Getting To Grips With The Green Gap
Compositional fluctuations in InGaN alloys are partly to blame for the dramatic drop in efficienty when moving from blue to green LEDs BY MATTHIAS AUF DER MAUR FROM THE UNIVERSITY OF ROME "˜TOR VERGATA' AND SERGEY YU KARPOV FROM STR GROUP
The revolution in solid-state lighting is well underway. GaN-based LEDs are gaining an ever-greater share of the general lighting market, and LED lamps are on sale in many outlets, enabling them to slowly replace conventional light bulbs.
However, there are still some fundamental issues to be solved. One of these is why does the LED's quantum efficiency plummet when its emission is shifted from the blue to the green (see Figure 1). And another key question is why does the efficiency of this device, regardless of its wavelength, fall as the current that passes through it is cranked up.
Figure 1. The dramatic drop in LED efficiency in the green and yellow is widely referred to as the "˜green gap'. The plot of external quantum efficiency is based on reports for different commercial and research grade nitride and phosphide LEDs. The black lines are guides to the eye.
It is important to realise that both of these issues, which are referred to as the green gap and droop, respectively, are not merely problems of an academic nature. Instead, they have far reaching consequences for LED-based technology and applications.
Take droop, for example. It is an impediment to ultra-efficient lighting, because applications such as generic illumination and car headlights require a high luminosity "“ and as luminosity is at most proportional to the drive current, high light intensity implies high current. So, when LEDs are used for white-lighting applications, they often have to operate at currents where droop is already significant, and this impairs the overall efficiency.
At first glance, the green gap appears to be far, far less of an issue than droop. After all, white LEDs generate their output by using a blue-emitting LED to pump a yellow-emitting phosphor, and the emission is combined. However, green LEDs are very relevant when trying to form a superior source of white light through the mixing of emission of differently coloured LEDs. This approach, which typically combines the output of red, green and blue LEDs "“ it has the moniker RGB "“ has two significant advantages over the incumbent technology: it eliminates phosphor conversion, which is hampered by an additional energy and thus efficiency loss; and thanks to the opportunity to control the single-colour components, it is possible to control the emission spectrum.
The challenge is that to ensure an efficient RGB white-light source, a sufficiently efficient green LED with a wavelength in the very middle of the green gap must be adopted. Note that it's not an option to shift the emission of red LEDs to shorter wavelengths, as efficiency nosedives, due to a switch from a direct to an indirect bandgap. As efficient green LEDs based on any material system are not available, phosphor-converted white LEDs are used, as they currently outperform any direct mixing approach. But this could change by uncovering the origin of the green gap, which is in fact just as big an issue as droop.
Today, the consensus of opinion is that the primary contributor to droop is non-radiative Auger recombination. When it comes to the green gap, its origin remains a mystery. It is well-established that as AlGaInP-based LEDs shift to shorter wavelengths, efficiency falls, due to a hike in electron leakage into the p-side of the LED structure. This increases with current and operating temperature. With nitride LEDs, the finger points at changes in the active region, which requires an increase in indium content in the InGaN quantum wells to reach longer wavelengths. There is a large mismatch in lattice constant between the InN and GaN crystals that constitute the InGaN alloy, and this leads to the introduction of more defects. They have been blamed on an increase in non-radiative recombination.
However, this explanation is flawed: recent experimental evidence rules out defects as a main cause of the green gap. Experiments undertaken by researchers at Osram Opto Semiconductors GmbH and the Technical University of Berlin on high-quality LEDs with different colours show that that the defect-related recombination coefficient is roughly constant from blue to green. Green LEDs have a lower efficiency, however, indicating that other effects must be behind the green gap.
What is beyond doubt is that increasing the indium content in the quantum well increases the internal electric field due to the presence of strong spontaneous and piezoelectric polarization fields in III-nitride wurtzite crystal structures. This electric field pulls apart the electrons and holes in the quantum well, pushing the former towards one InGaN/GaN interface, and driving the latter to the other (see Figure 2). This phenomena, known as the quantum confined Stark effect, reduces the overlap of electron and hole wave functions as the indium content in the well increases. As the radiative recombination rate and the optical emission are proportional to this overlap, they fall, along with the efficiency, as wavelength increases.
Figure 2. Electron (e0) and hole (hh0) ground-state wave functions (lines) in a polar InGaN single quantum well LED. The grey shadow indicates the band alignment in the structure. Here, the structure is grown along the  crystal direction, coinciding with the wurtzite crystal's c-axis. Â±s indicate the effective surface charges originated from the polarization discontinuity. This induces an internal electric field in the quantum well that separates electrons and holes along the growth direction.
When this reduction in efficiency with wavelength is measured, it is significantly larger than one would expect from the quantum confined Stark effect. So other factors must be at play "“ a view supported by experiments with non-polar nitride LEDs. These devices have no built-in fields, so they are not expected to have a green gap. But they do.
One explanation for the missing piece to this conundrum is random statistical fluctuations in the InGaN alloy composition, which lead to subtle effects in the electronic structure of the material. Such fluctuations are present in even the best quality material with a completely homogeneous composition, since the exact position of every single indium atom in the GaN host material is governed by statistics. The situation mirrors the classical experiment of drawing, on several occasions, a certain number of balls from a pot filled with many white and black balls: you know that the mean number of white balls that you pull out should equal the ratio of white to black ones, but the exact number will fluctuate from experiment to experiment.
Going back to the InGaN alloy, the implication is that there is a statistical spread in the local InGaN composition obtained by counting indium and gallium atoms (see Figure 3 for an illustration). So, there will be spatial fluctuations on the atomic scale in the energy landscape experienced by electrons and holes. In turn, this will lead to a spread in the energy levels of the electronic states, spatial inhomogeneity in the wave functions, and, as will be shown below, a reduction in the photon emission rate.
Figure 3. The statistical distribution of the local indium concentration (top panel) is obtained by counting atoms in small control volumes of defined size, shown schematically in the one-dimensional random gallium-indium sequence. Different control volumes give different local concentration, and the random atomic disorder produces a random potential landscape seen by the carriers (black curve), so that the envelope of the wave functions (blue curve) become inhomogeneous, with fluctuations that are correlated with the atomic disorder.
One question naturally arises: can these atomic-scale fluctuations have a measureable impact on macroscopic device performance, and ultimately contribute to the green gap or droop? We have been seeking an answer to this in our investigations at the University of Rome "˜Tor Vergata' and the STR Group. We know that today's experimental methods are incapable of uncovering the effect of atomic-scale alloy fluctuations on electronic and optical properties, so we have used theoretical calculations to consider their possible contribution to the green gap and droop.
The most natural way to study the influence of atomic scale features is by atomistic models, which explicitly use the detailed information on the atomic structure of the device, such as atomic species and positions. But the downside of this approach is that these atomistic simulation models can be computationally quite demanding, because simulated structures can contain hundreds of thousands of atoms or more.
A reasonable compromise is to use an empirical tight-binding model. For these calculations, we construct electronic wave functions from a linear combination of atomic orbitals, and only account for interactions between nearest neighbour atoms. This model is "˜empirical', because the parameters are obtained by fitting to the known band structures of bulk materials. The strength of such a model is that it combines reasonable computational efficiency with the ability to treat random atomic structures "“ and it is such a structure that lies at the heart of the matter.
We began by considering the effect of random compositional fluctuations on the maximum LED efficiency. To do this, we employed standard simulation models, and evaluated realistic LED structures under operating conditions near the efficiency maximum. We then constructed statistical sets of random atomistic structures, with efforts restricted to just the relevant part of the LED, which obviously includes the quantum well with randomly distributed indium atoms (see Figure 4).
Figure 4. The single quantum well LED structure that is analyzed: The InGaN quantum well (QW) is sandwiched between n- and p-type doped GaN layers, and an AlGaN layer follows the active region to block electrons from reaching the p-doped region. The 3 nm-thick QW and 4 nm of the surrounding GaN barriers has been modeled atomistically with random structures, as shown on the right. The randomly distributed indium atoms are highlighted as red dots. A 10 nm by 10 nm section of the quantum well has been simulated. This corresponding to roughly 100,000 atoms.
By using a classical valence force field method, we could then calculate exact atomic positions. After this, we projected the electrostatic potential obtained formerly in the device simulation onto the atomic positions and constructed the tight-binding Hamiltonian "“ and then from this, we obtained the energies and wave functions of a subset of all electron and hole states via an efficient implementation of an eigenvalue solver, using modern graphics processing units. Finally, from the electron and hole states, we calculated the radiative recombination rate and the carrier densities. To determine the impact of LED performance on wavelength, we performed these calculations for different indium contents between 15 percent and 35 percent, and for each of those, we considered several dozen random samples, each with a different random distribution of indium atoms. Taking this approach allowed us to extract the statistical mean values for both the radiative recombination rate and the carrier densities.
Figure 5. In the plane of the quantum well, the electron and hole wave functions that are represented by yellow and green in the figure, respectively, have their maximum where the local indium concentration is highest (see panels on the right). The side view on the left shows the vertical spatial carrier separation along the quantum well thickness. Electrons and holes are separated by the quantum-confined Stark effect.
For comparison, we repeated the calculations, but this time we ignored the random structure of the alloy. In this case, we treated InGaN as an effective medium, with effective parameters for the construction of the tight-binding Hamiltonian. This approach is comparable to that adopted for a standard device simulation, where electronic states are calculated using a continuous effective medium approximation.
It is immediately apparent from these simulations that the energies of electron and hole states are not the same in every randomly disordered sample. Instead, they show statistical scattering. This state of affairs is also seen in the momentum matrix elements, which are the most relevant quantities for radiative recombination. This insight indicates a variation of electron-hole wave function overlap from sample to sample, which will surely impact the final emission efficiency.
What's more, comparing these results with those employing an effective medium approximation reveals that the inclusion of compositional fluctuations reduces the momentum matrix elements, and that the reduction becomes stronger for higher indium content that equates to longer wavelengths. This implies that the effective medium based models tend to overestimate the radiative recombination rate, and thus the efficiency.
Implications of fluctuations
It's easy to intuitively grasp the implication of the simulations by looking at the wavefunctions of the electron and hole ground states, and relating them to the local indium concentration. Both the electron and hole wavefunctions accumulate at the peaks in the indium concentration on the relevant atomic planes in the quantum well (see Figure 5). So, to put it another way, electrons and holes head to where more indium is present. However, due to their spatial separation in the growth direction, this happens largely independently, with wave functions for electrons and holes peaking at different positions in the quantum well plane. The upshot is an additional reduction of overlap.
Even more interesting than this qualitative insight are comparisons with actual measured data. To do this, we have compared our results for the effective radiative recombination parameter with those extracted experimentally on high-quality, single-quantum-well LEDs emitting at different wavelengths . What we find encouraging is that our theoretical prediction for the wavelength-dependency of the peak LED efficiency accurately matches that obtained with experimental parameters (see Figure 6). If, on the other hand, compositional fluctuations are neglected, they will predict a smaller reduction in efficiency . This is a strong indication that random compositional fluctuations play a critical role in the green gap.
Figure 6 Left panel: Peak internal quantum efficiency for different emission wavelengths, due to differences in indium content, assuming constant non-radiative recombination constants borrowed from . The model, which includes random alloy fluctuations, accurately reproduces the wavelength dependence of the internal quantum efficiency obtained by using the measured radiative recombination parameter. Alloy fluctuations contribute up to 30 percent of the green gap. Right panel: IQE as a function of current density for different indium contents. Compared to the model based on the effective medium approximation, random alloy fluctuations decrease peak efficiency and tend to slightly increase droop.
We have also pursued a semi-empirical simulation [4,5]. With this approach, we assume that electrons are completely delocalised in the conduction band and holes are strongly localised by composition fluctuations in a bulk InGaN alloy. This difference between the two types of charge carrier results from the far higher effective mass for holes than electrons. Unlike the atomistic model, the strength of carrier localisation is controlled by the hole localisation energy as a fitting parameter.
Figure 7. Spectral dependency of the radiative (B, left panel) and Auger (C, right panel) recombination coefficients from measurements  (balls) and from the semi-empirical model (lines), estimated for different electron energies Ee.
One of the strengths of this semi-empirical model is that it provides a good fit to the spectral dependencies of both the radiative and the Auger recombination coefficients reported by the team from Osram and the University of Ulm  "“ so it ultimately provides an insight into the efficiency drop in the green gap (see Figure 7). However, this model overestimates the absolute values of the coefficients because it neglects the effects of polarization fields. Yet, despite this discrepancy, it is still able to highlight the crucial impact of carrier localisation on both radiative and Auger recombination in InGaN-based LEDs. It is this phenomenon that plays a role in the green gap.
A good test of any model is whether it can explain temperature-dependent behaviour. Recent measurements have revealed that radiative recombination coefficients show an anomalous increase with temperature , which can be attributed neither to bulk materials nor to quantum wells (Figure 8). Now an explanation is needed that offers a natural interpretation, in terms of carriers localised by compositional fluctuations, using both the semi-empirical model  and atomistic simulations.
Figure 8. Temperature dependence of the radiative recombination coefficient in blue and green multi-quantum-well LEDs measured in  (symbols). The atomistic simulation accounting for compositional fluctuations (solid lines) reproduces qualitatively the observations (the simulated values for the green LED have been divided by a factor of 8 to match the range of measured values), while conventional theory predicts the opposite trend both for bulk materials and quantum wells (dashed lines).
Have our efforts finally laid bare the nature of the green gap? Well, not just yet "“ there is still much work to be done. What we can say is that we have undoubtedly reached a better understanding of the role of random alloy fluctuations and associated carrier localisation, and we are now confident that they play an important role in the green gap. We know that statistical compositional fluctuations are present in even in the best quantum well with highly homogeneous InGaN composition, and this results in a fundamental limitation for the efficiency of long-wavelength nitride LEDs. So, it is not possible to overcome the green gap through technology optimization. However, if there is indium clustering in InGaN quantum wells "“ this may arise due to non-optimal growth conditions and processes "“ the efficiency will plummet faster as the emission lengthens from blue to green and yellow.
The closing of the green gap will require a more detailed, quantitative understanding of the effects of alloy composition fluctuations. Heading the list of issues that must be clarified in future is the effect of alloy fluctuations on the efficiency of green non-polar and semi-polar LEDs, as the electron and hole localisation in these devices is no longer spatially uncorrelated, thanks to the absence or severe reduction in the quantum confined Stark effect. From a practical perspective, there is also great interest in understanding the influence of compositional fluctuations on the emission spectra, and the impact of indium clustering on the optical spectra.