Review—The Physics of Recombinations in III-Nitride Emitters

at (attributed to Coulomb interaction) ﬁeld

III-nitrides have historically been a surprising material, emitting light with a high efficiency despite challenging properties such as a high defect density and the existence of strong polarization fields which separate electrons and holes. Early on, improvements in III-nitride efficiency were chiefly achieved by progress in epitaxy and material quality. As these material science challenges were addressed, the corresponding device physics became easier to study, with its understanding also becoming more relevant to performance improvements.
Today, the peak internal quantum efficiency (IQE) of the best violet and blue light-emitting diodes (LEDs) exceeds 90%. 1 Such devices offer little room for performance improvement, but on the other hand offer an ideal testbed to study the physics of recombinations in highquality samples. At other wavelengths, there are still vast opportunities for performance improvements -especially in the yellow-to-red and deep-ultra-violet ranges -with limitations stemming from an interplay of material constraints and physical effects. It is the hope of the authors that this Review will summarize the state of the art in understanding the recombination physics of III-nitride emitters, and provide useful guidance to tackle these future challenges.
This Review focuses on active region recombinations in visible LEDs with InGaN quantum wells (QW); it leaves aside the important discussion of carrier transport effects. It is composed of two parts. In the first section, we discuss techniques for measuring recombination dynamics. In the second section, we present key findings on recombination physics. The Review focuses on results obtained by our group over the past decade, while placing them in the context of overall work in the field.

Measuring Recombination Dynamics
Motivation -why measure dynamics?-The experimental study of recombinations in semiconducctor light emitters relies chiefly on measuring two quantities: the IQE and the dynamics of emission. IQE metrology is a complex task, whose detailed discussion is outside the scope of this Review. Its determination can be attempted by various techniques. A popular approach is low-temperature photoluminescence, 2 although the underlying assumption of nearunity IQE at cryogenic temperature has been shown to fail in some cases, making this technique unreliable. 3 It has also been proposed to extract the absolute IQE from a shape analysis of the uncalibrated IQE curve; 4 as we will show, however, the underlying assumptions are often not met. Perhaps the most robust approach to determine IQE is from calibrated external quantum efficiency (EQE) measurements in LEDs whose extraction efficiency is known accurately -this is possible if the geometry is simple and planar, 5 or in more complex devices by using a sophisticated light extraction model. 1,6,7 z E-mail: aurelien.david@polytechnique.org Once the IQE is known, it is commonly analyzed by the well-known ABC model:

IQE =
Bn 2 An + Bn 2 + Cn 3 , [1] with n the carrier density, A the Shockley-Read-Hall (SRH) coefficient, B the radiative coefficient and C often identified as an Auger coefficient. This model often provides good qualitative guidance regarding the relative influence of recombination channels on IQE. For example, it is generally true that a variation in SRH defects affects the low-current side of the IQE curve; that an increase in the radiative rate broadens the IQE curve and raises the peak IQE; and that a reduction in carrier density (e.g. thanks to better carrier spreading) delays the onset of efficiency droop. These qualitative trends are illustrated in Fig. 1. Yet, the quantitative insight gained from the ABC model is imperfect. Indeed, the model is merely a textbook approximation of recombination behavior -as this Review will show, actual recombinations show significant departure from such behavior. Besides, ABC fitting relies on a number of uncontrolled assumptions which lead to very large error bars on the resulting coefficients, as discussed in Ref. 8.
Therefore, to get further insight into recombination physics necessitates additional information. Specifically, it is desirable: to  . The thin dashed lines shows that a fit by a standard ABC model can yield excellent agreement with the TRPL trace, while severely misestimating the IQE and the corresponding breakdown of dynamics between radiative and non-radiative paths. disentangle the contribution from radiative and non-radiative processes to IQE; to gain experimental access to the carrier density n; and to analyze the IQE curve without forcing a priori the limiting assumptions of the ABC model in order to (i) verify the validity of this model and (ii) if warranted, derive the coefficients A, B and C. All these requirements can be fulfilled by proper measurements of the carrier dynamics (i.e. their recombination lifetime). This has prompted a prolonged effort in the III-nitride community to achieve such measurements. Therefore, we begin with a discussion of lifetime metrology, and offer comments on the limitations of some commonlyused approaches.
Ultrafast time-resolved photoluminescence (TRPL).-In TRPL, an ultrafast laser excites the sample and the time-dependent luminescence decay I (t ) is collected. The TRPL method is appealing in principle, as all the information on dynamics is encoded in a single decay curve. However, this conceptual advantage also turns out to present a significant practical drawback: extracting the underlying information from this decay curve is quite difficult. The decay has a non-exponential time dependence due to the contributions from various channels: SRH, radiative and droop. 9 A phenomenological bi-exponential fit is often employed, [10][11][12] although there is often no underlying physical model justifying this fit, leaving ambiguity in the interpretation of the resulting lifetimes.
Instead, a proper fit of the decay curve requires a physical model of the recombination rate. For instance, in a simple AB model (considering SRH and radiative recombinations only), the decay law is I (t ) = Bn 2 with: . [2] A more realistic case is the full ABC model, whose decay can also be expressed analytically or modeled numerically, then fitted to the experimental decay curve to yield the recombination parameters (A, B, C). 13 Unfortunately, such fitting cannot account for any departure from the assumed model. As an example, Fig. 2 shows the experimental properties of a high-efficiency single-quantum well (SQW) sample having carrier-dependent recombination coefficients (as will be discussed further below), and shows that a fit by an ABC model can be misleading. In particular, if the sample's IQE is ignored, the fit to the decay curve can be excellent despite very poor estimates of the recombination coefficients.
Corrective physical effects (e.g. field screening and phase-space filling) can be included in the decay model to improve the accuracy of the fit. 14,15 Nonetheless, this procedure only accounts for whichever ef-fects are explicitly included in the model. As we will see, this approach is challenging due to the variety and complexity of such effects.
Alternatively, it has been proposed to use the instant lifetime τ 0 at the beginning of the decay (i.e. by treating the decay as monoexponential at t = 0) as an indicator of the recombination time. 16 This approach has the merit of yielding a well-defined quantity. However, the physical meaning of the resulting lifetime is usually not clarified. In fact, if the radiative recombination is exactly bimolecular, analysis reveals that τ −1 0 = 2G/n 0 , where G is the total recombination rate and n 0 the instant carrier density. In the case of an ideal ABC model, Thus τ 0 is indeed indicative of recombination dynamics, but its proper interpretation requires an estimate of n 0 . Interestingly, if radiative recombination dominates other channels (e.g. in high-quality samples near peak efficiency) then τ −1 0 = 2Bn 0 , which is exactly the differential radiative lifetime (see below). Conversely, at low density, τ 0 is half the SRH lifetime 1/A.
Besides the difficulty of fitting the decay curve, other complications of the TRPL technique include the imperfect knowledge of the initial carrier density, and the diffusion dynamics of excited carriers to the active region.
In summary, extracting accurate information from TRPL curves is a significant challenge, whose complexities are frequently overlooked. In our view, this motivates the use of differential lifetime techniques as an alternative.
Small signal / Differential lifetime.-Small-signal measurements proceed by exciting the device with a constant signal of large amplitude, and adding to it a time-dependent small signal. Given a recombination rate G(n) and a small-signal population dn, the differential recombination lifetime is defined as τ −1 = dG/dn (i.e. it is the time it takes for the population dn to cause an extra recombination dG). To derive τ, the response to the small-signal excitation is measured; this response can be optical (variation in light output) or electrical (variation in current). In the most common implementation of this scheme, the large and small signals are provided by electrical injection, usually as a sum of a large DC bias and a small sine AC bias. The AC bias leads to an AC modulation of the optical output, whose phase delay and/or amplitude are measured. This scheme is also referred-to as time-resolved electroluminescence (TREL). Often, a network analyzer provides both the source signal and the measurement (which can either be the optical response from a photodetector, or the electrical response as an impedance measurement).
Differential lifetime measurements offer several advantages. First, no assumptions on the recombination model are required, as τ is a purely experimental quantity, which can be compared to a recombination model a posteriori. Second, once τ is known, one can compute the carrier density as: where G is related to the injected current density by J = et G (with e the electron charge and t the active region thickness). Eq. 3 gives access to the carrier density (a quantity which is otherwise difficult to estimate) without any assumption. In principle, the integral in Eq. 3 runs from zero to the current density of interest. In practice however, the measurement of τ is only available down to a minimum current density J min ; to deal with this, one can assume that τ is constant below J min (this is valid if the measurement reaches the SRH-dominated regime), or use a more accurate approximation as discussed in Ref.

17.
Such measurements were first introduced to characterize homojunction devices in conventional III-V's. In this simple case, the only relevant lifetime is the carrier recombination at the junction, and the response can be expressed as: This response describes both the optical and electrical signalseither can therefore be collected to measure τ. In some cases, τ is simply determined from the −3 dB roll-off frequency of either signal rather than by fully fitting the frequency-dependent response. Eq. 4 can be seen as the transfer function of a single-pole system, i.e. it is equivalent to an RC circuit, with τ = RC, R the junction's dynamic resistance and C its capacitance.
Subsequently, differential lifetime measurements were applied to more advanced heterostructures, including double-heterostructure and quantum-well lasers. There, dynamics are more complicated since they combine carrier-transport effects (e.g. diffusion through the junction, capture by QWs...) and recombinations. 18 This is manifested by a more complex electrical and optical response, departing from that of Eq. 4. A number of studies through the 80's and 90's attempted to model these effects. [19][20][21][22][23][24] The most successful model in our opinion is that of Refs. 22,23, which includes the effects of junction capacitance, carrier capture, and thermionic emission; it accurately reproduces the electrical response and temperature-dependence of GaAs lasers below threshold.
A variation of the TREL scheme uses a square pulse as the small signal, and measures the decay of luminescence versus time after the pulse is switched off. This scheme is fully analogous to the one previously discussed -they are respectively the frequency-domain and time-domain version of the same measurement. Therefore, in a single-pole case (Eq. 4) the time decay is exponential, whereas in the presence of transport effects it is more complex. However, any deviations from an exponential behavior are difficult to observe in the time decay of a small signal, whereas they are obvious when decomposed in the frequency domain (as will be shown in Fig. 3). This makes frequency-domain measurements preferable to detect transport effects.
Starting in the 2000's, small-signal measurements were applied to III-nitrides. However, for many years, these studies ignored the impact of transport effects and used the single-pole analysis of Eq. 4 to analyze results. 4,17,[25][26][27][28] Recently some of the present authors investigated the detailed TREL frequency response of III-nitride LEDs, finding pronounced evidence of transport effects. 29 To provide accurate characterization, the electrical and optical responses were measured simultaneously. Both displayed a complex multi-pole behavior, shown in Fig. 3. This data was successfully interpreted with an extension of the transport model of Refs. 22,23. In particular, our study in Ref. 29 revealed dynamics related to the capture-and-escape physics of carriers by the QW. At high temperature, the TREL data shows clear evidence of thermionic carrier escape from the QW -similar to observations in conventional III-V devices. At low temperature on the other hand, the dynamics reveal the existence of a second carrier capture process (in addition to the conventional ultrafast phonon-assisted capture, on a timescale of a few ps) which competes with thermionic emission. We proposed that this could be caused by an electron-electron Coulomb scattering process. Subsequently, theoretical investigation of this Coulomb-induced capture was performed in Ref. 30, with good agreement with our experimental observations. Such measurements have since been successfully replicated by other groups, confirming that a full recombination-transport model is necessary to properly fit TREL data. 31,32 Even so, the recombinationtransport model we proposed is only suited for SQW samples. In more complex epitaxial structures (e.g. multiple-quantum wells with interwell transport effects, or electron blocking layers causing large carrier accumulation), we have observed an even more convoluted frequency response, whose study would call for further improvements to the transport model.
It should be stressed that ignoring transport effects can in some cases have a significant impact on the quantitative interpretation of TREL measurements. Many studies simply measure the −3 dB cutoff frequency of the optical or electrical response, assuming it is indicative of the recombination lifetime. Using the electrical roll-off can be especially misleading, as it may be dominated by the junction capacitance (especially if the capacitance is high). Conversely, the optical roll-off approximately characterizes a composite lifetime given by: τ −1 = τ −1 rec + τ −1 esc (with τ rec the recombination lifetime and τ esc the thermionic escape time). Therefore, its value can be strongly affected by thermionic escape when τ esc is commensurate with τ rec . In our samples, we have found this to be the case at elevated temperature (T ∼ 100 C) -indicating that studies of high-temperature dynamics can be particularly compromised by ignoring transport effects. This is illustrated in Fig. 3c.
Variations on the conventional TREL scheme exist. For instance, in an attempt to circumvent transport effects, it has been proposed to combine a constant large-signal electroluminescence (EL) bias with a small-signal resonant photoluminescence (PL) excitation, and to monitor the time response of the PL signal. 33,34 Unfortunately, this 'hybrid EL-PL' scheme still suffers from transport-related artifacts. Namely, photo-generated carriers can escape from the active region. This artifact is especially likely at low power density, where the recombination lifetime is slow and carrier escape can be predominant -as revealed by separate biased-photoluminescence studies in Ref. 35. Therefore, the low-current escape time can be mistaken for the SRH lifetime, as confirmed in Refs. 36,37. This artifact led to a report of a wavelengthindependent SRH lifetime, which in turn was taken as a basis for a model explaining the green gap. 38 In our opinion, this model is questionable as it is based on unreliable data. At higher power density on the other hand, the EL-PL method is less likely to suffer from carrier-escape artifacts, because the recombination time becomes fast and precludes significant carrier escape. Accordingly, this method has recently been used to study droop in deep-UV AlGaN LEDs, and has produced suggestive evidence that Auger scattering is also responsible for droop in this wavelength range. 39 In summary, conventional TREL measurements have been used in a number of studies of III-nitride recombinations. However, many of these should be taken with caution as they ignore transport effectsan oversight which can lead to large inaccuracies, especially at low current or high temperature. Nonetheless, with a proper model and a suitable sample structure, successful data extraction from the TREL technique remains possible, as further illustrated in Appendix A.

Optical differential lifetime (ODL).-
The complications associated with carrier transport effects in conventional small-signal measurements call for a different technique. Recently, the present authors introduced a small-signal measurement using an all-optical excitation scheme: the QW is excited resonantly by a laser whose output is the sum of a continuous large signal and a small sine signal. 40 This scheme circumvents the aforementioned injection effects, enabling a direct measurement of the active region's recombination dynamics. It also provides high dynamic range, with lifetime measurements accessible over four decades of optical current density.
We found that the epitaxial structure should be optimized to enable proper ODL measurements. First, the QW should be placed near the center of a pin junction, so that it is neither p-nor n-doped and intrinsic recombination dynamics can be probed (as will be shown in the following section). Second, the intrinsic region should be thick enough to reduce the junction field and avoid field-induced carrier sweep-out at low excitation density. In practice, an intrinsic region of 200 nm is suitable.
Thanks to a calibrated IQE measurement, the total recombination rate R can be separated into its radiative and non-radiative parts: By combining these with the carrier density n derived from the lifetime (Eq. 3), three recombination quantities can be calculated: [5] These quantities assume a simple interpretation in the framework of the ABC model. At low current density, a should converge to the SRH coefficient A; whereas b and c should be current-independent quantities, respectively equal to the B and C coefficients (i.e. the radiative and droop currents exactly have n 2 and n 3 dependence). As we will see hereafter, b and c actually display a carrier dependence, which constitutes a departure from the ABC model and encodes some important physics. Nonetheless, we will identify them with B and C for convenience.

Recombination Physics -Selected Results
This section summarizes some important results on the recombination physics of visible InGaN LEDs. The majority of these are derived from ODL measurements, and performed on high-quality samples grown by metal-organic chemical vapor deposition on bulk GaN substrates. SQW samples are studied, in order to preclude inter-well transport effects and reduce the uncertainty on the active region's thickness; nonetheless, the underlying recombination physics of course applies equally well to multiple-QW (MQW) samples.
Radiative recombination.-As a preliminary to presenting results on radiative recombination, we first discuss a challenge in measuring dynamics accurately: the influence of doping. Many III-nitride structures feature doped active layers -either intentional doping, or more often modulation doping since the quantum well is very close to n-doped layers. A typical example is a so-called PL test structure (Fig. 4a), for which modulation doping can be on the order of 10 19 cm −3 . This causes the radiative rate to transition from a standard bimolecular behavior to a mono-molecular behavior: G r = B(n + n 0 )p ≈ Bn 0 p. Doping effects are not restricted to PL test structures, and can also occur in full LED structures, depending on the doping profile.
To circumvent this, we employ pin test structures (as discussed above) which remove doping artifacts by ensuring the QW is in the center of the intrinsic region (Fig. 4b). The effect of modulation doping on dynamics is demonstrated in Figs. 4c, 4d, where such a pin structure is used, and the QW position is varied across the intrinsic region. When the QW is in the middle of the intrinsic region, its background doping is very low and intrinsic dynamics prevail. In contrast, when the QW is moved toward the n or the p side, modulation doping occurs, leading to a shortened lifetime and a transition to monomolecular radiative dynamics. We believe that this doping effect has previously been misinterpreted as evidence of excitonic recombination dynamics. 41 Although not shown here, doping effects also affect the other (non-radiative) recombination channels. 42 Accordingly, in all the following, results will be shown for pin structures with centered QWs to avoid doping artifacts. In a conventional PL test structure, the QW is often grown right next to n-doped layers and capped with a thin undoped GaN layer, leading to significant modulation n-doping in the QW. (b) In a pin structure with a centered QW, modulation doping is avoided. (c-d) Differential lifetime τ and corresponding B coefficient for a series of pin samples, where the QW position is varied across the intrinsic region (the legend indicates the relative QW position, with 0% and 100% corresponding to the n and p sides respectively). When the QW is off-center, the radiative recombination becomes mono-molecular.  These samples are all of high quality, with a peak IQE of about 85%. Importantly, the peak IQE is maintained throughout the series but it is shifted to lower currents when composition increases -this trend will be examined in the section on polarization fields & overlap effects. As the composition increases, the polarization fields separate the electron and hole wavefunctions, reducing B -a manifestation of the quantum-confined Stark effect. 43 In addition, for each sample, B displays a clear carrier dependence, contrary to the assumption of the ABC model (i.e. a carrier-independent bimolecular coefficient B). Qualitatively, this dependence is interpreted as follows. At low density the radiative rate is enhanced by Coulomb interaction; this effect is screened as the density increases; at even-higher density, free carriers screen the polarization charges across the QW, leading to an improved electron-hole overlap and an increase in B. Fig. 6a summarizes these regimes. Note that, at even-higher density, B should again decrease due to phase-space filling. This regime cannot be reached in ODL experiments due to the time resolution of our setup; however, it has been observed in previous work using TREL measurements. 17 Coulomb enhancement (a type of many-body effect) is caused by the attractive Coulomb interaction between electrons and holes, which increases their interaction beyond the bimolecular free-carrier regime. 44,45 To further illustrate this effect, we measured a series of SQW samples with a constant In composition of 12% and varying thickness in the range 2.5-4.5 nm. The resulting B coefficients are shown in Fig. 6b. As the QW becomes thinner, the increased electronhole overlap leads to stronger Coulomb interaction and to a more pronounced enhancement of B at low current. This enhancement goes well beyond the conventional increase of electron-hole overlap in thin QWs (modeled as dashed lines in Fig. 6 for reference). As already discussed, the use of pin test structures is necessary to avoid doping artifacts and observe this low-current behavior. Note that the carrier dependence shown in Fig. 6 does not follow a mono-molecular trend; this stands in contrast to Fig. 4, confirming the observed effect is distinct from a doping artifact. It also precludes an interpretation of the radiative process as being caused by recombination from a population of excitons -in accordance with estimates from Saha's equation, 14,46 which indicates that only a small fraction of the carriers may form excitons at room temperature (Fig. 6c).
Coulomb enhancement was investigated in Refs. 48,49, by considering one-dimensional (homogeneous) InGaN QWs. However, an added theoretical difficulty is brought by the existence of random alloy fluctuations, whose spatial and energy scales are very similar to those of Coulomb interaction. Recently, we showed in Ref. 50 that Coulomb interaction and alloy disorder should be treated on equal footing to be quantitatively accurate. The resulting predictions, shown in Fig. 6b, agree with the experimental data remarkably well without the use of fitting parameters.
Despite the complexity of modeling Coulomb enhancement, it plays an important role at low current density, as the enhanced radiative rate competes with non-radiative recombinations to boost the IQE. For instance, in a 2.5 nm QW, the radiative rate is completely dominated by Coulomb enhancement, which leads to a tenfold increase in B at low carrier density. It is likely that such effects will be of importance in future applications where low-current efficiency is crucial, e.g. micro-LED displays.
In many studies, the carrier-dependence of B is ignored and a single value is reported. This corresponds to taking a cross-section of the curves B(n) shown in Figs. 5-6 at a particular (and often un-specified) value of n. Depending on this value, the relative magnitude of the aforementioned effects (separation by polarization fields, Coulomb enhancement, carrier screening) can vary.
In summary, the radiative rate of III-nitride QWs shows a complex behavior. Its magnitude is dictated by the wavefunction separation (itself a function of polarization fields and of nanoscale alloy disorder), and is further affected by Coulomb enhancement at low density and field screening at high density. The interplay of these effects, whose balance depends on the details of the QW structure, can lead to a notable departure from the often-assumed constant bimolecular rate. To optimize LEDs for a given application, this complexity should be taken into account, together with the behavior of non-radiative processes which will now be discussed. SRH recombination.-SRH recombinations, which cause a decrease in efficiency at low current, remain relatively less studied than other channels. Often, they are treated empirically by assuming a recombination rate G SRH = An, with A a heuristic coefficient indicative of material quality. The study of SRH recombinations in the QW itself is complicated by the existence of parasitic transport-related recombination pathways -such as tunnel-assisted recombinations across the pn junction. 51,52 Transport-induced recombinations are weak or absent in the high-quality material investigated here; 53 in this case, SRH recombinations in the active region itself remain the limiting contribution to IQE.
We presented an in-depth study of SRH recombinations in Ref. 54. We showed that, contrary to common expectation, a series of QWs with varying thickness had a near-identical EQE (with a high peak value of ∼75%) irrespective of thickness: this is a priori surprising since B decreases by about two orders of magnitude from the thinnest to the thickest QW. To account for the maintained EQE, the other recombination rates must vary along with B. To confirm this, dynamics were measured on numerous QW samples of varying In content and thickness. As shown in Fig. 7a, we found that the decrease of B is indeed accompanied by a nearly-commensurate reduction of A. Since all these samples have a high peak IQE of ∼85%, variations in A cannot be attributed to a change in crystal quality, and instead illustrate a coupling between A and B. Experimentally, we find A ∼ B 0.9 (the value of this exponent is subject to some inaccuracy, in part due to the carrierdependence of B; here, we use the value of B at n = 3 × 10 18 cm −3 , corresponding to the trough seen in Fig. 6). This coupling between A and B confirms that wavefunction-overlap affects the SRH coefficient. Such a trend was first hypothesized early in Ref. 55; it received experimental support in Ref. 26 and further qualitative scrutiny in Ref. 56. We fully characterized this effect and proposed a quantitative model in Ref. 54. The physical underpinning of this dependence is that a full SRH cycle requires both an electron and a hole to be captured by the same defect. Point defects in GaN have a very small Bohr radius (∼1 Å), making them quasi-punctual on the scale of a QW. Therefore, the exponential tails of both carriers need to overlap with a defect to complete an SRH cycle (Fig. 7b). Such capture processes can be seen as field-assisted point defect recombinations, or equivalently as defect-assisted tunneling from the conduction to the valence band. The resulting local SRH rate can be expressed as: with ψ e,h the wavefunctions and c e,h the capture cross-sections for electrons/holes. As shown in Ref. 54, this formula effectively leads to a power-law dependence of A with the wavefunction overlap, and hence with B (with an exponent which depends on the properties of the SRH defect and can take values close to unity, consistent with the experimental observations of Fig. 7a).
The overlap-dependence of SRH recombinations has important implications. In particular, the value of A cannot simply be taken as a proxy for defect density, since overlap can affect it by orders of magnitude for a given defect density. This also means that it is difficult to compare values of A across samples with different QW designs. In addition, it follows from Eq. 6 that the SRH rate is dominated by the contribution at a position where the product ψ e ψ h is maximal, which occurs close to the middle of the QW -in contrast to the common belief that SRH recombinations may take place at defective QW/barrier interfaces. Note that, in addition, SRH transitions between adjacent QWs may occur in structures with thin QW barriers.
Besides the overlap-dependence of SRH recombinations, the nature of the defects causing such recombinations is of utmost interest. The effect of threading dislocations has often been discussed, with studies concluding that they should not dominate the SRH rate once their density is below ∼10 8 cm −2 (a condition which is met in highquality GaN LEDs), 57 because the lateral diffusion length of carriers in InGaN QWs (on the order of 100 nm) makes capture by dislocations unlikely. Other extended defects (e.g. misfit dislocations) are strongly deleterious for IQE, but are not usually found in high-quality LEDs.
In addition to extended defects, there is clear indication that the IQE remains limited by point defects, whose signature can be observed in deep-level optical spectroscopy. 58 It has long been known empirically that the introduction of an InGaN underlayer (UL) beneath the active region has a strong beneficial effect on IQE -ULs are commonplace in commercial LEDs, either as a continuous InGaN layer or as an In-GaN/GaN superlattice. The physics behind this beneficial effect were only investigated more recently. Ref. 59 showed that the UL reduces the non-radiative carrier lifetime, and Refs. 60,61 proved that the UL acts by burying point defects. The specific identification of the point defects causing SRH recombinations remains an open question. A few studies have pointed to the deleterious role of specific extrinsic defects (e.g. calcium in low-IQE MBE samples); 62 however, there is no similar evidence that a specific species dominates SRH recombination in high-quality MOCVD samples. It has also been argued that SRH recombinations may chiefly be caused by intrinsic defects, in particular vacancies and vacancy complexes; this line of thinking has received suggestive evidence recently. 60,61 Work is also ongoing on the theoretical front to identify possible candidates for SRH centers -especially from ab-initio modeling. 63,64 Importantly, the expected mechanism for dissipating energy is multiphonon emission; for large-bandgap III-nitride emitters, this requires the emission of ten phonons or more, making such a process generally unlikely. Therefore, for any defect to cause significant SRH recombination, it must display a strong carrier-phonon coupling constant. A value of S ω of ∼1 eV (with S the Huang-Rhys factor and ω the phonon frequency) fits the data of Fig. 7 -however, it is as-yet unclear what specific defects could provide such coupling.
Finally, it has been proposed that low-current recombinations, rather than consisting of two successive multiphonon-emission steps, might in fact be caused by a multiphonon step and an Auger scattering event to the defect level (with the former step being rate-limiting, to retain a linear carrier dependence), as sketched in Fig. 8e. Investigations of this hypothesis are ongoing. 65 Droop and Auger recombination.-Efficiency droop refers to the non-thermal reduction in IQE at high current density. The mechanism underlying droop was a topic of particular contention for many years. It is now generally accepted that Auger scattering plays a large role in causing droop. As previously discussed, this is compatible with basic experimental data, since an ABC fit (with C identified as an Auger coefficient) can match LED IQE curves reasonably well. This observation alone, however, is inconclusive, as IQE curves can also be fitted with other models. 66,67 A convincing argument for Auger scattering was built over time, upon consistent evidence from various investigations, including: • Proof that droop is a phenomenon intrinsic to the active region, present even under PL excitation and thus not caused primarily by carrier leakage. 68 From this work, there is compelling evidence that the majority of droop comes from an intrinsic effect in high-quality LEDs; however, an additional contribution from transport effect cannot be excluded in some structures (e.g. tunneling leakage due to defects) or at high temperature (as discussed in the section on radiative recombinations).
• Lifetime measurements confirming the roughly-n 3 carrier dependence of droop. 17,40 This observation, which requires a proper estimate of n, is an important test: in contrast to a simple phenomenological fit by the ABC model, it explicitly shows that the carrier-dependence is that of an Auger process.
• The observation of 'hot' Auger carriers by electron spectroscopy. 69 These measurements are a crucial observation; although they are not fully quantitative, they provide explicit evidence of the existence of an Auger process.
• Theoretical predictions with good match to experimental data. 70 Various models have been investigated; to date, the most compelling work points to indirect Auger scattering (enhanced by phonon emission and alloy disorder) as the likeliest underlying process.
In recent years, the cumulative evidence from these various arguments has led many in the III-nitride community to consider interband Auger scattering as a primary explanation for efficiency droop. Despite this relative consensus, some aspects have remained unclearsuch as the large scatter in reported C coefficients across literature.
Recently however, we reported a surprising result: droop consists of two distinct contributions, which were previously not distinguished. 71 One is an intrinsic process, whose magnitude is determined by the active region structure, and is most likely caused by an interband Auger  71) where the defect density was intentionally varied. Blue dots: larger set of 160 samples, with [In] varying between 10% and 20% (for this set, the values of A and C were first corrected to remove the inherent variation with [In], leaving only variations due to changes in point defect density). The slope for both data sets is close to unity (line), indicating that the extrinsic droop current scales linearly with the density of SRH defects. The large sample set shows some scatter, as these samples were grown over several years, with some uncontrolled variations.
process as described above. The other is a hitherto-ignored extrinsic process, whose rate scales linearly with the SRH defect density. To demonstrate this, we prepared a series of SQW samples where the active region was kept identical (thickness 4 nm, [In]=13%), but the SRH coefficient A was intentionally varied by over two orders of magnitude by modulating the defect density. ODL analysis of these samples showed that the radiative coefficient B was unchanged (as expected) whereas the droop coefficient C displayed a significant variation, scaling linearly with A. The relationship between A and C is shown in Fig. 9. Therefore, the droop current can accurately be decomposed as: where C 0 is the intrinsic Auger coefficient, a and k is an empirical coupling constant for the extrinsic process. Importantly, in these experiments, the n 3 -dependence of both the intrinsic and extrinsic droop currents is derived experimentally (from lifetime measurements), rather than being assumed. We found that a value of k = 2 × 10 38 cm 6 accurately predicted the droop coefficient of a large variety of samples with varying [In] and concentrations of point defects.
To further corroborate this observation, we considered a muchlarger set of 160 SQW samples (with a nominal thickness 4 nm). This set was grown over several years, and thus displays uncontrolled variations in composition ([In] = 10-20%) and material quality, and most likely small variations in thickness. The variation in composition and thickness leads to a large variation in electron-hole overlap, which affects all recombination coefficients (as will be discussed in the section on Polarization fields & overlap effects). This effect must therefore be corrected-for (which is done by deriving the empirical variation of A and C versus wavelength in high-quality samples) so the samples can be compared, revealing the effect of defect density on A and C. As shown in Fig. 9, after this correction is applied, the same correlation between A and C as above is observed (albeit with more scatter than in the dedicated series of samples of Ref. 71). Here again, a slope a Namely C 0 = 2.5 × 10 −33 cm 6 s −1 for a 4 nm SQW with [In]=13%. of one is observed (showing again that the extrinsic droop current is proportional to A), and a near-identical value of k is found.
These findings indicate that a part of the droop current is mediated by a defect-assisted process. Owing to the well-defined cubic dependence reported in Ref. 71, we tentatively propose that a defect-assisted Auger step may be at play. Fig. 8 illustrates several non-radiative processes, with defect-assisted processes (d-f) being possible candidates. Note however that the carrier-dependence of processes (d-e) does not appear compatible with our experimental observations.
The magnitude of this defect-assisted droop process can be significant: in lower-quality samples (IQE < 50%), it is much larger than intrinsic Auger scattering. This phenomenon constitutes a significant departure from the conventional understanding of efficiency in the framework of the ABC model, and is likely to have meaningful practical implications. In particular, as we will argue in the section on the green gap, we believe that defect-assisted droop is an important contributor to the green gap.
Understanding the breakdown between these two contributions to droop also enables a finer analysis of the intrinsic process. To do so, we consider high-quality samples (peak IQE ∼85%) where droop is dominated by intrinsic Auger scattering; these samples have a varying active region thickness and composition, causing B to vary by two orders of magnitude. ODL measurements reveal a strong correlation between the radiative coefficient B and the intrinsic Auger coefficient C 0 , with an empirical dependence C 0 ∼ B 1.2 , as shown in Fig. 10a. This power law also applies to the carrier-dependence of B and C for a given sample: as shown in Figs. 10b-10c, C(n) ∼ (B(n)) 1.2 .
Qualitatively, this scaling law between B and C can be understood in terms of wavefunction overlap effects: both the radiative and Auger rates involve overlap integrals, and a decrease in overlap should affect both channels. This was first argued in Ref. 26, with further discussion in Ref. 56. However, a quantitative explanation for the scaling law reported here has long remained lacking; we presented such a model in a recent publication. 72 Polarization fields & overlap effects.-It has often been argued that the strong polarization fields separating electrons and holes are a major cause of efficiency reduction in III-nitride emitters. However, common experimental evidence shows that the impact of wavefunction separation on IQE is much less drastic that one may expect, with relatively thick QW samples still achieving high IQE. For instance, we demonstrated in Ref. 54 a series of SQW LEDs of varying thickness having a maintained peak EQE, and we observed a peak IQE of about 93% in 4 nm SQW samples with [In]=13%. Likewise, high peak IQEs are observed in commercial LEDs throughout the violet-to-cyan range, despite a significant change in field intensity with wavelength.
This robustness against wavefunction separation stems from the fact that all recombination processes scale with the wavefunction overlap, with somewhat-similar exponents. As previously shown in Figs. 7 and 10, for high-quality samples, we find empirical scaling laws A ∼ B 0.9 and C 0 ∼ B 1.2 . This results in a partial compensation, whereby a reduction in B is accompanied by near-commensurate reduction in the non-radiative processes. This phenomenon was reported experimentally in Ref. 26; it was given qualitative theoretical grounding in Ref. 56, and a more rigorous justification for these scaling laws -which are in fact a general property of recombinations in systems where wavefunctions are separated-was recently presented in Refs. 54,72. Note however that these trends only apply so long as the density of SRH-causing defects remains constant. With higher defect density, A of course rises, which in turn leads to an increase of the defect-assisted droop process.
Therefore, in the regime of high-quality samples, polarization fields have a modest effect on the peak IQE. On the other hand, as shown in Ref. 26, they can have a strong effect on the relationship between carrier density n and current density J, owing to Eq. 3. For instance, if all recombination rates are reduced tenfold due to wavefunction separation, the relationship between n and J is also shifted by a decade. This contributes to the well-known trend whereby droop becomes 'worse' at longer wavelength: in fact, the relationship between droop and carrier density is unchanged, but carriers accumulate faster in the active region due to the slower recombination time, causing droop at a lower current. This effect is illustrated in Fig. 5, where samples with higher [In] droop earlier.
An additional implication of this wavefunction-overlap effect is that it can be difficult to compare values of the ABC recombination coefficients across literature, since they may vary by 1-2 orders of magnitude depending on the details of the active region design. We believe this effect, compounded by the carrier-dependence of B and C for a given sample, contributes to the scatter often reported in literature (exemplified for instance in Ref. 73).
III-nitrides grown along other crystal directions (non-polar or semipolar) have been proposed to reduce or remove the impact of polarization fields. 74 Early on, it was hoped that the resulting increase in oscillator strength would enable very high peak IQE and potentially solve the challenge of droop. However, droop is also observed in high-quality non-polar and semi-polar LEDs. 75 This is easily understood in view of the present results: field reduction affects all recombinations. Therefore, crystal planes with reduced fields may not necessarily be transformative for peak IQE, although they should help delay the onset of droop to a higher current density. Ultimately, the benefits of specific growth planes may have more to do with epitaxial properties, and especially with their varied ability at integrating high indium contents while maintaining a good material quality. 76 The green gap.-The 'green gap' refers to the general trend of reduced IQE in longer-wavelength III-nitride emitters -although today, efficient green LEDs are available commercially, and this performance gap is most pronounced for yellow-red emitters. The green gap is likely to be a complex phenomenon, caused both by device physics effects and material challenges -in particular, the breakdown of material due to excessive strain. The contribution and interplay of these effects is still under investigation.
It has been proposed that the green gap may be caused by in-plane carrier localization: as the composition increases, wavefunctions become more separated, which may affect various recombination coefficients. However, in more recent studies, conclusive proof of a link between localization and the green gap remains elusive. It is therefore unclear whether localization per se is a sufficient explanation for the green gap -as will be discussed further in the section on other effects.
Recently, we proposed that the green gap may in part be caused by the defect-assisted droop process discussed in the section on droop. Namely, we observed in Ref. 71 that at higher In composition, SRHcausing defects seem to incorporate more readily in InGaN QWs, which in turns causes a drastic increase in the defect-assisted droop current.
Motivated by this hypothesis, we have investigated the effect of defect reduction on long-wavelength emitters, with experimental success. Fig. 11a shows the IQE of a series of QWs with identical active regions ([In]=25%): as the defect density is reduced, the IQE improves In SQW samples, the wavelength shifts at high current density; the plots of (c) show the locus of wavelength and IQE as the current density is varied, and demonstrate that a high IQE can be extended to longer wavelength. both at low and high current. Notably, the high-current improvement is incompatible with a standard ABC model, and can only be justified by a reduction in the droop coefficient.
By applying such improved growth conditions to QWs of varying content, we were able to maintain a relatively high IQE up to fairly long wavelength. As show in Figs. 11b-11c, a peak IQE of 70% is maintained up to [In]=28%, with little change in the shape of the efficiency curve (apart from an overall shift of the curve to lower current, as explained in the section on polarization fields & overlap effects). These recent results were only obtained under PL excitation in SQW samples, and further work would be required to translate them to realistic LED structures under electrical injection. Nonetheless, the data of Fig. 11 clearly demonstrates that the reduction in IQE at long wavelength is not a fundamental limitation, and that a high IQE can be obtained at long wavelength thanks to high-quality epitaxy.
Therefore, we believe this data constitutes a solid proof-of-concept that defect reduction is a central strategy to tackle the green gap. Of course, growing low-defect material with even-higher In content (e.g. to achieve red emission) remains a major challenge in material science, and may require unconventional epitaxial approaches.
Thermal droop.-Thermal droop (TD) describes the reduction in IQE observed at elevated temperature. It remains a poorlyunderstood phenomenon, and has received less scrutiny than conventional (current-induced) droop. Nonetheless, TD is an important effect from a practical standpoint: it typically reduces LED performance by 10-15% at real-world operating temperatures. 77 To date, one of the most detailed studies of this phenomenon is Ref. 52, which showed that at low current (<0.5 A.cm −2 ) it is caused by defect-assisted carrierescape processes. However, TD in this model acts as a shunt path which should saturate at high current density, contrary to experimental evidence. A clear explanation for TD at operating current is lacking.
The understanding of TD is complicated by a general ambiguity: to what extent is the phenomenon caused by intrinsic variations in recombinations, or by transport/injection effects (i.e. carriers either not being captured by the active region, or escaping from it and recombining elsewhere)? Hereafter, we offer a few comments, supported by our recent experiments in high-quality samples (where the defectassisted process of Ref. 52 is absent, as indicated by an examination of current-voltage characteristic at low current).
First, the arguments generally used at room temperature to justify that current droop is intrinsic rather than transport-related (e.g. the shape of the efficiency curve under electrical and optical injection is very similar, and estimates of IQE and light extractrion leave little room for an injection efficiency below-unity) do not apply as clearly at high temperature. For instance, EL-and PL-based efficiency curves sometimes show different shapes, with a worse TD observed in EL measurements.
Second, an examination of the total recombination lifetime reveals that it only has a very weak temperature dependence above room temperature, across the whole range of low-to high current density. This was reported in Refs. 29,54, and is also illustrated in Fig. 3. It is difficult to reconcile this observation with a model whereby TD is mainly due to active region recombinations (as was proposed e.g. in Ref. 28).
Third, TD can vary significantly with the design of the active region (including the number of quantum wells and the electron blocking layer design) -unlike current droop, which displays a more universal behavior. In single-QW LEDs, TD is often pronounced. However, by optimizing the design of MQW LEDs, we have achieved a near-suppression of TD, with the ratio of high-temperature to roomtemperature EQE (the so-called hot/cold ratio) reaching nearly unity at high current density. Fig. 12 illustrates this result.
All these considerations indicate that transport/injection effects play a key role in TD. A finer analysis of these arguments will be presented in an upcoming publication. 78 Besides, electron spectroscopy offers additional opportunities to understand TD, by enabling observation of carrier leakage; 79 a recent study using this technique confirms the predominant effect of carrier escape. 80 Carrier localization, caused by random-alloy fluctuations in compound QWs, has been shown to affect the wavefunctions of carriersin particular low-energy hole states, which are localized on a scale of 1-2 nm. While the existence of this effect is not in doubt, its impact on device properties remains more disputed. It has been argued that localized carriers are less likely to diffuse to dislocations, which may explain the good efficiency of InGaN LEDs even with rather high dislocation densities (although these are no longer a concern in state-of-the-art LEDs). 57 Conversely, it has been argued that localization increases with In content, and that the ensuing lateral separation of electrons and holes may be a cause for the green gap. 81 However, there are reasons to doubt this model. First, it is based on the assumption that wavefunction separation is only detrimental to the radiative rate, and does not affect non-radiative recombinations; however, as we have shown, the contrary is true, and in-plane separation is likely to affect of channels, leading to a first-order compensation. Second, strong localization only occurs for low-energy wavefunctions (in the Urbach tail of the density of states), whereas higher-energy states are delocalized. 82 Taking into account a thermal population, it is found that a significant fraction of the populated states are delocalized (as shown in the supplemental material of Ref. 50), so that statements about recombinations cannot be made simply on the basis of a few localized states. Calculations including many states, as in Ref. 83, instead conclude that the radiative rate is increased by localization.
As discussed in the section on radiative recombinations, our results suggest that Coulomb enhancement plays an important role in determining the radiative rate. From this standpoint, the leading role of localization is to modulate the strength of the electron-hole interaction, and therefore the magnitude of Coulomb enhancement; we believe this effect may be much more significant than changes in electron-hole overlap in the free-carrier picture.
Transport properties play an important role in device efficiency. Several effects are relevant.
Electrical efficiency, the ratio of photon energy to electrical voltage, characterizes the efficiency with which carriers are brought to the active region; interestingly, this quantity can reach values above unity at low current (by harnessing thermal energy from the lattice). In Ref. 53, we derived a theoretical limit for electrical efficiency, which is caused by the active region recombination rate. Today, this limit is achieved in state of the art violet and blue LEDs, and progress is being made in green devices. Voltage remains high in longer-wavelength devices, on behalf of their large band offsets.
Alloy disorder may have an impact on electrical efficiency, separate from its often-discussed effect on recombinations. It was argued in Ref. 84 that alloy fluctuations play a key role in enabling inter-well transport in MQW LEDs. Including disorder in drift diffusion models indeed leads to a more realistic current-voltage characteristic than with conventional models. Studies of this effect are ongoing.
An additional challenge is to obtain a uniform carrier distribution across quantum wells. Ideally, a MQW structure with many active QWs would reduce the impact of droop. However, the heavy mass of holes causes their accumulation in a few QWs. In earlier devices, it was shown experimentally that only the p-side QW was active, 85 thus calling for an optimization of the QW and barrier designs to improve spreading across the active region. In principle, guidance from modeling should be useful; yet even sophisticated models struggle to properly account for the low turn-on voltage of InGaN LEDs 86,87 -possibly because of the complex contribution of alloy disorder to transport. 84,88 Nonetheless, such optimization has been achieved experimentally in state-of-the art commercial devices: these benefit from better vertical spreading -although spreading across an arbitrary number of QWs remains elusive, and droop is still a limitation.

Conclusions
The experimental study of recombinations in high-quality IIInitride emitters, enabled by proper measurements of carrier dynamics, gives access to a wealth of physical effects beyond the commonlyconsidered ABC model. In addition to its well-known dependence on wavefunction overlap, the radiative rate is also strongly affected by a combination of Coulomb enhancement, disorder-induced localization, and carrier screening, giving it an intricate carrier-dependence. Radiative emission competes with non-radiative channels (SRH recombinations and droop), but these also display a similar wavefunction dependence, leading to a remarkable robustness of the internal quantum efficiency to details of the active region. Efficiency droop, a crucial limit to high-current efficiency, is composed of two contributions: conventional Auger scattering and a defect-assisted droop process. The latter is more prevalent at long wavelength, making defect reduction an important strategy to reduce the green gap.
Most of the results presented in this Review were obtained on single-quantum-well LEDs, but the underlying physical effects are the same in more advanced structures; their complex interplay determines the internal quantum efficiency of real-world devices. Importantly, our results show that some of the main challenges still limiting some devices (e.g. deleterious effects of fields and defects, and longwavelength behavior) are not fundamental and can be mitigated with proper epitaxial design and improved material quality. Therefore, by understanding and harnessing the physics of III-nitride recombinations, we believe that high-efficiency will be achievable over a wider range of wavelengths, enabling III-nitrides to fulfill their promise as a universal material for ultra-violet to red emission.

Appendix A -Comparison of TREL and ODL
As discussed above, ODL enables a direct measurement of the active region's recombination dynamics, whereas conventional TREL must deconvolve the recombination response from the transport response. This can make the extraction of the recombination time difficult, especially at low current where the large LED impedance sometimes dominates the optical response. In addition, in some epitaxial structures, additional transport effects (e.g. transport across the EBL, or across QWs in a MQW structure) can complicate the TREL response beyond the model we proposed in Ref. 29. TREL analysis is therefore limited to some epitaxial structures where the model is suited -in practice, SQW LEDs with an appropriate doping profile.
Nonetheless, by employing such epitaxial structures and with careful fitting, good agreement can be demonstrated between the two techniques. As an example, in Fig. 13, we compare two 4 nm SQWs with a similar composition [In] = 13%. The TREL sample is an LED, with the SQW embedded at the center of a 30 nm-thick undoped region, and features an EBL. The ODL sample features the same QW, at the center of the 200 nmthick undoped region of a pin structure. The two sample therefore have slightly different electrostatics -in particular the ODL sample sees more polarization field drop across the QW at low current, leading to a higher electron-hole separation.
The IQE of both samples has a similar shape, with a slight offset in current density (which may be due to the different electrostatics, and may also stem from the difficulty to calibrate the absolute magnitude of the ODL photocurrent). To first order, the total recombination lifetime is quite similar for both samples, leading to similar trends for the B and C coefficients. In particular, the carrier-dependence of B and C determined from ODL (discussed in the section on radiative recombination) is also observed in TREL, with a very similar behavior. The low-current value of these coefficients is slightly lower in the ODL sample, in accordance with the reduced wavefunction overlap. Given this, it is possible to reproduce with TREL measurements some of the results obtained with ODL. As an example, we have studied the dynamics of a series of LEDs with SQWs ([In]=13%) of varying thickness in the range 2.5-5 nm. The resulting B and C coefficients are shown in Figs. 13e-13f. The same trends are observed as in ODL. Regarding B, we observe a substantial increase for thinner QWs, and a pronounced carrierdependence, with an uptick at low current (attributed to Coulomb interaction) and at high current (due to field screening). This is similar to Fig. 6. For C, we observe a similar current-dependent behavior as for B, in accordance with the scaling law discussed in the section on droop. Note that some of the TREL data is noisy, due to the difficulty of perfectly fitting the data for some samples.