where energetic disorder in the CT state is either ignored or assumed to lead to a manifold of CT states with an energy-independent optical transition matrix element (12) (see SI Appendix, Note S1 for an explanation why this assumption is not necessarily valid in disordered organic semiconductors). For brevity we will refer to this scenario as recombination from “relaxed,” “equilibrium,” or “lowest-energy” CT states in the DOS manifold as schematically shown in Fig. 2 C, Left.

Fig. 2.

CT-EL is simultaneously governed by DOS disorder, kinetic effects, and the Franck–Condon principle. (A) Nonequilibrium case accounting for kinetic and energetic disorder effects. (Left) Schematic shows the higher-lying DOS site distributions (shaded red) governing CT-EL at disordered D/A interfaces. The higher-lying DOS site distributions are situated above their respective DOS equilibrium energies σ²/kT (marked by the horizontal dashed line) and govern the 0–0 lineshape broadening and 0–0 peak emission energy at E_{CT-EL 0–0}. The central CT energy E_CT centr marks |E_HOMO D − E_LUMO A|. (Right) The Franck–Condon principle schematic, describing the vibrational overtones m = {0, 1, …, M − 1} spaced by the phonon-mode energy ħw_ph governing CT luminescence. The blurring of the free energy surfaces illustrates the influence of the energetic disorder; the vibronic levels will broaden correspondingly (not shown for clarity). (B) Schematically shows how kinetic and energetic disorder effects (A, Left) and the Franck–Condon principle (A, Right) contribute to the final CT-EL spectrum. (C) Conventional CT-EL picture assuming emission from equilibrium DOS sites and not accounting for kinetic effects. (Left) Schematic shows the quasi-equilibrium DOS site distributions (shaded blue) situated at σ²/kT below their respective DOS centers. In this case, which is not consistent with our experiments as described in the main text, the 0–0 lineshape emission would occur at E_{CT Eq 0–0}. (Right) The conventional Franck–Condon principle schematic ignoring the influence of energetic disorder. (D) Schematically shows the 0–0 CT-EL peak energetic position with respect to the joint electron and hole DOS (dashed black trace) in the case of thermal site distribution (blue, C) and nonthermal site distribution (red, A) governing CT-EL. Note that all of the above energy values must be corrected by the CT exciton binding energy E_CT bind following Eqs. 2 and 3.

However, adjacent equilibrium-energy DOS sites are extremely rare—most low-energy sites are spatially separated as schematically shown in Fig. 1B (dashed circles) and explicitly calculated using 3-dimensional (3D) kinetic Monte Carlo (kMC) for MDMO-PPV:PC₆₁BM in Fig. 1C. In the 3D kMC simulations we have assumed uncorrelated DOS site energies, as experimentally confirmed in earlier work for the systems investigated here (13), which results in most low-energy D/A sites being spatially separated by distances larger than 5 to 10 nm (see SI Appendix, Fig. S1 for the radial distribution function). This renders CT-EL from “lowest-energy” CT states extremely unlikely. Having both the electron and hole at their DOS equilibrium energy is also inconsistent with the notion that at least one of the carriers must be mobile to meet its counterpart—for CT-EL to take place at least one of the carriers must be (thermally) excited toward the transport level (14, 15) (Fig. 1B). However, quantitatively accounting for such kinetic effects while fitting CT-EL experiments has not yet been demonstrated.

In case of equilibrium-site occupation (Fig. 2C), the CT-EL spectrum would merely reflect a convolution of equilibrium electron and hole populations in the DOS. In this case, the CT-EL spectra would also be expected to strongly depend on current density due to the DOS state-filling prevalent in organic semiconductors (14), but this is generally not the case—CT-EL spectra are typically only weakly (16), if at all, current-density-dependent (Fig. 1A). Instead, the absence of state-filling effects could be rationalized by nonequilibrium DOS site occupation (Fig. 2A). Since considerably more DOS sites are accessible for transport at higher energies, as shown in Fig. 1D, carrier-density effects would be significantly diminished (17), rationalizing the typically weak current-density dependence of the CT-EL spectra (Fig. 1A). The hypothesis that CT-EL originates from higher-lying DOS sites (higher-energy CT states), as opposed to the widespread notion that CT-EL occurs at the “relaxed and lowest energy” CT states, was first proposed in ref. (16) on the basis of semiquantitative analysis of the weak current-density dependence of CT-EL spectra. However, this analysis relied critically on the assumption of an exponential DOS and does not generalize to other DOS shapes, such as a Gaussian DOS, as described below.

In an exponential DOS, state-filling effects are prevalent at all carrier densities (18), necessitating the use of higher-lying DOS sites to rationalize the weak current-density dependence of CT-EL spectra. In contrast, state-filling effects are absent in a Gaussian DOS in the Boltzmann limit, that is, at similarly low carrier densities that are typically encountered in well-functioning OPV devices, as shown in SI Appendix, Fig. S2. Hence, the experimentally observed weak bias dependence of CT-EL could instead be explained using equilibrium-site occupation in a Gaussian DOS, without needing to account for higher-lying DOS sites as proposed in ref. 16 (see SI Appendix, Fig. S2 and Note S2 for further explanation). As such, there currently is no consensus or general quantitative proof whether CT-EL at organic heterointerfaces originates from lower- or higher-lying DOS sites in any reasonable DOS shape relevant to organic semiconductors. This inconsistency remains unresolved and questions both the ubiquitous use of E_CT as a reference energy as well as the interpretation of the CT-EL spectra.

In addition to the ill-defined E_CT, the spectral broadening of CT-EL, which is expected to simultaneously depend on the energetic disorder of the constituent materials (16) (inhomogeneous broadening; Fig. 2 A and C, Left) and on the molecular vibrations governing emission via the Franck–Condon principle (homogeneous broadening; Fig. 2 A and C, Right) (19, 20), is not fully understood. For example, CT-EL spectral broadening is material-specific and could be used to rapidly compare the D/A interface of various blends, as shown in Fig. 3A. Although the energetic disorder contribution to CT-EL spectral broadening has been suggested to be both negligible (20) and, recently, to be significant (21), explicit quantification of the relative contributions due to energetic disorder versus Franck–Condon broadening remains elusive due to the absence of a quantitative CT-EL model—the reasons for the differences between CT-EL spectra as in Fig. 3A are not entirely clear. As such, despite the widespread use of CT-EL for OPV device characterization, the lack of a consistent framework and the resulting lack of quantitative understanding limit its versatility.

Fig. 3.

CT-EL spectral broadening. (A) CT-EL peak broadening σ_CT-EL is material-dependent. The spectra have been centered at 1.2 eV for comparison and are compared at a current density of J = 200 mA cm⁻². (B) Comparison of experimental CT-EL peak broadening (filled squares) and 0–0 transition lineshape broadening estimated by DOS convolution σ_SCLC 0–0 (open circles) for different donor polymers blended with either PC₆₁BM or PC₇₁BM. The actual DOS-induced broadening of the 0–0 transition is σ_{3D kMC 0–0} ∼70 to 80 meV (filled circles), as calculated by 3D kMC using energetic disorder values measured by T-dependent SCLC.

Here, we utilize current-density-dependent CT-EL experiments, fluorescence line narrowing (FLN) spectroscopy (22, 23), and temperature-dependent charge transport measurements in combination with 3D kMC simulations to demonstrate a quantitative model for CT-EL at disordered organic heterointerfaces. Our model quantitatively and simultaneously accounts for charge-transport physics in an energetically disordered DOS and Franck–Condon broadening. As input to the model we use experimentally determined D/A blend parameters, such as Gaussian DOS energetic disorder (spanning the range σ ∼60 to 100 meV), as measured by temperature-dependent space-charge limited currents (SCLC) and phonon-mode energies (ħw_ph D/A ∼120 to 145 meV) governing CT luminescence, as selectively/directly measured by FLN spectroscopy at cryogenic temperatures (7 K). The experimentally calibrated model enables us to elucidate the commonly observed CT-EL features: emission peak position in energy, spectral broadening, and weak current-density dependence in terms of material DOS, which is important for the understanding of OPV device physics and for the definition of E_CT. Complementary optical modeling confirms that the measured CT-EL bias dependence is a D/A materials property and not an optical cavity effect. We explicitly quantify that CT-EL is governed by a nonequilibrium distribution of DOS sites which, depending on the energetic disorder, are situated ∼50 to 320 meV above equilibrium donor HOMO or acceptor LUMO DOS sites (highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively), as schematically shown in Fig. 2A. For the resulting CT-EL emission, we find that the lowest-energy CT states are situated ∼180 to 570 meV below the 0–0 CT-EL transition—CT-EL does not originate from the lowest energies in the total manifold of available DOS sites.

The quantitative and experimentally calibrated analysis of CT-EL distinguishes our model from prior work (16, 21), where energetic disorder and vibronic progression effects were either discussed in isolation or only semiquantitatively accounted for, while kinetic effects governing charge transport to the recombination sites as well as the likelihood of CT emission at those DOS sites during CT-EL remained unquantified. In contrast, we use experimentally measured energetic disorder and CT phonon-mode energy values, while simultaneously accounting for kinetic charge transport/recombination effects using 3D kMC simulations, which is not possible using earlier employed analytical and drift-diffusion based models. The major distinguishing feature of our CT-EL model is not in the separate experimental validation of the above features but in the fact that our model not only simultaneously and quantitatively accounts for kinetic, energetic, and vibronic effects but also successfully fits bias-dependent CT-EL experiments. It is this combination that allows to provide a quantitative understanding of CT-EL.

Results and Discussion

Energetic Disorder of D and A Materials.

To estimate the energetic position of the recombining carriers in their respective DOS during CT-EL, the DOS energetic disorder σ of the constituent D/A materials must be known. We have performed temperature (T)-dependent SCLC transport measurements (200 to 300 K) on 6 different D/A blends and have extracted their σ for both electrons and holes following the Gaussian disorder model (GDM) (15), which has been effectively used to explain charge motion in a large variety of organic semiconductors in the past (13, 24) (see SI Appendix, Figs. S3 and S4 for SCLC data). We have assumed uncorrelated Gaussian DOS site energies, as experimentally confirmed in earlier work (13). For the materials studied here, the extracted uncorrelated Gaussian DOS disorder values for the donor HOMO and acceptor LUMO, σ_HOMO D and σ_LUMO A, respectively, span the typical range encountered in organic semiconductors σ ∼60 to 100 meV (SI Appendix, Table S1). While the CT state disorder σ_CT at the D/A interface governing CT-EL may differ from the DOS site disorder sampled by SCLC measurements, σ_CT estimates in the literature span a comparable σ_CT ∼60 to 100 meV range (12) and would otherwise not affect the conclusions of this work (see SI Appendix, Note S2 for details).

Upper Limit for Inhomogeneous 0–0 Lineshape Broadening.

To show that the energetic disorder of the constituent materials does not suffice to explain the CT-EL spectral broadening, we quantify the upper limit for inhomogeneous CT-EL broadening via the convolution of the hole and electron Gaussian DOS, which equals the geometric mean of the experimentally determined σ_HOMO D and σ_LUMO A. Fig. 3B shows that the experimentally observed CT-EL broadening σ_CT-EL, obtained by Gaussian fitting to CT-EL measurements in Fig. 3A, is larger than CT-EL broadening estimated by DOS convolution σ_SCLC 0–0 (Fig. 3B, open circles). The lack of correlation between experimental CT-EL spectral broadening and DOS energetic disorder becomes apparent when comparing materials with overall similar energetic disorder but different CT-EL broadening, for example rr-P3HT:PC₆₁BM and MDMO-PPV:PC₆₁BM in Fig. 3B.

The actual disorder contribution to CT-EL broadening is even less than estimated above, since the distribution of spatially neighboring donor HOMO and acceptor LUMO DOS sites contributing to CT emission is much narrower than assumed by DOS convolution. This is due to kinetic effects governing the likelihood of charge carriers meeting at specific DOS sites, as well as the likelihood of CT emission at those DOS sites. To explicitly account for such kinetic effects, we have performed numerical 3D kMC simulations of the 0–0 lineshape, which are described in detail below. These simulations confirm that the resulting 0–0 lineshape broadening (Fig. 3B, filled circles) is indeed less than obtained by DOS convolution (Fig. 3B, open circles).

Three-Dimensional kMC Simulations of the 0–0 Lineshape.

We explicitly calculate the energetic position of the recombining carriers in their respective DOS during the CT-EL experiment by numerical OPV device simulations using 3D kMC based on the extended GDM (25), which describes nearest-neighbor charge hopping in an energetically disordered Gaussian DOS (we also consider an exponential DOS; see SI Appendix, Fig. S5). It is an extension of a previously described model, shown to successfully describe photoinduced charge separation (26) and transient picosecond–microsecond carrier dynamics for polymer:fullerene (27–30), polymer:polymer (30), and small-molecule:fullerene (31) blends in illuminated OPV devices, now used to model OPV devices operating as LEDs.

Briefly, our model accounts for charge injection/extraction at ohmic contacts, charge transport, electric field, and state-filling effects, all Coulomb interactions including the contribution of the ∼220-meV CT binding energy on CT-EL spectra, image charges in metal electrodes, and electron–hole recombination. The ∼220-meV CT binding energy estimate corresponds to an intersite D/A distance of 1.8 nm where CT recombination occurs, but the electron–hole interaction is distance-dependent as numerically calculated by the model. Charges forming a CT complex do not necessarily recombine and can meet/separate multiple times during their lifetime under favorable conditions. This implicitly accounts for the experimentally observed CT state diffusion by correlated electron and hole hopping motion (32). While extending our model to quantitatively account for the quantum yield of CT luminescence is also possible and important for understanding energy losses in optoelectronic devices, this is beyond the scope of this work.

As in our previous work, we use the simplest treatment with uncorrelated electron/hole Gaussian DOS sites, which was independently confirmed for the investigated D/A blends by T-dependent SCLC measurements in earlier work (13). To demonstrate model generality, we have also performed 3D kMC simulations assuming an exponential DOS. See SI Appendix, Fig. S5 and Note S3 for a discussion of possible alternatives, such as accounting for reorganization effects (found to be insignificant), DOS site correlations, and an exponential DOS, all showing that the conclusions of our work are expected to be general to any organic semiconductor.

The model accounts for equilibrium as well as nonequilibrium phenomena (carrier populations characterized by a temperature equal to the lattice temperature or an effective temperature higher than the lattice temperature, respectively). For brevity, the terms “nonequilibrium,” “nonthermal,” or “nonrelaxed” are used interchangeably and refer to charges at DOS sites with significantly higher energy than DOS equilibrium site energies.

While the investigated D/A blends in this work have moderate energetic offsets for charge separation [with the exception of APFOgreen9:PC₇₁BM, which was deliberately chosen due to its low energetic driving force (33)], the CT-EL mechanism outlined in this work also applies to state-of-the-art systems with small energetic offsets (34). As long as the materials do not have strong DOS site energy correlations or anticorrelations, neither CT-S₀ nor S₁-S₀ emission is likely to occur from sites that have both a low hole energy and a low electron energy. The influence of spatial correlations is insignificant, as shown previously (35). Even in the limit of negligible energy offsets at the D/A interface, resulting in charge transfer and subsequent luminescence from the pristine D or A components, the model conclusions remain valid, but the model would have to be adapted to accommodate specific details.

The 0–0 Lineshape Broadening.

We simulate the 0–0 lineshape broadening for the materials shown in Fig. 3 using experimentally determined σ_HOMO D and σ_LUMO A, and applied bias corresponding to J = 200 mA cm⁻² (2 to 3 V across devices to ensure a sufficient signal-to-noise ratio for D/A blend comparison). Fig. 3B shows that the resulting inhomogeneous 0–0 lineshape broadening obtained by 3D kMC is σ_{3D kMC 0–0} ∼70 to 80 meV across different D/A combinations and depends only weakly on the DOS disorder of the constituent materials, spanning the range σ ∼60 to 100 meV for the investigated D/A blends.

To highlight the weak dependence of the 0–0 CT-EL lineshape broadening on DOS disorder, we simulate a hypothetical low-disorder case σ_HOMO D = 50 meV and σ_LUMO A = 60 meV, resulting in 62-meV inhomogeneous 0–0 lineshape broadening, whereas a high-disorder case σ_HOMO D = 100 meV and σ_LUMO A = 120 meV leads to 85 meV (Fig. 4A shows the corresponding 0–0 CT-EL spectra, 0–0 broadening was estimated at 2 V). We propose that the weak 0–0 lineshape broadening dependence (σ_0–0 ∼60 to 85 meV) on D/A energetic disorder (σ ∼50 to 120 meV) is a general feature of D/A organic semiconductor blends with a Gaussian DOS. As a result, the experimental CT-EL spectral broadening cannot be explained by the 0–0 transition broadening as the latter is almost invariant on D/A combination (Fig. 3B, filled circles). From a practical standpoint, this means that CT-EL broadening is mostly indicative of Franck–Condon broadening (19, 20) but is a poor indicator of the energetic disorder present in the BHJ mixture.

Fig. 4.

Three-dimensional kMC simulations of the 0–0 lineshape reveal nonequilibrium DOS site distributions governing CT-EL and enable CT-EL model fitting to experiments. (A) Bias-dependent 0–0 lineshape simulations using 3D kMC for materials with σ_HOMO D = 50 meV and σ_LUMO A = 60 meV (black) and σ_HOMO D = 100 meV and σ_LUMO A = 120 meV (blue). (Inset) The calculated spectra for different active layer thicknesses, confirming that the calculated spectra are a thickness-independent D/A materials property and not an artifact present only in devices with a thin active layer. (B) Energy-resolved HOMO (red) and LUMO (blue) site energies leading to CT-EL obtained by 3D kMC for σ_HOMO D = 100 meV and σ_LUMO A = 120 meV, at an applied bias of 1 V (filled area) and 5 V (shaded area). The horizontal colored lines mark the equilibrium σ²/kT energy in a Gaussian DOS (lightly shaded gray area, not to scale). (C) Global model fit (blue dashed lines) to experimental CT-EL spectra (red lines) for MDMO-PPV:PC₆₁BM (1:4) at the indicated current-densities. The 0–0 peak (shaded gray) is obtained by 3D kMC, whereas the lower energy 0–1 and 0–2 vibrational peaks (lighter shade of gray) are generated using Eq. 1. (D) PL spectra following 1.39 eV (890 nm) excitation of MDMO-PPV:PC₆₁BM (1:4) (red) and pristine MDMO-PPV (black) measured by FLN spectroscopy at 7 K. These measurements confirm the characteristic vibrational energy ħw_ph D/A = 137 meV, independently obtained by model fits to CT-EL spectra in C, where ħw_ph fit = 143 meV.

Energetic Position of the Recombining Carriers in Their DOS.

Fig. 4B shows the simulated energetic position of the recombining carriers in their respective Gaussian DOS for a material with σ_HOMO D = 100 meV and σ_LUMO A = 120 meV. Clearly, most CT-EL photons originate from carriers recombining at sites ∼180 meV and ∼330 meV (holes and electrons, respectively) above their DOS equilibrium energy σ²/kT.

To rule out the influence of charges residing at artificially high-lying DOS sites following injection from the contacts, we have carried out reference simulations of OPV devices with different active layer thicknesses (Fig. 4 A, Inset). These reference simulations confirm that the simulated nonequilibrium DOS site distribution governing CT-EL (Fig. 4B) is indeed a thickness-independent D/A materials property and is not an artifact present only in devices with a thin active layer. Complementary transfer matrix model (TMM) simulations (36), accounting for optical interference effects in the device stack during CT-EL (37) and the possible bias dependence of the emission zone position in the active layer, reveal that cavity effects also cannot explain the experimentally observed and simulated CT-EL bias dependence (see SI Appendix, Fig. S6 for details). Both of these results confirm that the observed and simulated CT-EL bias dependence as well as the simulated nonequilibrium DOS site distribution in Fig. 4B are a D/A materials property.

The gap between the sites contributing to CT emission and the equilibrium energy sites decreases with decreasing energetic disorder and tends to zero for low disorder. Indeed, our model predicts that for σ_HOMO D = 50 meV and σ_LUMO A = 60 meV the hole recombination energies center around σ²/kT, whereas the nonequilibrium gap for electrons decreases to ∼20 meV. However, to the best of our knowledge, a combination of such low D/A disorder values is rare for OPV materials, and the gap remains significant for typically encountered disorder values (13, 30, 31) (see SI Appendix, Note S2 for further discussion regarding disorder values). For example, CT-EL in a device with modest disorder values σ_HOMO D = 81 meV and σ_LUMO A = 73 meV, as found in the T-dependent SCLC analysis of MDMO-PPV:PC₆₁BM, is governed by nonequilibrium gaps of ∼80 meV and ∼60 meV, respectively, with a total difference of ∼140 meV compared to an equilibrium picture (Fig. 2C)—a significant number for a photovoltaic device.

The observed disorder dependence highlights that the CT-EL peak position only in part depends on the donor HOMO and acceptor LUMO levels—D/A disorder also governs the 0–0 CT-EL peak position. For example, materials with identical donor HOMO and acceptor LUMO levels but different disorder values will have their 0–0 CT-EL peaks situated at different energies. This renders the commonly employed E_CT assignment on basis of Gaussian fitting of the high-energy CT-EL tail and the low-energy tail of subgap external quantum efficiency (EQE) spectra (38) ambiguous, as will be shown below.

Most importantly, our model also reproduces the weak bias or current-density dependence of CT-EL (Fig. 4A), which originates due to the nonequilibrium DOS site distribution in Fig. 4B. SI Appendix, Fig. S5 shows that the same conclusion holds true when accounting for reorganization effects (found to be insignificant), DOS site correlations, and an exponential DOS, meaning that the conclusions of our work are expected to be general to any organic semiconductor. As such, these results resolve the inconsistencies present in the literature regarding which DOS sites govern CT-EL and show that CT-EL originates from considerably higher than equilibrium energy DOS site distributions (Fig. 2A), in contrast to the widespread equilibrium picture (Fig. 2C).

Three-Dimensional kMC Model Accounting for Energetic Disorder, Kinetic Effects, and the Franck–Condon Principle.

To demonstrate the full consistency of our newly proposed model, we develop, as a next step, a framework that quantitatively and simultaneously combines the 3D kMC model (accounting for kinetic charge transport/recombination physics in an energetically disordered DOS; Fig. 2 A, Left) together with the Franck–Condon progression (accounting for the vibronic replicas of the 0–0 emission; Fig. 2 A, Right). To avoid ambiguous fits to the experimental CT-EL current-density dependence, we describe the full CT-EL spectrum with the least number of experimentally unknown parameters following ref. 19. by

$$I_{\text{CT}-\text{EL}}\left(\hslash \omega \right)={\left[n\left(\hslash \omega \right)\hslash \omega \right]}^3·{\displaystyle \sum _{m=0}^{M-1}\frac{e^{-S}{S}^m}{m!}·\mathit{\Gamma }\left(\hslash \omega -\left(\hslash {\omega }_{0-0}-m\hslash {\omega }_{\text{ph}}\right)\right)},$$ [1]

where n(ħw) is the refractive index at photon energy ħw, S is the Huang–Rhys factor, m = {0, 1, …, M − 1} denotes the vibrational energy level and M the total number of vibronic peaks, ħw_0–0 is the 0–0 transition energy (lowest D⁺/A⁻ excited state to D/A ground state), and ħw_ph is the effective phonon-mode energy governing CT luminescence (see Fig. 2 schematic). We account only for emission from the lowest-energy D⁺/A⁻ excited state to the vibrational D/A ground state manifold (Fig. 2 A, Right); since on-site thermalization is considerably faster (<1 ps) (39, 40) than intersite thermalization in the DOS, the latter may take as long as 1 µs to complete (29). The model can be extended to account for the D⁺/A⁻ excited state manifold but this is outside the scope of this work.

The first term [n(ħw)ħw]³ accounts for the influence of the photon DOS (in the medium surrounding the emitter) to the emission spectrum. Since n(ħw) of D/A blends is typically only weakly energy-dependent at CT-EL emission energies, as confirmed by spectroscopic ellipsometry for the blends in this work (SI Appendix, Fig. S7), the energy dependence of n(ħw) has only a minor effect on the overall shape of the CT-EL emission spectrum. Differences in the CT-EL emission spectra between the D/A blends mainly originate from the remaining terms in Eq. 1.

The last term Γ(ħw – (ħw_0–0 – mħw_ph)) describes the 0-m transition lineshape, which is red-shifted by m × ħw_ph with respect to the 0–0 transition. To limit the number of unknown parameters, the 0-m and 0–0 lineshapes are assumed to be identical. The 0–0 lineshape is calculated by 3D kMC simulations as shown in Fig. 4A, using experimentally determined disorder values σ_{HOMO D/LUMO A} whenever appropriate (see SI Appendix, Note S2 for details) and using the experimental set of bias or current-density conditions. The experimental CT-EL bias or current-density dependence is then fully captured by the simulated bias dependence of the 0–0 transition (Fig. 4A), accounting for the nonequilibrium DOS distribution governing CT-EL (Fig. 4B). We simulate a volume of 90 × 90 × d nm³, where d is the measured OPV device thickness, spanning the range 80 to 220 nm for the investigated D/A blends, optimized for photovoltaic performance.

We incorporate the Franck–Condon progression (19) (second term in Eq. 1) by simultaneously fitting the entire set of experimental current-density- or bias-dependent CT-EL spectra, severely constraining the model and ensuring reliable fits. All current-density conditions in Fig. 4C are globally fitted by a single parameter set, resulting in a 3-parameter fit (ħw_ph fit, S, and M). Fig. 4C shows excellent fits to the experimental CT-EL current-density dependence of MDMO-PPV:PC₆₁BM, whereas SI Appendix, Fig. S8 shows equally good fits for the other D/A blends.

Model fits for MDMO-PPV:PC₆₁BM suggest that the vibrational energy responsible for CT-EL is ħw_ph fit = 143 meV (1,150 cm⁻¹), which is characteristic for C–C skeletal vibration. Note that the typical vibrational energy governing S_n-S₀ transitions is not necessarily dominant in the CT_n-S₀ case discussed here due to the different geometry and coupling involved (intra- versus intermolecular transition). Indeed, using FLN spectroscopy (22, 23) we found that the phonon-mode energy ħw_ph D/A governing CT luminescence differs from the phonon mode that governs the S₁-S₀ transition (see the next section and SI Appendix, Figs. S9 and S10 for details).

Measurements of the Phonon-Mode Energy Governing CT Luminescence.

To ensure our framework is fully consistent with experiments, we independently confirm the number of vibronic peaks M and the characteristic phonon-mode energy ħw_ph governing CT luminescence using FLN spectroscopy (22, 23). Briefly, we rely on D/A blend photoexcitation using low-energy photons (1.39 eV for MDMO-PPV:PC₆₁BM and 1.49 eV for most of the other blends) to predominantly excite the low-energy part of the CT manifold, leading to line narrowing such that the 0–0 and 0-m transitions can be clearly resolved. We record D/A blend film photoluminescence (PL) at cryogenic (7 K) temperatures to further suppress inhomogeneous broadening, which allows us to reveal the underlying vibronic progression at the D/A interface governing CT luminescence. The resulting spectra are shown in Fig. 4D for MDMO-PPV:PC₆₁BM and in SI Appendix, Fig. S9 for the other blends. Comparison to pristine MDMO-PPV spectra clearly shows that the blend FLN spectra correspond to CT emission (Fig. 4D). This is the case for all except one of the other blends, APFOgreen9:PC₇₁BM, which has a low energetic driving force for charge separation (33). While the strong overlap with pristine D luminescence in the case of APFOgreen9:PC₇₁BM has prevented us to unambiguously assign the blend FLN spectra to CT luminescence, we were able to extract the characteristic vibrational energy governing CT-PL in all of the other blends, as shown in SI Appendix, Fig. S9.

To estimate the characteristic vibrational energy governing CT-PL in MDMO-PPV:PC₆₁BM we use multipeak Gaussian fitting (Fig. 4D, dashed black lines). To minimize the number of experimentally unknown fit parameters, we allow the 0-m lineshape broadening to increase for the lower-energy 0-m transitions, which we tentatively attribute to the presence of more than one vibrational frequency (41) or dispersion in molecular vibrations; both warrant further study but are outside the scope of this work. The characteristic energy obtained from the fits (ħw_ph D/A = 137 meV) and the number of clearly visible vibronic peaks (M = 3) are in good agreement with the model fits to CT-EL in Fig. 4C, confirming the validity of our proposed CT-EL model by independent experiments. SI Appendix, Figs. S8 and S9 and Table S2 show equally good agreement for the other blends. This is a selective and direct measurement of the characteristic phonon-mode energy ħw_ph D/A governing CT luminescence in organic semiconductors, without being overshadowed by the otherwise dominant pristine D and/or A luminescence.

CT-EL Recombination Mechanism.

To gain further insight, we have investigated whether charges residing at higher-lying DOS sites, which could be considered “more mobile,” perform the last hop to recombine with charges residing at lower-energy DOS sites, which could be considered as temporarily “immobile/trapped.” This mechanism would be consistent with multiple-trapping-and-release type models often used to describe charge carrier recombination in OPV devices. To the contrary, as detailed in SI Appendix, Fig. S11, our 3D kMC simulations indicate that CT-EL is instead governed by events in which the charge carrier that makes the last hop resides at a lower-energy DOS site to recombine with a charge carrier residing at a higher-lying DOS site. A similar recombination mechanism was proposed earlier in the context of charge separation of photogenerated charge pairs (42).

Relation to Subgap Absorption Measurements and the CT Energy.

Having developed a fully consistent framework for CT-EL, we now demonstrate the importance of CT emission occurring from a nonequilibrium DOS site distribution (Fig. 2A) in relation to subgap absorption and EQE measurements. For comparison, we determine the conventionally defined CT energy E_CT conv following the procedure outlined by Vandewal et al. (38), which relies on Gaussian fitting of appropriately-normalized reduced EQE and CT-EL spectra, as shown in SI Appendix, Fig. S12. The following analysis indicates that there are many available DOS sites for charge generation, transport, and recombination below the conventionally defined CT energy. The energy definitions used in subsequent analysis are schematically shown in Fig. 2.

Fig. 5 shows the experimental and modeled CT-EL spectra together with subgap EQE spectra measured by Fourier-transform photocurrent spectroscopy (FTPS). The remaining discrepancy between the measured and modeled CT-EL spectra (red and blue traces in Fig. 5, respectively) likely originates due to the material DOS not being exactly Gaussian, and possibly due to S₁-S₀ emission from the pristine D and/or A at higher energies, which is not accounted for by the model. Nevertheless, model fitting at low energies is already superior to the conventional approach (see also SI Appendix, Fig. S12) and sufficiently accurate for subsequent analysis. Since the position of the recombining carriers in their respective DOS is known from our model (Fig. 4B), Fig. 5 enables us to estimate the position of E_CT conv in relation to the joint DOS of the D/A blend (Fig. 2D).

Fig. 5.

Relation between nonequilibrium DOS sites governing CT-EL and subgap EQE measurements and the definition of E_CT. Comparison of CT states with DOS equilibrium site energy E_CT eq (black dashed lines, Eq. 2) to the conventionally defined E_CT conv (38) reveals a substantial E_CT conv − E_CT eq ∼290- to 800-meV difference depending on the D/A blend, highlighting the importance of the high-energy DOS site distributions governing CT-EL. Here, the experimental CT-EL spectra (red lines), the CT-EL model fit (blue dashed line), and the corresponding vibronic progression are scaled by a constant, such that the experimental EQE-FTPS spectra (open circles) and the CT-EL spectra would coincide at E_CT conv, which was independently determined following the definition proposed by Vandewal et al. (38) (see SI Appendix, Fig. S12). E_CT centr marks the central CT energy |E_HOMO D − E_LUMO A| corrected by the CT exciton binding energy E_CT bind ∼220 meV following Eq. 3. (A) MDMO-PPV:PC₆₁BM (1:4), (B) APFO3:PC₆₁BM (1:4), (C) TQ1:PC₇₁BM (1:2.5), and (D) APFOgreen9:PC₇₁BM (1:3).

The importance of high-energy DOS sites governing CT-EL becomes apparent when comparing E_CT conv and CT states with equilibrium DOS site energies E_CT eq across different D/A combinations, calculated for an electron and hole pair forming a CT state at DOS equilibrium sites as

$$E_{\text{CT}\hspace{0.17em}\text{eq}}=\left(\left|E_{\text{HOMO}\hspace{0.17em}\text{D}}\right|-\frac{{\sigma }_{\text{HOMO}\hspace{0.17em}\text{D}}^2}{kT}\right)-\left(\left|E_{\text{LUMO}\hspace{0.17em}\text{A}}\right|-\frac{{\sigma }_{\text{LUMO}\hspace{0.17em}\text{A}}^2}{kT}\right)-E_{\text{CT}\hspace{0.17em}\text{bind}},$$ [2]

where E_HOMO D and E_LUMO A are the donor HOMO and acceptor LUMO energies, respectively, and E_CT bind is the CT exciton binding energy ∼220 meV, corresponding to an intersite D/A distance of 1.8 nm as used in 3D kMC. Fig. 5 shows that E_CT eq is situated significantly below the conventionally defined CT energy. The E_CT conv − E_CT eq difference is substantial and spans the range ∼290 to 800 meV for the investigated D/A blends (see SI Appendix, Table S3 for values and SI Appendix, Note S4 regarding E_{HOMO D/LUMO A} determination). This implies that there are many available DOS sites for charge generation, transport, and recombination below the conventionally defined CT energy. Fig. 5 also illustrates that the conventionally determined E_CT conv is not a good measure of the central CT energy E_CT centr, corresponding to the difference between the donor HOMO and acceptor LUMO levels corrected by the CT binding energy, determined as

$$E_{\text{CT}\hspace{0.17em}\text{centr}}=\left|E_{\text{HOMO}\hspace{0.17em}\text{D}}-E_{\text{LUMO}\hspace{0.17em}\text{A}}\right|-E_{\text{CT}\hspace{0.17em}\text{bind}}.$$ [3]

The main purpose of Fig. 5 is not to redefine E_CT but to highlight that CT-EL originates from considerably higher-lying DOS sites than previously thought, as depicted schematically in Fig. 2. The many available DOS sites below E_CT conv also rationalizes the experimentally observed photogenerated carrier thermalization to these low-lying DOS sites (29), even following photoexcitation at E_CT conv (31), when the OPV device is operated as a solar cell rather than as an LED.

Generally speaking, blend DOS describes all hopping sites, whereas the manifold probed in CT emission corresponds to a subset of the total DOS sites forming a D/A interface, weighted by the probability of an electron and a hole meeting and recombining. Although the D/A interface can be probed by spectroscopic tools such as CT-EL and EQE-FTPS, the relation of the probed sites to the total DOS, relevant for carrier transport and thermalization, is typically elusive in such measurements. Figs. 4 and 5 reveal how DOS sites relevant to subgap photovoltaic action and CT-EL relate to the total DOS.

Relation to Device Open-Circuit Voltage and Energy Losses.

In view of the higher-energy DOS subset sampled during CT-EL, it is not evident why the conventionally defined E_CT conv has been so effective in quantitatively describing V_OC (6, 7) and photovoltaic energy conversion losses in OPV devices, since models based on E_CT conv generally assume that equilibrium DOS site occupation governs CT-EL. While the results presented here do not invalidate established E_CT to V_OC relations, the nonequilibrium DOS site distribution governing CT-EL adds a layer of previously overlooked complexity.

Comparing the measured EQE-FTPS spectra to the EQE spectra derived from CT-EL measurements, we find that the commonly employed reciprocity analysis (6, 7) also applies to the investigated systems, as detailed in SI Appendix, Fig. S13; nevertheless, detailed understanding of how charge transport/recombination kinetics and energetic disorder affect reciprocity relations requires further investigation. Analysis accounting for these factors, while outside the scope of this work, is expected to deepen the insight into the underlying photophysics, especially in the origins of energy losses in OPV devices. In addition, extended treatment of photogenerated charge energetics could reveal novel routes to improved V_OC. As such, these results pose fundamental questions, particularly regarding the meaning of reciprocity relations relating light emission to photovoltaic action, and could serve as motivation for the continued pursuit of fully understanding V_OC and energy losses in organic optoelectronic devices.

Materials and Methods

Full material names, photovoltaic device fabrication and characterization using CT-EL, FLN and EQE-FTPS measurements are described in SI Appendix, Materials and Methods.

Material Choice.

To accurately model experimental CT-EL spectra, material combinations showing only CT emission and no pristine material emission (over the investigated current-density range) were selected. The APFOgreen9 based BHJ system was deliberately chosen due to its low energetic driving force for charge separation (33).

Three-Dimensional kMC Simulations.

The most comprehensive model description can be found in ref. 29. The Miller–Abrahams formalism was used to quantify with the least number of unknown parameters the nearest-neighbor hopping rate of a charge carrier in a cubic lattice with an uncorrelated Gaussian DOS (an exponential DOS was also considered; see SI Appendix, Fig. S5). The BHJ active layer was treated as an effective medium, meaning that the used hopping parameters represent average values over the entire BHJ film—local variations in the physical properties of the nanoscale morphology are not explicitly accounted for. The experimentally measured BHJ active layer thickness was used in the simulations. Additionally, ohmic contacts were included as quasi-infinite reservoirs for charge injection and extraction. The corresponding Miller–Abrahams rates were calculated with respect to the contact Fermi level; site energies in the semiconductor were corrected for the image potential. The number of particles in 3D kMC simulations of the 0–0 lineshape in Fig. 4A spans the range of 113× to 303× particles for the 1 V and the highest voltage simulations, respectively, corresponding to a reasonable carrier density range of 1.5 × 10¹⁷ cm⁻³ to 4.4 × 10¹⁷ cm⁻³.

Data Availability.

All data are provided in the main text and SI Appendix.