Various exciton annihilation processes are known to impact the efficiency roll-off of organic light emitting diodes (OLEDs); however, isolating and quantifying their contribution in the presence of other factors such as changing charge balance continue to be a challenge for routine device characterization. Here, we analyze OLED electroluminescence resulting from a sinusoidal dither superimposed on the device bias and show that nonlinearity between recombination current and light output arising from annihilation mixes the quantum efficiency measured at different dither harmonics in a manner that depends uniquely on the type and magnitude of the annihilation process. We derive a series of analytical relations involving the DC and first harmonic external quantum efficiency that enable annihilation rates to be quantified through linear regression independent of changing charge balance and evaluate them for prototypical fluorescent and phosphorescent OLEDs based on the emitters 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4*H*-pyran and platinum octaethylporphyrin, respectively. We go on to show that, in most cases, it is sufficient to calculate the needed quantum efficiency harmonics directly from derivatives of the DC light versus current curve, thus enabling this analysis to be conducted solely from standard light-current-voltage measurement data.

Bimolecular annihilation reactions involving energy transfer from one exciton to another or to a charge carrier are an important factor for both efficiency roll-off^{1–4} and operational degradation of organic light emitting diodes (OLEDs).^{5–8} Annihilation loss in OLEDs is frequently quantified by modeling their external quantum efficiency (EQE) roll-off versus current density;^{1,2,4,9,10} however, it is notoriously difficult to identify which mode of annihilation is dominant (e.g., triplet-triplet vs. triplet-polaron) and subsequently to disentangle its magnitude from competing roll-off processes such as declining charge balance that manifest with a similar (often unknown) functional dependence.^{3,4,11} This uncertainty makes it difficult to determine the presence and impact of annihilation in the course of routine device testing and therefore stands as an important characterization need for continuing OLED development.

Here, we introduce a simple analytical method to directly identify and extract OLED annihilation rates from standard light-current-voltage (LIV) measurement data. The foundation of this approach lies in a frequency domain EQE analysis and is most easily understood in analogy to impedance spectroscopy, where in this case both the current and electroluminescence intensity are measured using a lock-in amplifier at different harmonics of the sinusoidal dither superimposed on the DC device bias [see Fig. 1(a)]. In the presence of annihilation, the relationship between recombination current and light output (proportional to exciton density) becomes nonlinear, thereby mixing the different EQE harmonics in a manner that depends uniquely on the type and magnitude of annihilation as illustrated schematically in Fig. 1(b).

While OLED impedance spectra are generally difficult to interpret due to complicating factors such as space charge accumulation, unknown trap distributions, and injection/heterojunction barriers,^{12,13} the relationship between electroluminescence and current density is accurately described via a simple rate equation. Consider applying a voltage, $V=V0+V1ei\omega t$, to an OLED under the usual condition that the dither is much smaller than the DC bias ($V1\u226aV0$). Because the IV relationship is nonlinear (i.e., diode-like), the resulting current density will contain additional harmonics, $J=J0+J1ei\omega t+J2e2i\omega t+\cdots $, as will the emitted electroluminescence intensity, $\u2009L=L0+L1ei\omega t+L2e2i\omega t+\cdots $, which has units of photons emitted per unit area per second. Assuming for simplicity a uniform recombination zone of width, $a0$, in the OLED emissive layer (EML), the exciton density, $X$ (e.g., triplets in a phosphorescent OLED), can be described by the rate equation^{1,2}

Here, $Jf$ accounts for current that does not contribute to exciton formation (i.e., recombination at traps or charge imbalance leakage current), $\tau $ is the natural exciton lifetime, $q$ is the electronic charge, $P$ is the polaron density, and $kXX$ and $kXP$ are the exciton-exciton (XXA) and exciton-polaron annihilation (XPA) rate coefficients, respectively. The exact nature of the excitons and polarons (e.g., singlet/triplet, host/guest exciton, electrons, or holes) is left general at this stage.

Because the light emission is directly proportional to the exciton density through the photoluminescence quantum yield, $\eta PL$, and out-coupling efficiency, $\eta OC$, via $L=a0\eta PL\eta OC\tau \u22121X\u2261\alpha X$, Eq. (1) can be rewritten and expanded harmonically in terms of $L$ and $J$. Focusing initially on the simplest case, where XXA is dominant and XPA is negligible ($kXP\u21920$), the quadratic annihilation term leads to harmonic mixing and contributions of the form $L02+L12/2+L22/2+\cdots \u2009$ at DC and $2L0L1+L1L2+\cdots $ at the first harmonic (1$\omega $). Retaining only the lowest order term for each frequency and dividing by the respective current density yields

at DC and

at the first harmonic. Here, $EQE0=qL0/J0\u2009$ and $EQE1=qL1/J1$ are the EQEs defined at each respective frequency and the terms $\varphi B$ are the corresponding charge balance factors that represent the fraction of total current at each harmonic that recombines to form excitons.

If the modulation frequency is low compared to the natural exciton decay rate ($\omega \u226a\tau \u22121$), then the $i\omega $ phase term in Eq. (3) is negligible and the ratio of $EQE0$ to $EQE1$ becomes

where $\gamma 10=\varphi B1/\varphi B0$ is defined as the $1\omega $ to DC charge balance ratio and is approximately unity when $\varphi B0$ varies slowly with current density as discussed below. In this case, Eq. (4) predicts that the EQE ratio will be equal to one at low brightness and will asymptote to two at high brightness when annihilation becomes dominant, providing a simple qualitative measure of how significant annihilation is at a given brightness. Quantitative extraction of $kXX$ is subsequently facilitated by rearranging Eq. (4) as

which predicts that plotting the difference $EQE0\u2212EQE1$ versus $(2EQE1\u2212EQE0)L0$ will result in a linear relationship with slope equal to $kXX\tau 2/a0\eta PL\eta oc$ when $\gamma 10\u22481$.

If the modulation frequency is also slow compared with the electrical response of the device (in practice ≲kHz modulation frequencies relative to the ∼$\mu $s scale turn-on time of the OLED),^{14,15} then DC steady-state conditions are reached at every moment in the slowly varying modulation. In this quasi-static regime, and in the limit of small modulation amplitude, $EQE1$ is equivalently given by the derivative of the DC light vs. current curve, that is, $EQE1=q[dL0/dJ0]\u2261dEQE$. Thus, the harmonic analysis above can be performed using standard LIV measurement data provided the point spacing is sufficient to accurately resolve $dEQE$.

Figure 2 explores these predictions numerically for a realistic OLED dominated by XXA in the presence of changing charge balance. A typical power law IV characteristic is assumed with recombination ($JR0$) and leakage current ($Jf0$) components divided according to the fictitious charge balance simulated in Fig. 2(a) (right-hand scale). The light output is calculated according to Eq. (1) at DC assuming $\eta PL=0.4$, $\eta oc=0.2$, $a0=5$ nm, $\tau =88$ *μ*s, and $kXX=4\xd710\u221214$ cm^{3} s^{−1}, typical for a classic phosphorescent OLED based on platinum octaethylporphyrin (PtOEP).^{1,16,17} In Fig. 2(a), red lines indicating the $1\omega $ EQE and charge balance quantities have been calculated both via direct numerical Fourier transform from Eq. (1) using a dithered bias voltage and as derivatives of the DC LIV (i.e., as $dEQE$), which were verified to be equivalent.

Figure 2(b) displays the $EQE0/EQE1$ ratio for the simulated device together with the reference case of unchanging charge balance (dashed line). Extension of the EQE ratio outside of the annihilation-relevant y-axis range between one and two occurs when $\gamma 10\u22601$, providing a clear visual indication that annihilation is not the dominant loss mechanism at these current densities. Despite the clear impact of changing charge balance, Fig. 2(c) shows that re-plotting the simulation data in the form of Eq. (5) still yields a clearly recognizable linear region from which it is possible to obtain an accurate extraction of $kXX$ within 10% of the true value defined by the $\gamma 10=1$ reference.

The robustness of this extraction method in the presence of changing charge balance can be understood more clearly from reference to the quasi-static derivative definition of $\gamma 10$ via $\varphi B1=1\u2212dJf0/dJ0$. From this, $\gamma 10=1$ whenever $dJf0/dJ0=Jf0/J0$ (i.e., when $Jf0$ is linearly related to $J0$), which defines a region of locally unchanging charge balance ($d\varphi B0/dJ=0$). This condition is always met when the charge balance reaches a peak, and thus an approximately linear slope still manifests in this region to reveal the magnitude of annihilation in Fig. 2(c). Further inspection of Eq. (5) shows that, to first order, deviation of $\gamma 10$ from unity translates to a y-intercept in the fit line, with $\gamma 10<1$ resulting in a positive intercept that marks the onset of declining charge balance.

Once the bracketed annihilation term in Eq. (4) is known from the linear fit of Eq. (5), the “annihilation-only” component of the EQE ratio [the green dashed line in Fig. 2(b)] can be reconstructed, enabling $\gamma 10$ to be determined as a function of current density by dividing it by the actual EQE ratio [i.e., by dividing the green dashed line by the black line in Fig. 2(b)]. Using the quasi-static derivative definition of $\gamma 10$ subsequently leads to a first order differential equation

that can be used to solve for $Jf0$ by numerical integration of $J0$ and the (now known) function for $\gamma 10$. This requires an initial condition value for $Jf0$ which, although not rigorously known, should be well-approximated in OLEDs with near-unity internal quantum efficiency^{18} as $Jf0IC\u22480$ (i.e., perfect charge balance) when $\gamma 10$ crosses/or is equal to unity.

The same methodology can be applied to the situation when XPA is the dominant annihilation process. In this case, it can be argued on general grounds (see Sec. S1 of the supplementary material^{19}) that the relationship between exciton and charge carrier density in the recombination zone can be expressed in the form $X=CP\beta $, where $\beta \u22651$ and $C$ is a constant. Proceeding in the same manner as above, it is straightforward to show for $\beta =2$, and subsequently to verify numerically for general $\beta $, that the EQE ratio is given by

and can be linearized for $kXP$ extraction as

Equations (7) and (8) are the XPA analogs of Eqs. (4) and (5) for XXA and therefore XPA device analysis would proceed similar to that described above. In the case of space charge limited (SCL) transport^{20} assumed in previous XPA analyses,^{2,9}$\beta =2$ as described in the supplementary material,^{19} and thus Eq. (7) predicts that the EQE ratio of an OLED dominated by XPA would asymptote toward a value of $3/2$, in contrast with the value of 2 for XXA.

Initial testing of these predictions was carried out on classic double heterostructure fluorescent and phosphorescent OLEDs shown in the inset of Fig. 3(a) using emissive layers doped, respectively, with 2 wt. % DCM laser dye [4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4*H*- pyran] or 8 wt. % PtOEP. Efficiency roll-off in these devices has been studied previously,^{1,3} where it was shown that the PtOEP device is strongly affected by triplet-triplet XXA, whereas annihilation is negligible for its DCM counterpart. Devices were fabricated via vacuum thermal evaporation on pre-patterned indium tin oxide anodes with LiF (0.7 nm)/Al (80 nm) cathodes (4 mm^{2} active device area) and were subsequently packaged in a N_{2} glove box using a thin layer of ultraviolet-curable adhesive (Norland, NOA 63) and a glass cover slip.

Figure 3(a) shows the external quantum efficiency ($EQE0$) for each OLED and Fig. 3(b) plots their respective $EQE0$ and $dEQE$ in the form of Eq. (5). The PtOEP device exhibits a linear region similar to that predicted in Fig. 2(c) for current densities ranging approximately 0.1–4 mA/cm^{2}. At higher currents, the data display a characteristic bend back toward the ordinate axis, indicating that another roll-off mechanism becomes dominant. By contrast, the DCM data are never quite linear and display a continuous downward curvature. Attempting an approximate linear fit in spite of the curvature leads to an unphysical singlet-singlet XXA rate coefficient that is several orders of magnitude too large, indicating that XXA is not significant for these DCM devices over the measured current density range.^{3} In addition to DC characterization, the same analysis was performed using the dithered AC EQE lock-in amplifier experimental configuration shown in Fig. 1(a) and reasonable agreement between the two methods was confirmed; the comparison is detailed in Sec. S2 of the supplementary material.^{19}

Assuming $\eta OC\u22480.2$, the slope of the fitted PtOEP linear region in Fig. 3(b) together with its phosphorescent lifetime $\tau =88$ *μ*s [Ref. 16] and quantum yield $\eta PL=0.4$ [Ref. 17] results in $kXX/a0=(1.6\xb10.2)\xd710\u22128$ cm^{2} s^{−1}, which is a figure of merit indicative of the annihilation intensity in the device. If the recombination zone is subsequently taken to be the emissive layer width ($a0=10$ nm), then this result yields $kTTA=2kXX=(3.2\xb10.4)\xd710\u221214$ cm^{3} s^{−1}, in agreement with the triplet-triplet annihilation (TTA) coefficient found previously for this device (the factor of 2 is inserted for consistency with the notation of previous work).^{1}

Figure 4(a) shows the EQE ratio for the PtOEP device together with the annihilation-only contribution [dashed green line cf. Fig. 2(b)] derived from the linear fit of $kXX/a0$. It is evident from extension of the measured EQE ratio above two that, while XXA is significant in this device, declining charge balance also becomes a factor at high current densities of >10 mA/cm^{2}, which is confirmed by the rising shoulder of N,N′-di(1-naphthyl)-N,N′-diphenyl-(1,1′-biphenyl)-4,4′-diamine (NPD) emission shown in Fig. 4(b).^{3} Determining $\gamma 10$ from the quotient of the measured EQE ratio and the extracted XXA contribution, and assuming an initial condition $Jf0IC=0$ when $\gamma 10$ crosses unity, numerical integration of Eq. (6) results in the normalized charge balance of the device ($\varphi B0$) plotted on the right hand scale.

Joule heating or field-induced exciton dissociation^{21} could also play a role in the efficiency roll-off. However, because field-induced dissociation mainly affects the exciton formation process (i.e., before a Coulombically correlated carrier pair has recombined to form a tightly bound exciton), its distinction from charge imbalance is ultimately superficial since one cannot distinguish between a carrier pair that forms and is dissociated versus one that never forms at all. The effect of field-induced dissociation is thus implicitly included in $\varphi B0$. Alternatively, Joule heating effects could be identified from pulsed LIV measurements.

Although the expressions derived above assume a uniform recombination zone, accounting for a spatially varying recombination zone does not materially change the analysis. Its primary effect is to change the slope of the linearized EQE plot (reflective of an average annihilation intensity) and to soften the transition between the asymptotes in the EQE ratio plot (their values remain unchanged). A recombination zone that changes significantly with current density would lead to curvature in the linearized EQE plot, making extraction of the annihilation intensity difficult. However, it still would not violate the asymptotes of the annihilation-limited EQE ratio, making this type of plot a robust identifier of devices in which annihilation is *not* the dominant cause of roll-off.

It is important to note that the harmonic EQE analysis developed here does not produce any “new” information not already contained in a standard LIV measurement. Rather, it simply repackages the data in a form that is more sensitive and conducive to identifying the effects of annihilation. Mixing among higher EQE harmonics (e.g., relationships involving $EQE2$, $\gamma 20$, and so on) should provide additional constraints on the type and magnitude of annihilation and can, in principle, be determined from higher order LIV derivatives, $dnL/dJn$. However, in practice, measurement noise and finite point spacing of the DC LIV amplify fluctuations in the higher derivatives, preventing them from being recovered accurately. The value in direct frequency domain EQE harmonic analysis [i.e., as in Fig. 1(a)] is thus that the lock-in amplifier more effectively averages and recovers higher harmonics from the noise. The disadvantage is that the measurement takes longer to perform.

Looking forward, the added information contained in higher EQE harmonics may be sufficient to reliably deconvolve the magnitudes of two different annihilation processes such as, e.g., triplet-triplet and triplet-polaron annihilation when both are significant in a device. This goal is particularly important for understanding roll-off in state-of-the-art phosphorescent OLEDs known to be affected by both processes^{2,9} and is currently the subject of ongoing work. Similarly, increasing the modulation frequency beyond the quasi-static regime to obtain additional phase information in the form of frequency-dependent complex EQE spectra could help to characterize dynamic processes in the emissive layer such as charge trapping and exciton formation on emissive dopants. Finally, the analysis here should find straightforward extension to inorganic LEDs, where it may help to resolve ongoing questions about the role of Auger recombination in InGaN/GaN LED efficiency droop.^{22,23}

This work was supported in part by the Dow Chemical Company and by the Air Force Office of Scientific Research Young Investigator Program.