Main

Organic light-emitting diodes (OLEDs) are now widely used in displays and are being developed for applications in lighting, sensing and communications1,2. They consist of layers of charge transporting and light-emitting organic semiconductors in between two electrodes, at least one of which is transparent. When the injected charges recombine, they form both singlet and triplet excitons. Spin statistics suggest three triplets form for each singlet, a ratio that has been verified for evaporated OLEDs using low molecular weight emitters21. In OLEDs using fluorescent materials, only the singlets emit light. Phosphorescent OLED materials were therefore developed to obtain light emission from the triplets as well22. These work very well for red and green emission, but there is not yet a blue phosphorescent emitter meeting all commercial requirements23. Consequently, there is currently great interest in thermally activated delayed fluorescence (TADF) as an alternative approach to obtaining light from triplets3,4. Following the pioneering work of Adachi and coworkers in 2011, there have been more than 4,000 papers with the keyword thermally activated delayed fluorescence24,25 (based on results from 16 February 2024 that mention thermally activated delayed fluorescence or TADF since 2011).

A problem in both organic and inorganic LEDs is that as the drive current is increased for more light output, the efficiency decreases26. This is known as efficiency roll-off and is illustrated in Fig. 1a, which shows the efficiency as a function of current density for prototypical examples of fluorescent, phosphorescent and TADF OLEDs3,27,28. Figure 1a shows that the phosphorescent and TADF OLEDs have more than four times the efficiency of the fluorescent OLEDs, but that their efficiency decreases and particularly severely for the TADF OLEDs as the current density is increased. To compare the behaviour of a wide range of OLEDs of each type, we define J90 as the current density at which the external quantum efficiency (EQE) falls to 90% of its peak value, as illustrated in Fig. 1b.

Fig. 1: Examples of efficiency roll-off.
figure 1

a, Efficiency roll-off of prototypical fluorescent (Alq3), phosphorescent (Ir(ppy)2acac) and TADF (4CzIPN) OLEDs3,27,28. b, Schematic graph showing definition of J90. c, Graph of the relation between J90 and EQE1,000 for TADF5,38,39,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113, fluorescent (Fluo.)28,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129 and phosphorescent (Phos.)86,87,88,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155 devices emitting in the red (R), green (G) and blue (B) regions of the spectrum (for the references, see Methods).

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We have extracted J90 from published data on a wide range of OLEDs together with their EQE at a practical luminance of 1,000 cd m−2. These are plotted for each class of OLEDs and each colour in Fig. 1c. The ideal behaviour would be high J90 (for low efficiency roll-off) and high EQE: that is, the top right quadrant of the graph. Most fluorescent OLEDs fall in the green rectangle (A), which is a region of high J90 but low EQE. Most phosphorescent OLEDs (and a few others) fall in the blue rectangle (B). The upper half of this rectangle represents OLEDs with high efficiency and fairly high J90. TADF OLEDs fall mainly in region C. Notably, there is a much wider spread of both EQE and J90 than for the other classes of OLEDs, possibly because TADF OLEDs is a much younger field. The upper right part of region C shows that there are some reports of TADF OLEDs with high EQE and moderately high J90, although lower than for good phosphorescent devices. However, region C also extends to extremely low values of J90 that is, there are many TADF devices experiencing significant efficiency roll-off at current densities below 0.1 mA cm−2. Even for a green device, this would correspond to a brightness of at most 100 cd m−2, whereas typical displays run at 400 cd m−2 and their individual pixels often run at much higher brightness to achieve an average 400 cd m−2 on the display29. Hence, many reported TADF OLEDs have severe efficiency roll-off and even the best have significant efficiency roll-off (J90 of a few mA cm−2).

Efficiency roll-off in TADF devices

This brings us to the central question of this analysis, which is, what can be done in terms of emitter design to reduce efficiency roll-off (that is, increase J90)? In other words, which photophysical processes need to be tuned by molecular design to minimize inherent limitations of the emitter that contribute to efficiency roll-off? Efficiency roll-off arises both from the emitter design and the device design, but for an optimized device design (for example, balanced charge carriers and wide recombination zone) it will ultimately be limited by the properties of the emitter.

To identify the crucial parameters for emitter design, we first need to consider what causes efficiency roll-off. Studies in phosphorescent OLEDs have shown that triplet–triplet annihilation (TTA) and triplet–polaron annihilation (TPA) are the main loss mechanisms as the current density is increased29,30,31. A similar understanding is developing in TADF OLEDs in which TTA, TPA and singlet–triplet annihilation (STA) may all contribute32,33,34. These are all bimolecular processes and thus much more severe at higher excitation densities. Furthermore, as all these processes involve triplets, they can be mitigated by reducing the triplet lifetime and hence reducing the triplet population. This has been achieved successfully in phosphorescent OLEDs by engineering the light-emitting material (for example, by using an iridium complex) to show a large radiative rate constant from the triplet state and thus achieving a relatively short triplet lifetime of around 1 µs. For comparison, delayed fluorescence lifetimes in organic TADF materials range from 1 µs to beyond 500 μs. We briefly note that as well as reducing efficiency, bimolecular processes involving triplets are also a main mechanism of device degradation, providing a further reason to reduce the triplet population in operating devices35.

Hence to reduce efficiency roll-off, we need to reduce triplet lifetime or, more precisely, the triplet population during device operation. This, however, is not as simply achieved as in the case of phosphorescence. The key photophysical processes in a TADF emitter are shown in Fig. 2. Singlets are converted to triplets via intersystem crossing (ISC) with rate constant kISC, and triplets to singlets via reverse intersystem crossing (RISC) at rate constant kRISC. There is potentially radiative and non-radiative decay of both triplets and singlets, although in a good TADF material \({k}_{{\rm{r}}}^{{\rm{S}}}\) will be much larger than any of \({k}_{{\rm{nr}}}^{{\rm{S}}}\), \({k}_{{\rm{r}}}^{{\rm{T}}}\) and \({k}_{{\rm{nr}}}^{{\rm{T}}}\) (refs. 36,37). The main approach advocated in the literature for reducing efficiency roll-off is to increase kRISC, commonly by reducing the energy difference between singlet and triplet excited states (∆EST) through molecular design by reducing the exchange integral between the highest occupied and lowest unoccupied molecular orbitals. In addition, there have been other attempts to increase kRISC, for example by the use of heavy atoms, to increase spin–orbit coupling (SOC)5,38. The emphasis on kRISC is so strong that since 2016 there have been 16 publications in the Nature family alone exploring kRISC (refs. 5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20). However, the expected improvement in J90 has not always materialized.

Fig. 2: Simplified Jablonski diagram of a TADF emitter.
figure 2

The excited singlet and triplet states S1 and T1, respectively, are shown in equilibrium due to the occurrence of both intersystem crossing (kISC) and reverse intersystem crossing (kRISC) enabled by the small energy gap (ΔEST) between S1 and T1. A TADF OLED emits light by radiative decay from \({\text{S}}_{1}({k}_{\text{r}}^{\text{S}})\), whereas non-radiative decay from both \({\text{S}}_{1}({k}_{\text{nr}}^{\text{S}})\) and \({\text{T}}_{1}({k}_{\text{nr}}^{\text{T}})\), as well as the negligible radiative decay from \({\text{T}}_{1}({k}_{\text{r}}^{\text{T}})\) are other deactivation pathways of excited species.

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To understand how J90 depends on kRISC, we have plotted the graph shown in Fig. 3. There is some correlation (Spearman correlation ρ = 0.638) in so far as there is a tendency towards higher J90 for higher kRISC, but there is an enormous spread of the data (considering this is a log–log plot). For example, the blue dashed rectangle shows that J90 of roughly 2 mA cm−2 can be achieved with kRISC from 2 to 20 × 105 s−1. The insufficiency of kRISC as a guide for molecular design is vividly demonstrated by the red dashed rectangle that shows J90 for molecules designed with a high kRISC of 8–15 × 105 s−1. The values of J90 range from 0.03 to 40 mA cm−2, that is, by more than three orders of magnitude, showing kRISC alone is inadequate as a predictor of efficiency roll-off.

Fig. 3: Data analysis.
figure 3

J90 of the reported TADF OLEDs with respect to kRISC (Spearman correlation ρ = 0.638). Red crosses present TADF molecules containing heavy atoms that benefit from enhanced SOC to increase kRISC. Data inside the dashed boxes are for comparison.

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Derivation of FOM

To develop guidelines for TADF materials design to reduce efficiency roll-off in OLEDs, we should first look more closely at Fig. 2 and the mechanism of TADF.

The physics of TADF is often studied using transient photoluminescence (PL) measurements in which the emitter is excited using a laser pulse. Excitons are generated in the excited singlet state (S1), and the decay of the excited state is slowed by cycling to and from the triplet state (T1) by ISC and RISC, respectively. In an OLED, charge injection leads to a buildup of both S1 (25%) and T1 (75%) excitons as well as polarons. There is a dynamic equilibrium between S1 and T1 facilitated by the ISC/RISC cycling. To ascertain on which side the dynamic equilibrium lies, an equilibrium constant Keq is defined as

$${K}_{\text{eq}}=\frac{\left[{\text{S}}_{1}\right]}{\left[{\text{T}}_{1}\right]}.$$
(1)

For a three-level TADF OLED under low constant current electrical excitation, the equilibrium constant is given as follows (see Methods for the derivation)

$${K}_{\text{eq}}=\frac{3{k}_{\text{RISC}}+{k}^{\text{T}}}{3{k}^{\text{S}}+{k}_{\text{ISC}}},$$
(2)

where kS is the sum of the rate constants for ISC (kISC), radiative \(({k}_{\text{r}}^{\text{S}})\) and non-radiative \(({k}_{\text{nr}}^{\text{S}})\) decay from S1, and kT is the sum of the rate constants for RISC (kRISC), radiative \(({k}_{\text{r}}^{\text{T}})\) and non-radiative \(({k}_{\text{nr}}^{\text{T}})\) decay from T1. As explained earlier, to minimize the EQE roll-off a low T1 population is necessary to suppress TTA and to a lesser extent STA and TPA. For an OLED operated at high brightness this translates to the requirement of maximizing the S1 population relative to the T1 population, which can be achieved by maximizing Keq. Furthermore, according to Le Chatelier’s principle, an equilibrium can be moved to a desired product by removing the product from the equilibrium. Here, the radiative decay of S1 excitons is the desired product. Therefore, to minimize the fraction of triplet excitons in the steady-state OLED emitters should be developed (or selected) to maximize the product of radiative rate constant and equilibrium constant. In a good OLED, nearly all electrically excited excitons decay radiatively, that is \({k}_{\text{nr}}^{\text{S}}={k}_{\text{nr}}^{\text{T}}=0\). Thus, for a TADF emitter with photoluminescence quantum yield near unity and no phosphorescence contribution \(({k}_{{\rm{r}}}^{{\rm{T}}}=0)\), which is reasonable for good organic emitters, a figure of merit (FOM) for efficiency roll-off can be formulated as

$${k}_{{\rm{r}}}^{{\rm{S}}}{K}_{{\rm{e}}{\rm{q}}}=\frac{4{k}_{{\rm{r}}}^{{\rm{S}}}{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}}{3{k}_{{\rm{r}}}^{{\rm{S}}}+4{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}}.$$
(3)

Figure 4 shows J90 plotted as a function of this FOM. There is a stronger correlation with J90 (ρ = 0.700) than kRISC with J90 (ρ = 0.638). Higher \({k}_{\text{r}}^{\text{S}}{K}_{\text{eq}}\) leads to higher J90. Accordingly, maximizing \({k}_{\text{r}}^{\text{S}}{K}_{\text{eq}}\) and thus minimizing the T1 population under electrical excitation is a better strategy for improving efficiency roll-off than considering kRISC alone. Figure 4 compares efficiency roll-off as a function of our FOM (black circles) with efficiency roll-off as a function of kRISC (small grey circles). The FOM has a narrower spread of values as would be expected for the improved correlation.

Fig. 4: FOM.
figure 4

Correlations between J90 and \({k}_{\text{r}}^{\text{S}}{K}_{\text{eq}}\) with a Spearman correlation coefficient of ρ = 0.700. Red crosses identify TADF molecules containing heavy atoms that enhance SOC, which leads to an increase in kRISC. In grey circles, the correlation of J90 and kRISC from Fig. 3 is shown for comparison.

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It is interesting to apply this FOM to recent attempts to increase kRISC by incorporating heavy atoms into the molecule to increase SOC5,38. These studies are shown by red crosses in Figs. 3 and 4. This strategy is broadly successful at leading to fast kRISC but does not necessarily lead to the highest J90 as kISC also increases, or \({k}_{{\rm{r}}}^{{\rm{S}}}\) decreases. This interplay between these parameters is captured by the FOM as can be seen from the red crosses in Fig. 4 being in the same region as other materials. At the same time, incorporation of these larger atoms that result in weaker bonds also leads to faster non-radiative pathways and potentially poor device stability. Here, we can see that kRISC and the proposed FOM give distinct assessments of the heavy atom approach, and that the latter is a better predictor of J90 for future molecular design. Another possible guide for design is (as for phosphorescent devices) short delayed fluorescence lifetime (τDF)39. The correlation of J90 with τDF is shown in Extended Data Fig. 1a. There is a good correlation (ρ = −0.685), although still some scatter. Actually, τDF has a much stronger correlation with the proposed FOM (ρ = −0.801) than with kRISC (ρ = −0.709). In other words, the proposed FOM not only predicts the efficiency roll-off, but also clarifies the key physical processes that need to be optimized to achieve low efficiency roll-off. Although measuring τDF would be an effective way of screening materials for low potential efficiency roll-off after they have been synthesized, our FOM gives more insight into how to design a material for low efficiency roll-off by showing the exact combination of rate constants that should be optimized.

For many TADF materials kISC is substantially faster than \({k}_{{\rm{r}}}^{{\rm{S}}}\), in which case the FOM can be simplified to

$${k}_{{\rm{r}}}^{{\rm{S}}}{K}_{{\rm{eq}}}\approx \frac{{k}_{{\rm{r}}}^{{\rm{S}}}{k}_{{\rm{RISC}}}}{{k}_{{\rm{ISC}}}}\,{\rm{for}}\,{k}_{{\rm{r}}}^{{\rm{T}}}={k}_{{\rm{nr}}}^{{\rm{T}}}={k}_{{\rm{nr}}}^{{\rm{S}}}=0\,{\rm{and}}\,{k}_{{\rm{ISC}}}\gg {k}_{{\rm{r}}}^{{\rm{S}}}$$
(4)

This simplified FOM highlights the competition between kISC, kRISC and \({k}_{{\rm{r}}}^{{\rm{S}}}\) very clearly. It is equivalent to \({k}_{\text{r}}^{\text{S}}{K}_{\text{eq}}\) in the regime where \({k}_{\text{r}}^{\text{S}}\) is smaller than kISC (Extended Data Fig. 2).

Other factors affecting efficiency roll-off

Although there is a good correlation between J90 and the proposed FOM, there is a significant spread of data points in Fig. 4. This can be understood to arise because efficiency roll-off involves a combination of the intrinsic properties of the emitting molecule with the extrinsic properties of the device. An analogous situation exists when using photoluminescence quantum yield as a predictor of device efficiency: whether a material realizes its full potential also depends on the device. Similarly, our FOM describes the best that could be achieved with a particular light-emitting material in a device limited by the triplet population. In real devices, many factors, especially imperfect charge balance, could lead to worse performance than this ideal case, and hence can explain the spread of the data in Fig. 4. In addition, at low current density some devices show efficiency increasing with current density, as can be seen for the devices in Fig. 1a. As J90 is taken as a reduction from peak efficiency, this will lead to higher values of J90 than in devices with peak efficiency at very low current. There is another example of this effect in Extended Data Fig. 3 that compares two 2CzPN devices3,40. It should also be noted that practice for determining rate constants varies37,41, which could also contribute to the spread.

Another important factor that could contribute to the spread of data is that the effect of a given triplet population depends on the material. In particular, reported TTA rate constants γTT are widely spread over eight orders of magnitude (10−18–10−10 cm3 s−1)32,33,42,43. So, increasing the FOM will reduce triplet population, and is beneficial (increases J90) but the improvement arising from the reduced triplet population depends on the value of γTT. Similar considerations apply to STA, in which again there is a range of γST, and the relative importance of STA and TTA depends on the relative values of γTT and γST. As these rate constants are not yet widely measured, we have not at this stage attempted to incorporate them into a FOM. However, we show their potential effect in Extended Data Fig. 4, which shows calculations of how J90 would depend on FOM for systems with \({k}_{\text{r}}^{\text{S}}\) between 105 and 1010 s−1, \({{k}_{\text{ISC}}/k}_{\text{r}}^{\text{S}}\) between 10−1 and 103, and \({k}_{\text{r}}^{\text{S}}{K}_{\text{eq}}\) between 102 and 108 s−1 for a range of values of γST, γTT and \({k}_{\text{r}}^{\text{S}}\). Extended Data Fig. 3a shows how for a given FOM, each order of magnitude change in γTT leads to an order of magnitude change in J90. Extended Data Fig. 4b shows the potential interplay between TTA and STA. The J90 value behaves nearly linearly with \({k}_{\text{r}}^{\text{S}}{K}_{\text{eq}}\) when only TTA is considered (red dots, the slope is 2). If only STA is considered, there is still a correlation; however, at the same FOM, higher J90 is achieved when \({k}_{\text{r}}^{\text{S}}\) is large. If both TTA and STA are significant, then the efficiency is limited by TTA at low FOM and by \({k}_{\text{r}}^{\text{S}}\) at high FOM.

We also note that the kinetics of thin films can result in a multi-exponential transient PL, which is caused by conformational disorder44. Such a decay can be analysed using a Laplace transformation of the three-level kinetics of each conformer45. Our analysis does not include conformational disorder but could be applied in a similar manner to the analysis of multi-exponential transient PL caused by conformational disorder.

Conclusion

Our analysis has important implications for the rapidly growing field of TADF OLED development. At present, many such devices suffer such severe efficiency roll-off that they are unsuitable for practical application and, as we have shown, current emitter design focusing on maximizing kRISC alone is not an effective strategy. On the basis of the insight from considering the quasi-equilibrium in TADF, we instead propose that the focus of materials design and development should shift to maximizing a FOM that combines the physical processes that determine efficiency roll-off. Target values of the FOM will depend on the requirements of particular applications, as well as device design and severity of bimolecular effects. We estimate values of FOM required for materials with chromaticity close to the BT2020 standard46, and with Gaussian emission spectra of width 15 nm in the blue, 30 nm in the green and 45 nm in the red. We use the calculation for Extended Data Fig. 4a with γTT = 10−13 cm3 s−1 and find the FOM required to achieve 90% of a peak EQE of 25% at a brightness of 1,000 cd m−2. We find that for a deep blue emitter (λmax = 467 nm, CIE 1931 colour space (0.131, 0.049)) a FOM of at least 1.5 × 105 s−1 is required. For a green emitter (λmax = 529 nm, CIE (0.169, 0.772)) a FOM of 5.1 × 104 s−1 would be required, and for red (λmax = 650 nm, CIE (0.708, 0.292)) an FOM of at least 1.3 × 105 s−1 is needed.

In terms of material design for low efficiency roll-off, it is not necessary to maximize kRISC, but it is very desirable to maximize kRISC relative to kISC (without sacrificing \({k}_{{\rm{r}}}^{{\rm{S}}}\)). It is also a useful strategy to seek materials with high \({k}_{{\rm{r}}}^{{\rm{S}}}\) (providing kRISC/kISC is not reduced), which is also the underlying physics for hyperfluorescent OLEDs47, where the rate constant of Förster resonance energy transfer takes the place of \({k}_{{\rm{r}}}^{{\rm{S}}}\) in the FOM and lowers the triplet population on the TADF sensitizer. At the same time, there is a need to understand which process dominates the efficiency roll-off. Whereas all main annihilation processes scale with the triplet population and thus inversely with our proposed FOM, the relative importance of these processes in each OLED is not sufficiently known. Therefore, there is a need to measure both γST and γTT in a wider set of devices to fully understand how the excited-state kinetics of the emitter need to be engineered to reduce efficiency roll-off.

We hope that our FOM and these insights will enable the field of TADF OLEDs to overcome the challenge of efficiency roll-off and advance more rapidly to applications in displays, lighting and beyond.

Methods

Data collection

We considered the reported efficiency roll-off behaviour of TADF OLEDs published in peer-reviewed journals between 2016 and 2022. The data of the OLED and emitter were included in the analysis if the following criteria were met:

  1. 1.

    The reported OLED was vacuum-processed, in a bottom-emitting device structure with a TADF emitter in a host material.

  2. 2.

    Photophysical characterization of the thin film used as the emission layer was reported.

  3. 3.

    The photoluminescence quantum of the emitter film was reported to exceed 60%.

  4. 4.

    The calculation of all TADF rate constants was clearly detailed.

  5. 5.

    Device data clearly showed J90 data or the presented device data allowed for a reasonable estimation of J90.

Applying these criteria led to a total of 66 devices from 46 publications being included in our analysis5,38,39,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,156,157,158,159,160.

For comparison, Fig. 1c shows the relation between J90 and the EQE at 1,000 cd m−2 (EQE1,000) for TADF OLEDs5,38,39,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113 with representative fluorescent28,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129 and phosphorescent86,87,88,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155 devices across red, green and blue colours.

Steady-state population of the excited states

The kinetics of a TADF emitter as shown in Fig. 2 under electrical excitation can be described by the rate equations for the excited singlet state (S1) and the triplet state (T1) when neglecting annihilation processes as follows.

$$\frac{{\rm{d}}}{{\rm{d}}{\rm{t}}}{[{{\rm{S}}}_{1}]}_{t}=-\mathop{\underbrace{({k}_{{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{I}}{\rm{S}}{\rm{C}}})}}\limits_{={k}^{{\rm{s}}}}{[{{\rm{S}}}_{1}]}_{t}+{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}{[{{\rm{T}}}_{1}]}_{t}+\frac{1}{4}\gamma {[{\rm{n}}]}_{t}^{2}$$
(5)
$$\frac{{\rm{d}}}{{\rm{d}}{\rm{t}}}{[{{\rm{T}}}_{1}]}_{t}=-\mathop{\underbrace{({k}_{{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}})}}\limits_{={k}^{{\rm{T}}}}{[{{\rm{T}}}_{1}]}_{t}+{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}{[{{\rm{S}}}_{1}]}_{t}+\frac{3}{4}\gamma {[{\rm{n}}]}_{t}^{2}$$
(6)

where kS is the sum of the rate constants for ISC (kISC), radiative \(({k}_{\text{r}}^{\text{S}})\) and non-radiative \(({k}_{\text{nr}}^{\text{S}})\) decay from S1, where kT is the sum of the rate constants for RISC (kRISC), radiative \(({k}_{\text{r}}^{\text{T}})\) and non-radiative \(({k}_{\text{nr}}^{\text{T}})\) decay from T1, and γ is the Langevin recombination rate. The derivative of the polaron population [n]t can be sufficiently approximated by not distinguishing between the charge of the polaron as

$$\frac{{\rm{d}}}{{\rm{d}}{\rm{t}}}{[{\rm{n}}]}_{{\rm{t}}}=\frac{J(t)}{ed}-\gamma {[{\rm{n}}]}_{{\rm{t}}}^{2},$$
(7)

where J(t) is the current density at time t, d is the thickness of the emission zone and e is the elementary charge.

In normal device operation, the OLED is driven at constant current density (J(t) = Jconst) so the excited-state populations reach a steady state given by equation (8).

$$[{\rm{n}}]=\sqrt{\frac{{J}_{{\rm{c}}{\rm{o}}{\rm{n}}{\rm{s}}{\rm{t}}}}{\gamma ed}}.$$
(8)

The steady-state population of S1 and T1 ([S1] and [T1], respectively) can be obtained by substituting the differential equation for S1 in steady state into the differential equation of T1 steady state

$$\frac{{\rm{d}}}{{\rm{d}}{\rm{t}}}[{{\rm{S}}}_{1}]=-{k}^{{\rm{S}}}[{{\rm{S}}}_{1}]+{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}[{{\rm{T}}}_{1}]+\frac{1}{4}\gamma {[{\rm{n}}]}^{2}=0$$
(9)
$$\Rightarrow [{{\rm{S}}}_{1}]=\frac{{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}[{{\rm{T}}}_{1}]+\frac{1}{4}\gamma {[{\rm{n}}]}^{2}}{{k}^{{\rm{S}}}}$$
(10)
$$\frac{{\rm{d}}}{{\rm{d}}{\rm{t}}}[{{\rm{T}}}_{1}]=-{k}^{{\rm{T}}}[{{\rm{T}}}_{1}]+{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}[{{\rm{S}}}_{1}]+\frac{3}{4}\gamma {[{\rm{n}}]}^{2}=0$$
(11)
$$\Rightarrow [{{\rm{T}}}_{1}]=\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}\frac{{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}[{{\rm{T}}}_{1}]+\frac{1}{4}\gamma {[{\rm{n}}]}^{2}}{{k}^{\text{S}}}+\frac{3}{4}\gamma {[{\rm{n}}]}^{2}}{{k}^{{\rm{T}}}}=\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}}{{k}^{{\rm{S}}}{k}^{{\rm{T}}}}[{{\rm{T}}}_{1}]+\left(\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}}{{k}^{{\rm{S}}}}+3\right)\frac{\gamma {[{\rm{n}}]}^{2}}{4{k}^{{\rm{T}}}}$$
(12)
$$\begin{array}{l}\iff [{{\rm{T}}}_{1}]=\frac{\left(\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}}{{k}^{{\rm{S}}}}+3\right)\frac{\gamma {[{\rm{n}}]}^{2}}{4{k}^{{\rm{T}}}}}{1-\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}}{{k}^{{\rm{S}}}{k}^{{\rm{T}}}}}=\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}+3{k}^{{\rm{S}}}}{4({k}^{{\rm{T}}}{k}^{{\rm{S}}}-{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}})}\gamma {[{\rm{n}}]}^{2}\\ \,\,=\frac{3{k}_{{\rm{r}}}^{{\rm{S}}}+3{k}_{{\rm{n}}{\rm{r}}}^{{\rm{S}}}+4{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}}{4{k}^{{\rm{S}}}({k}_{{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{T}}})+4{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}({k}_{{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{S}}})}\gamma {[{\rm{n}}]}^{2}\end{array}$$
(13)
$$\begin{array}{l}\Rightarrow [{{\rm{S}}}_{1}]=\frac{1}{{k}^{{\rm{S}}}}\left({k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}[{{\rm{T}}}_{1}]+\frac{1}{4}\gamma {[{\rm{n}}]}^{2}\right)=\left({k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}+3{k}^{{\rm{S}}}}{({k}^{{\rm{T}}}{k}^{{\rm{S}}}-{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}})}+1\right)\frac{\gamma {[{\rm{n}}]}^{2}}{4{k}^{{\rm{S}}}}\\ \,\,=\,\left({k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}\frac{{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}+3{k}^{{\rm{S}}}}{{k}^{{\rm{S}}}({k}_{{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{T}}})+{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}({k}_{{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{S}}})}+1\right)\frac{\gamma {[{\rm{n}}]}^{2}}{4{k}^{{\rm{S}}}}\end{array}$$
(14)

By inserting equation (8) in equations (13) and (14) the steady-state populations are given as a function of the current density as

$$\left[{{\rm{S}}}_{1}\right]=\left({k}_{{\rm{RISC}}}\frac{3{k}_{{\rm{r}}}^{\text{S}}+3{k}_{\text{nr}}^{{\rm{S}}}+4{k}_{{\rm{ISC}}}}{{k}^{{\rm{S}}}\left({k}_{{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{nr}}}^{{\rm{T}}}\right)+{k}_{{\rm{RISC}}}\left({k}_{{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{nr}}}^{{\rm{S}}}\right)}+1\right)\frac{1}{4{k}^{{\rm{S}}}}\frac{{J}_{{\rm{const}}}}{{ed}}$$
(15)
$$\left[{{\rm{T}}}_{1}\right]=\frac{3{k}_{{\rm{r}}}^{{\rm{S}}}+3{k}_{{\rm{nr}}}^{{\rm{S}}}+4{k}_{{\rm{ISC}}}}{4{k}^{{\rm{S}}}\left({k}_{{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{nr}}}^{{\rm{T}}}\right)+4{k}_{{\rm{RISC}}}\left({k}_{{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{nr}}}^{{\rm{S}}}\right)}\frac{{J}_{{\rm{const}}}}{{ed}}$$
(16)

Calculation of J 90 for OLED examples

A set of 1,287 kinetic parameters for the equations (5) and (6) was generated, using the permutation of the input variables in Extended Data Table 1, with

$${k}_{{\rm{RISC}}}={k}_{{\rm{r}}}^{{\rm{S}}}{K}_{{\rm{eq}}}\left(\frac{3}{4}+\frac{{k}_{{\rm{ISC}}}}{{k}_{{\rm{r}}}^{{\rm{S}}}}\right)$$
(17)

and a thickness of the emission zone of d = 10 nm, a Langevin recombination rate161 of \(\gamma =6.8\times {10}^{-17}\frac{{\text{m}}^{3}}{\text{s}}\) as well as all other rate constants set to 0.

For the calculation of Extended Data Fig. 3a,b, the bimolecular rate constants were set to the values shown in the figure. J90 was obtained by minimizing equation (18) using the python package scipy162.

$${({\bar{\eta }}_{\text{EQE}}-0.9)}^{2}={\left(\frac{{\eta }_{\text{EQE}}(J)}{{\eta }_{\text{EQE}}^{0}}-0.9\right)}^{2}={\left(\frac{{\eta }_{\text{IQE}}(J)}{{\eta }_{\text{IQE}}^{0}}-0.9\right)}^{2}$$
(18)

where ηIQE(J) is the internal quantum efficiency (IQE) at current density J considering annihilation processes and \({\eta }_{\text{IQE}}^{0}\) is the IQE without considering annihilation processes. Both are given by

$${\eta }_{{\rm{I}}{\rm{Q}}{\rm{E}}}=({k}_{{\rm{r}}}^{{\rm{S}}}[{{\rm{S}}}_{1}]+{k}_{{\rm{r}}}^{{\rm{T}}}[{{\rm{T}}}_{1}])\frac{\gamma d}{J}$$
(19)

For \({\eta }_{\text{IQE}}^{0}\), the singlet population [S1] and triplet population [T1] were obtained from equations (15) and (16), respectively.

For ηIQE(J), [S1] and [T1] were obtained by minimizing the set of differential equations (20) and (21) with [n] given by equation (7), using the python package scipy161,162.

$$\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}{\rm{t}}}[{{\rm{S}}}_{1}]=-\mathop{\underbrace{({k}_{{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{S}}}+{k}_{{\rm{I}}{\rm{S}}{\rm{C}}})}}\limits_{={k}^{{\rm{S}}}}[{{\rm{S}}}_{1}]+{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}}[{{\rm{T}}}_{1}]+\frac{1}{4}{\gamma }_{{\rm{T}}{\rm{T}}}{[{{\rm{T}}}_{1}]}^{2}\\ \,\,\,\,-{\gamma }_{{\rm{S}}{\rm{T}}}[{{\rm{T}}}_{1}][{{\rm{S}}}_{1}]+\frac{1}{4}\gamma {[{\rm{n}}]}^{2}\end{array}$$
(20)
$$\frac{{\rm{d}}}{{\rm{d}}{\rm{t}}}[{{\rm{T}}}_{1}]=-\mathop{\underbrace{({k}_{{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{n}}{\rm{r}}}^{{\rm{T}}}+{k}_{{\rm{R}}{\rm{I}}{\rm{S}}{\rm{C}}})}}\limits_{={k}^{{\rm{T}}}}[{{\rm{T}}}_{1}]+{k}_{{\rm{I}}{\rm{S}}{\rm{C}}}[{{\rm{S}}}_{1}]-\frac{5}{4}{\gamma }_{{\rm{T}}{\rm{T}}}{[{{\rm{T}}}_{1}]}^{2}+\frac{3}{4}\gamma {[{\rm{n}}]}^{2}$$
(21)

Calculation of target value

The optical power flux Φ leaving an OLED relates to the current density J as

$${\varPhi }_{{\rm{e}}}=\int {\varPhi }_{{\rm{e}}}(\lambda ){\rm{d}}\lambda =\int \frac{hc}{\lambda }I(\lambda ){\eta }_{{\rm{EQE}}}\frac{J}{e}{\rm{d}}\lambda =\frac{hc}{e}{\eta }_{{\rm{EQE}}}\,J\int \frac{1}{\lambda }I(\lambda ){\rm{d}}\lambda $$
(22)

where I(λ) is the relative intensity of the OLED at wavelength λ, ηEQE is the ratio of photons leaving the OLED to the number of electrons flowing around the electrical circuit (EQE).

The total luminous flux ΦV can be calculated from the optical power flux using the photonic sensitivity curve V(λ) as

$${\varPhi }_{{\rm{V}}}={K}_{{\rm{m}}}\int {\varPhi }_{{\rm{e}}}(\lambda )V(\lambda ){\rm{d}}\lambda ={K}_{{\rm{m}}}\frac{hc}{e}{\eta }_{{\rm{EQE}}}\,J\int \frac{1}{\lambda }I(\lambda )V(\lambda ){\rm{d}}\lambda $$
(23)

where Km = 683 lm W−1 is a fudge factor called the peak response.

Under the assumption of Lambertian emission, the luminance LV of the OLED is then given as follows.

$${L}_{{\rm{V}}}=\frac{{\varPhi }_{{\rm{V}}}}{{\rm{\pi }}}={K}_{{\rm{m}}}\frac{hc}{{\rm{\pi }}e}{\eta }_{{\rm{EQE}}}\,J\int \frac{1}{\lambda }I(\lambda )V(\lambda ){\rm{d}}\lambda $$
(24)

Therefore, the current density required to generate a given luminance by an OLED with a given normalized spectrum and given EQE is given as follows.

$$J=\frac{1}{{K}_{{\rm{m}}}}\frac{{\rm{\pi }}e}{hc}\frac{{L}_{{\rm{V}}}}{{\eta }_{{\rm{EQE}}}}{\left[\int \frac{1}{\lambda }I(\lambda )V(\lambda ){\rm{d}}\lambda \right]}^{-1}$$
(25)

For the calculation of the target value, we have taken three assumed spectra for red, green and blue with a Gaussian shape and a full-width at half-maximum of 45, 30 and 15 nm, respectively. The centre wavelength was selected so that the colours of the three spectra are as close as possible to the primary colours of the BT.2020 standard in the CIE 1931 colour space, which are given by the coordinates (0.708, 0.292), (0.170,0.797) and (0.131,0.046), respectively46.

The calculation was performed at an EQE of 22.5% for Lv = 1,000 cd m−2, indicating a maximum EQE of 25%. The correlation for J90 to the FOM is taken from the simulated relationship shown in Extended Data Fig. 4a for γTT = 10−13 cm3 s−1 and γST = γTP = 0 as follows.

$$\log \,\left(\frac{{J}_{90}}{1\,{{\rm{Am}}}^{-2}}\right)=2\,\log \,\left(\frac{{k}_{{\rm{r}}}^{{\rm{S}}}{K}_{{\rm{eq}}}}{1\,{{\rm{s}}}^{-1}}\right)-8.3764$$
(26)
$$\iff {k}_{{\rm{r}}}^{{\rm{S}}}{K}_{{\rm{e}}{\rm{q}}}={10}^{\frac{\log \left(\frac{{J}_{90}}{1{{\rm{A}}{\rm{m}}}^{-2}}\right)+8.3764}{2}}{{\rm{s}}}^{-1}$$
(27)