Technical Insight
Investigating the LED's dark side
Novel LED model offers new insights into droop
BY WENG CHOW FROM SANDIA NATIONAL LABORATORIES
A revolution in lighting is well on its way. Rewind the clock a year or so and the prices of LED bulbs made many shoppers wince. But now it is possible to get a high-quality 60 W equivalent for well under $10, and that's allowing sales of LED bulbs incorporating chips from the likes of Cree and Philips Lumileds to take off.
Although these solid-state bulbs are much more pricey than incandescents, which have largely disappeared from shelves due to legislation, they more than make up for that additional up-front cost with a substantial trimming of the electricity bill. It is a more tricky decision, however, whether it makes more sense to buy an LED bulb or a cheaper compact fluorescent (CFL). In terms of durability, adaptability and environmental impact, the solid-state bulb is the clear winner. But both types of light are similar in the efficiency stakes, and thus the running costs.
The performance of the LED bulb is partly compromised by a power loss associated with the electrical conversion of the AC 110 V or 220 V source from a wall socket to the handful of volts of DC needed to drive the semiconductor chip. But this loss is overshadowed by a saturation of light output − or equivalently, a degradation of efficiency "“ that kicks in at a current density well below the desired operating range (Figures 1a and 1b). Identifying this mysterious power-sapping mechanism, known as LED efficiency droop, will aid attempts to combat it. This could lead to a single LED, operating a very high current density, replacing the battalion of chips currently crammed together in the bulb. Such a move would slash production costs and boost efficacy, giving the manufacturer a bigger profit and the consumer a bulb that retails for less and is cheaper to run.
Figure 1: One serious, puzzling weakness associated with LEDs manifests itself as (a) intensity saturation or (b) efficiency droop. Auger scattering (c) is a strong candidate for the cause of this problem, which is known as LED droop. Each Auger scattering event involves non-radiative recombination of an electron-hole pair and promotion of a carrier (shown here as an electron) to conserve energy.
The contenders
Attempts to uncover the origin of droop are well underway by researchers in industry, academia and in national laboratories. Efforts throughout the world have already led to the proposal of several competing explanations for the cause of this efficiency sapping mechanism.
One explanation that appears to be surviving the test of time, and perhaps even gaining ground, is carrier loss via Auger scattering. When a semiconductor emits light, electrons and holes have to come together and recombine. So, if there are more electron-hole pairs, more light is emitted. However, electrons and holes don't always recombine to emit light "“ instead, Auger scattering can take place, with an electron recombining with a hole to give the resulting energy to a second electron or a hole (this is depicted in Figure 1c). At low current densities, the Auger recombination process is far less likely than the radiative, light-emission process. However, as the current is cranked up, carrier density increases and Auger recombination takes off, with droop kicking in.
A great deal of controversy surrounds the question of whether Auger carrier loss is the primary culprit of LED droop. The disagreement is not really concerned with whether Auger scattering will eventually lead to efficiency loss "“ at some point it will make a contribution − but whether it can come into effect at current densities as low as tens of mill-Amps per square centimetre, where experiments show that droop is present. The two primary criticisms of an Auger explanation are that curve fitting of efficiency-verses-current density curves has to invoke Auger coefficients that are much larger than any calculated values, and have temperature dependence strongly contradicting first-principles physics.
Working at Sandia National Laboratories, I have been looking into these issues and have developed a new LED model that replicates the features seen in experimental current-efficiency curves. With this approach, the efficiency droop is predicted using Auger coefficients within the range calculated for phonon-assisted Auger scattering. Replicating the experimental temperature dependences is then possible, thanks to a model that features an Auger coefficient that increases with temperature.
Although the above quantitative agreement between experiment and microscopic theory helps strengthen the Auger argument, that was never the intent for pursuing this new modelling approach. Rather, my motivation came from noticing that efficiency droop occurs in many forms of LED "“ including those emitting at different wavelengths, those built on polar and non-polar substrates, those that do and don't have an electron blocking layer, and those with quantum wells and quantum dots.
As droop is observed in all these LEDs, it may be that instead of one cause dominating in all device architectures, there are several competing mechanisms. And it may be possible that their relative importance, or order of appearance with increasing excitation, changes under different experimental conditions.
To see if this is the case will require a modelling approach that systematically and consistently gathers all possible droop contributions − including Auger scattering, carrier leakage, plasma heating or defect losses − and incorporates them into a microscopic description of radiative and band-structure effects. Armed with such a model, researchers will be able to investigate and compare devices at the heterostructure-design level.
Conventional approaches
The approach that I have taken differs from the most common one for computing LED efficiency. This popular alternative will be referred to from now on as "˜Approach A', and is sketched in Figure 2. Its starting point is to choose the populations in the electron and hole states (carrier distributions), often by specifying a total carrier density and assuming thermal equilibrium at room temperature. Rates for various carrier loss processes, such as those associated with light-emission and defects, are then determined for that carrier population. Summing this, and then dividing the radiative rate by it, gives a value for the internal quantum efficiency (IQE). This can also be thought of as the rate of photon production, divided by the rate of carrier injection.
Figure 2: (Left) An outline of the widely adopted Approach A for determining LED efficiency versus pump rate. It is used to evaluate possible efficiency droop mechanisms, and to extract Shockley-Read-Hall, radiative and Auger coefficients from experimental data. (Right) Sketch of the ABC model, which is a simple implementation of Approach A.
One example of implementation of Approach A is the widely used ABC model, (see right side, Figure 2). Here, phenomenological constants A, B and C are introduced to account for defect (Shockley-Read-Hall), radiative-recombination, and Auger-scattering carrier losses, respectively. With this approach, the input parameter is total carrier density, and the defect, radiative and Auger rates are assumed to depend on this in a linear, quadratic and cubic manner, respectively.
Note that these relationships are consistent with the simplest functional dependences known for these processes. With this ABC model, the IQE, as indicated in the figure, declines at higher current densities. This approach can also include a term to account for carrier injection efficiency, which is the fraction of injected carriers ending up in the emitting region.
When A, B and C are treated as fitting parameters, the ABC model is very successful at reproducing practically all experimental curves involving a plot of IQE versus pump rate. However, concerns arise when A, B and C are associated with the physical processes of defect, radiative and Auger losses. Taking this approach oversimplifies carrier-density dependences and leads to disagreement with coefficient values from microscopic calculations that are far more rigorous − they directly account for the quantum mechanics of electrons and holes, the effects of band structure, and so on. This departure from the results of microscopic calculations is particularly alarming in the Auger case: Here, the values for the C coefficient for fitting experimental data can be several orders of magnitude higher than those provided by microscopic calculations.
It is also possible to implement approach A using less phenomenology than the ABC model. Greater rigour results from using rates for carrier losses that are determined from microscopic theory. This is possible with the radiative contribution, thanks to semiconductor quantum luminescence equations that provide a truly predictive treatment, and include many-body Coulomb effects. And for the defect contribution, it is possible to turn to microscopic models that predict deviations from the linear carrier-density dependence of the widely used Shockley-Read-Hall expression.
A similar capability exists for computing the Auger rate. However, here the challenge is not simply computing the actual rate: It is also employing accurate band-structure information, one band-gap energy removed from the conduction or valence band edges, for this is where the second electron or hole will end up.
One of the weaknesses of the microscopically based approaches is that they fail to replicate experimental results. Instead they predict that the efficiency droop is not as severe or as prevalent as that encountered in experiments. This discrepancy boils down to the microscopically calculated Auger contribution being too small to comfortably reproduce experimental droop observations.
An alternative approach
The alternative approach that I have pursued, which is referred to here as Approach B, is procedurely different (see Figure 3). This time, the starting point is the pump rate, which is the current for electrical injection. At the heart of these calculations is the determination of carrier distributions resulting from the pump, according to the band structure and loss mechanisms. Note, however, that when modelling optically pumped experiments − such as that conducted by a team at Philips Lumileds that led to the first suggestion that Auger scattering is a concern − the pump rate is equal to the pump-laser intensity.
Figure 3: Approach B, which is introduced by Weng Chow from Sandia National Laboratories, follows more closely the sequence of events leading to light emission in an experiment. Starting with the pump, carrier distributions are created, which in turn produce light and losses.
In an LED, carriers are injected into the barrier and cladding layers via carrier-carrier and carrier-phonon collisions. These interactions cause carrier capture and escape into and out of the quantum wells. What's more, they can cause heating of carrier distributions above the lattice temperature (see the illustration of carrier-carrier scattering in Figure 4a).
Figure 4: a) Carrier transport mechanisms are carrier-carrier and carrier-phonon collisions. b) Change in carrier density leads to changes in band structure, such as envelope wavefunctions for electrons and holes (red and blue curves for quantum wells and barriers, respectively), energy levels (vertical placements of envelope functions) and quantum-confinement potentials (black lines). c) Radiative transitions from quantum well and barrier.
Note that there is a complex interplay between the band structure and the carrier distributions (depicted by the two-way arrows in Figure 3). The band structure plays an important role in determining the carrier distributions "“ but in turn, the carrier distributions can alter the band structure, due to a screening of the internal electric fields (see Figure 4b).
These fields are strong in c-plane grown, wurtzite InGaN quantum wells, which are found in the light-generating region in the vast majority of LEDs. Note that in some other material systems, such as those used to produce telecom lasers, the carrier population exerts a far weaker influence on the band structure.
The strong internal electric fields produce a large distortion to the quantum-confinement potentials, and this can blur the distinction between quantum-well and barrier states (see hole energies and envelope functions in the left plot in Figure 4b). When this happens, the well and barrier can be equally occupied, with both contributing to optical emission (see Figure 4c). Emission from the quantum-well is then determined by the quantum-confined Stark effect, while barrier emission can occur due to the quantum-mechanical possibility of finding electrons and holes inside the band gap "“ this is known as the Franz-Keldysh effect.
When an Auger event occurs in a wide band-gap material, such as InGaN, transfer of a significant amount of energy to a second electron or hole can occur, leading to highly energetic carrier distributions. Due to rapid carrier-carrier scattering, any changes in the carrier distribution can then be described by changes in plasma temperature. With the ABC model, carrier losses are assumed to be fully accounted for by the Cn3 term (where n is the carrier density), and are based on a combination of the Auger process sketched in Figure 1c and carrier-phonon collisions depicted in Figure 4a. The latter is assumed to be infinitely fast, so that plasma temperature remains at the lattice temperature.
Modelling real devices
To see if the approach that I have developed provides a more accurate description of LED behaviour than other models, I have used it to calculate the behaviour of two LEDs. They were chosen to provide a very stringent test of my model, because the experiments conducted on these devices involved measurements of the IQE as a function of current density at a range of temperatures.
The devices that I have modelled emit at 523 nm and 465 nm, and have single-quantum-well active regions with a GaN barrier and either a 2 nm In0.37Ga0.63N quantum well or a 3 nm In0.20Ga0.80N quantum well, respectively. Results of the modelling show interesting similarities with the experimental data (see Figure 5). Changes in the shape of the IQE-versus-current-density curves due to differences in temperature are also seen in the model. At high lattice temperatures, both types of LED exhibit the familiar IQE-versus-current-density behaviour that can be described by the ABC model. However, at low temperatures, the device with the In0.37Ga0.63N quantum well has a second bump in its IQE profile that becomes more pronounced with decreasing temperature. An ABC model cannot describe the appearance of this feature, but it exists in the model that I have developed − and it might also be present in microscopic models based on Approach A that account for emission from barrier regions (this is shortly discussed).
Figure 5: IQE versus current density for LED with (a) 2nm In0.37Ga0.63N and (b) 3nm In0.20Ga0.80N quantum-well active media. Approach B is used to obtain the computed curves. The insets show the experimental results.
With my model, there is no double bump transition in the LED with the In0.20Ga0.80N quantum well, replicating the experimental result. The difference between the devices with In0.37Ga0.63N and In0.20Ga0.80N wells is fundamental to the quantum-well structures, and depends entirely on the band-structure differences that result from a smaller piezoelectric field in the active region with a lower indium concentration.
Attempts to model LED behaviour should involve plausible physical assumptions. That is the case in the model that I have proposed: magnitudes and temperature dependence features in the Shockley-Read-Hall and Auger coefficients, and the carrier-phonon scattering rate (in figure 6, these coefficients are plotted as a function of temperature).
Figure 6: (From top to bottom) The Shockley-Read-Hall coefficient, Auger coefficient and carrier-phonon scattering rate used in producing the curves in Figure 5. An important result is Auger coefficient magnitude range and increasing value with increasing temperature, consistent with first-principles physics.
The decrease in the Shockley-Read-Hall coefficient with decreasing temperature is expected for defect-related loss, and the increases in Auger coefficient and carrier-phonon scattering rate with increasing temperature are consistent with microscopic calculations. Meanwhile, the values for the Auger coefficient are within the range predicted for phonon-assisted Auger scattering, and are below values obtained from experimental curve fitting with the ABC model, which are thought to be unrealistically high.
A great strength of my model is that it provides physical insight into the double bump behaviour. In Figure 7a, there are plots for the quantum-well and barrier contributions to spontaneous emission at various current densities for the In0.37Ga0.63N device at a lattice temperature of 200K. The curves show that the first bump is primarily due to barrier emission, while the dominant cause of the higher excitation bump is quantum-well emission. An interplay of the quantum-confined Stark and Franz-Keldysh effects governs these relative contributions, and explains the blue shift of the emission peak with increasing excitation.
Figure 7: a) Spontaneous emission contributions from quantum wells and barriers (solid and dashed curves, respectively) versus current density. b) Contributions to IQE from recombination and scattering processes. The solid red curve is the sum of quantum-well and barrier emission, the dotted curve shows the defect loss, and the dashed curve shows the leakage contribution. c) IQE and d) plasma temperature versus current density. Differences between solid and dashed curves indicate the Auger contributions. All results are for an In0.37Ga0.63N device at 200K lattice temperature.
Also included in Figure 7 are the physical mechanisms integrated into Approach B. As excitation increases, their relative contribution changes. Figure 7b suggests that the required Auger coefficients are smaller, because the onset of Auger carrier loss is delayed to higher carrier densities with the help of carrier leakage. Loss due to leakage includes failure to capture, and loss of barrier carrier population through non-radiative recombination, as well as drift and diffusion out of the active region. Its domination at low carrier densities is due to the distortion of the confinement potential by strong internal electric fields. The solid red curve in Figure 7b is the sum of quantum-well and barrier emission: This has a slope change at low current density resulting from a switch between predominately barrier and predominately quantum-well emission. At higher excitation, a second slope change appears, due to the onset of Auger carrier loss. This appears as the difference between the solid black curve and the sum of all the other curves.
A more complete description of Auger effects is possible with the model based on Approach B, thanks to the tracking of carrier occupations over time in individual momentum-resolved states. By taking this approach, it is possible to model side effects from Auger scattering, such as increases in plasma temperature or carrier leakage.
A projected slide shows the equations of motion for a LED model following Approach B. Similar equations have been used to explore dynamical response and instabilities in quantum-dot and quantum-well lasers, under conditions such as the presence of optical feedback or injected signal. Also in the picture are (left to right) Alan Wright, Weng Chow and Jeff Nelson, who contributed to the early GaN research at Sandia National Laboratories. During the early 1990s, they welcomed the emergence of wide-band-gap lasers and solid-state-lighting as providing opportunities to continue exploring many-body physics, and how to handle defects and the d-shell electrons in density-functional-theory calculations.
The influence of the Auger coefficient on plots of IQE as a function of current density is illustrated in Figure 7c. Two values are employed: a C coefficient of 2.3 x 10-31 cm6 s-1, which was used to produce the 200K lattice temperature curve in Figure 5a; and a value of 10-34 cm6 s-1, which is what one would expect by extending an Auger coefficient calculation for near-infrared semiconductors to one with roughly a 2.7 eV band-gap energy.
In these plots in Figure 7c, there is no efficiency droop when a value of C equal to 10-34 cm6 s-1 is used for the current densities considered. These plots also indicate the efficiency loss from Auger scattering when the value for C is 2.3 x 10-31 cm6 s-1 "“ this is the difference between the solid and dashed curves. Plots of the plasma temperature versus current density are shown in Figure 7d. Here, the dashed curve shows the rise in temperature that primarily results from the capture of carriers from barrier to quantum-well states. Indicated by the solid curve is a significant additional rise in plasma temperature because of Auger scattering.
These results are encouraging, given that my work is still in its infancy. So far, it is a very rudimentary implementation of a new modelling approach that can successfully replicate experimental results with microscopic theory. There is much room for improvement, including more careful treatment of band-structure and carrier-dynamics coupling, the inclusion of many-body effects and a more detailed description of defect loss. If these refinements are made, hopefully this will deliver an even closer agreement between theory and experiment, and ultimately offer new insights into the origins of LED droop.
The described work is performed at Sandia's Solid-State Lighting Science Center, an Energy Frontier Research Center (EFRC) funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences. Weng Chow wishes to thank the Technical University Berlin for hospitality and travel support provided by SFB787.
Further reading
Further information on Modeling Approach B:
W. W. Chow Optics Express 19 21818 (2011)
W. W. Chow Optics Express 22 1413 (2014)
Information on experiments used to illustrate Approach B:
J. Hader et. al. Appl. Phys. Lett. 99 181127 (2011)
A. Laubsch et. al. Phys. Status Solidi C 6 S913 (2009)
K. Fujiwara et. al. Phys. Status Solidi C 6 S814 (2009)
