Carrier density dependent Auger recombination in c-plane (In,Ga)N/GaN quantum wells: insights from atomistic calculations

Understanding Auger recombination in (In,Ga)N-based quantum wells is of central importance to unravelling the experimentally observed efficiency ‘droop’ in modern (In,Ga)N light emitting diodes (LEDs). While there have been conflicting results in the literature about the importance of non-radiative Auger recombination processes for the droop phenomenon, it has been discussed that alloy fluctuations strongly enhance the Auger rate. However, these studies were often focused on bulk systems, not quantum wells, which lie at the heart of (In,Ga)N-based LEDs. In this study, we present an atomistic analysis of the carrier density dependence of the Auger recombination coefficients in (In,Ga)N/GaN quantum wells. The model accounts for random alloy fluctuations, the connected carrier localisation effects, and carrier density dependent screening of the built-in polarisation fields. Our studies reveal that at low temperatures and low carrier densities the calculated Auger coefficients are strongly dependent on the alloy microstructure. However, at elevated temperatures and carrier densities, where the localised states are starting to be saturated, the different alloy configurations studied give (very) similar Auger coefficients. We find that over the range of carrier densities investigated, the contribution of the electron-electron–hole related Auger process is of secondary importance compared to the hole-hole-electron process. Overall, for higher temperatures and carrier densities, our calculated total Auger coefficients are in excess of 10−31 cm6 s−1 and may reach 10−30 cm6 s−1, which, based on current understanding in the literature, is sufficient to result in a significant efficiency droop. Thus, our results are indicative of Auger recombination being an important contributor to the efficiency droop in (In,Ga)N-based light emitters even without defect-assisted processes.


Introduction
Indium gallium nitride ((In,Ga)N) quantum wells (QWs) are commonly used as the active region of modern lightemitting diodes (LEDs) operating in the violet-blue spectral range [1][2][3].Even though they have found widespread applications, their efficiency deteriorates as the current density driving the device is increased [2].Several factors have been discussed as the potential origin of this phenomenon, known as the efficiency 'droop' [2].As it has been observed that the efficiency 'droop' scales with the third power of the carrier density, Auger recombination has been identified as a likely cause of this effect [3][4][5][6]; this includes defect and trap-assisted Auger recombination (TAAR) processes [7,8].Initial theoretical predictions produced Auger coefficients much smaller than required to explain the experimentally observed results, thus raising the question of whether Auger recombination is really the origin of the efficiency 'droop' in (In,Ga)N/GaN QWs [9][10][11].However, these first studies widely neglected carrier localisation effects, which significantly affect the electronic and optical properties of (In,Ga)Nbased systems [9,10].First-principles calculations in bulk (In,Ga)N have shown that alloy disorder, and also phonons, can lead to a significant enhancement of Auger recombination rates and are thus important to be considered for an accurate description of these non-radiative processes [12].However, translating the first-principles results to (In,Ga)N/GaN QWs is not straightforward for several reasons.Firstly, a fully three dimensional description of the (In,Ga)N QW and GaN barrier system is required, including the intrinsic wurtzite nitridespecific (piezoelectric and spontaneous) polarisation fields.Furthermore, the dimensions of the simulation cell need to be large in the growth plane (c-plane) of an (In,Ga)N QW system in order to capture wave-function localisation effects [13].Thirdly, due to the presence of the intrinsic piezoelectric and spontaneous polarisation fields, increasing the carrier density in an (In,Ga)N well leads to a screening of the internal polarisation fields; therefore, this requires a self-consistent calculation on a large simulation supercell.Overall, this screening effect can lead to changes in the electronic structure and, thus, carrier localisation effects in the well, which in turn can affect the Auger rate.Moreover, to capture carrier localisation effects accurately, ideally an atomistic electronic structure theory is required.Taking all this together, density-functional theory (DFT) calculations are not feasible for the QW systems under consideration, and empirical (atomistic) models need to be used.Given all these challenges, only recently attempts have been undertaken to meet all these requirements for a detailed understanding of Auger recombination rates in (In,Ga)N QW systems [10,14,15].For instance, we have presented an atomistic theoretical analysis of the temperature dependence of the Auger rate in (In,Ga)N/GaN QWs at a fixed carrier density.Indeed our investigations revealed significant Auger coefficients in these systems [15].However, these calculations were performed at a moderate carrier density in the well, such that the screening of the built-in field was of secondary importance.Thus, our previous studies did not give insight into situations relevant to the current (efficiency) droop observed at high carrier densities in (In,Ga)N-based LEDs.
In this work we extend our model to address the question of how the Auger rate in (In,Ga)N/GaN QWs changes with carrier density.We have selected here systems with 15% In, since this In content is in the range of contents often used in blue to green emitting (In,Ga)N QW systems (depending also on well width) [13,16,17].Our calculations show that at low temperatures (T = 10 K), and in the lower carrier density range, the alloy microstructure of the well leads to strong variations in the Auger coefficients.This stems from the fact that, in this situation, states near the 'band' edges are being populated, which are strongly localised, especially for holes.However, when increasing the carrier density, the impact of the alloy microstructure is diminished.At room temperature (T = 300 K) these effects are further reduced and at (very) high carrier densities (n = 10 20 cm −3 ) the different microscopic alloy configurations (Cfs.)result in (very) similar Auger coefficients.Moreover, we observe that the Auger rate starts to increase due to the screening of the built-in field at a similar carrier density as reported for instance in the experimental studies in [18].Also our calculations show that with increasing carrier density the electron-electron-hole (eeh) Auger rate is of secondary importance when compared to the hole-holeelectron rate (hhe).Finally, on average we find a total Auger coefficient >10 −31 cm 6 s −1 , which is usually regarded as large enough to facilitate the efficiency 'droop' in (In,Ga)N based LEDs [2,19] and is further supported by our recent theory experiment comparison [20].
The paper is organised as follows.Section 2 describes the theoretical background of our studies, while section 3 presents our results.A summary and conclusion is given in section 4.

Theory
In this section we give an overview of the theoretical framework employed to obtain the Auger rates and coefficients in (In,Ga)N/GaN QWs.We start in section 2.1 with a brief overview of the method used to calculate the Auger recombination rate.Section 2.2 describes the electronic structure model underlying our calculations, which is an atomistic tightbinding (TB) model.This section also includes a discussion of the self-consistent method used to account for the screening of the built-in polarisation field with increasing carrier density.

Calculation of Auger rates
For the calculation of the Auger rate we follow the approach discussed in [12,15].Overall, we focus on the alloy-enhanced Auger recombination since in the band-gap/wavelength range relevant to our studies, contributions from phonon-assisted Auger processes are smaller compared to alloy-enhanced scattering effects as shown in the DFT-based results for bulk (In,Ga)N in [12].More details are also given in our previous work [15].In the following we briefly summarise the approach that underlies the Auger rate investigations.
Starting from Fermi's golden rule, the Auger recombination rate, R, can be evaluated using the expression Bold indices run over the involved electronic states.Temperature and carrier density dependence are included in the model in two places, the first of which is the Fermi factor, (1), where the factors f m denote Fermi-Dirac functions.The second of which arises from the screening of the Coulomb interaction, described by the (screened) Coulomb matrix elements (CMEs), M 1234 .The CMEs between different states, ψ m , are given by the expression The wave-functions, ψ m , required for evaluating the above matrix elements are obtained from our TB model, which will be described below.
The screening factor (i.e. the inverse screening length), α, is determined separately for electrons (α e ), and holes (α h ), by using the Debye-Hückel equation.For electrons this is α e = 4π ne 2 ϵ0ϵ∞kBT , and similar for holes (with p replacing n).
Here, k B is the Boltzmann constant, T is temperature, ε 0 is the vacuum permittivity, n (p) is the electron (hole) carrier density, ϵ ∞ the high frequency dielectric constant and e is the elementary charge.The total inverse free-carrier screening length is then given by α 2 = α 2 e + α 2 h .The calculation of the CMEs, when using TB wave-functions, is detailed in [15].
Based on the Auger recombination rate, R, the Auger coefficient, C, is determined by where V is the volume of the QW and n is the carrier density.
In the following we distinguish between two Auger process coefficients, one for electron-electron-hole Auger recombination, C eeh , and one describing the hole-hole-electron Auger recombination, C hhe .The total Auger coefficient is then Finally, all calculations have been performed under the assumption that p = n, i.e. that the density of holes is equal to that of electrons, an assumption that is commonly employed in the literature [12,18,21].

Electronic structure model for (In,Ga)N QWs
To obtain the electronic structure required for the Auger rate calculations discussed above, we use a nearest-neighbour sp 3 TB model that takes input from a valence force field and local polarisation theory model to account for alloy fluctuations in (In,Ga)N/GaN QWs on an atomistic level [22,23].
As described in our previous work, the model has been parameterised and benchmarked against hybrid-functional (HSE) DFT and experimental literature data.We note that, for bulk structures, the TB bands are in good agreement with HSE-DFT results not only near the Γ-point but also away from Γ; this is also true for energetically higher (lower) lying conduction (valence) band states, as shown in [22].This aspect is particularly important since, in contrast to radiative recombination processes, for Auger recombination not only states near the band edge but also higher/lower lying states are relevant.Moreover, our atomistic theoretical framework has shown good agreement with experimental data on (In,Ga)N/GaN QWs, for instance in terms of photolumninescence (PL) linewidths or carrier localisation lengths [13], highlighting that the model captures alloy-induced carrier localisation effects accurately.
For low carrier densities in (In,Ga)N/GaN QWs, the model briefly outlined above is sufficient to determine their electronic structure since screening of the internal polarisation field is on secondary importance.As a result, it is possible to directly use the obtained single particle TB states and energies to calculate, e.g.radiative recombination rates [24], excitonic effects [25], or the temperature dependence of the Auger rate [15] in the low carrier density regime.However, at higher carrier densities this is no longer the case as the electrostatic built-in field will be screened.Previous theoretical and experimental works indicate that for the QW systems targeted here (see discussion below on the QW model system), the screening effect becomes important at carrier densities on the order of n = 10 19 cm −3 and above [6,26,27].Thus, a fully self-consistent approach is required.However, for a supercell that contains approximately 82 000 atoms (see discussion below), it is numerically very demanding to carry out such a self-consistent TB-Poisson calculation, since it has to be performed for each carrier density and for a large number of electronic states.In general, we note that even without having to evaluate the electronic structure self-consistently, Auger calculations are already numerically extremely demanding due to the large number of states and connected CMEs required for such an investigation, see section 2.1.To keep the information about alloy-induced carrier localisation effects, but at the same time take built-in field screening effects into account, we proceed as follows.We start from a one dimensional Schrödinger equation describing a wurtzite (In,Ga)N QW grown along the c-axis: where the confining potential U b (z) for electrons, b = e, or holes, b = h, is given by [28] The different contributions to U b (z) are the bare (band offset) confining potential, U b 0 (z), the intrinsic (spontaneous and piezoelectric related) electrostatic potential, U b p (z), and the screening potential U b scr (z), respectively.Here, U b scr (z) arises from the spatial separation of electron and hole wave-functions in the well due to the presence of the intrinsic built-in field.Following Haug and Koch [29], the screening potential is calculated via: Equations ( 3)-( 5) are evaluated self-consistently for a given carrier density, n sys , within the well.We use band offsets, average strain fields, spontaneous and piezoelectric coefficients as well as effective masses from our TB model as input for ( 3)- (5).Solving these equations self-consistently allows us to determine U e,h scr (z) for a given carrier density, n sys .As a result, one can determine by how much the intrinsic electrostatic built-in field is reduced at a given n sys .This information is then fed back into the TB model by adjusting/scaling the built-in potential across the well and diagonalising the Hamiltonian for each carrier density n sys .In doing so, we obtain, for a given n sys , the (screened) TB wave-functions and energies for our supercell, while still including local alloy contributions.The obtained wave-functions and energies then form the input for the Auger recombination rate calculations detailed above.We note that a similar screening approach has been used by Auf der Maur et al [30], to study the impact of alloy fluctuations on the radiative recombination rate in cplane (In,Ga)N/GaN QWs.At elevated temperatures and/or high carrier densities the above chosen method should be sufficient to capture built-in field screening effects, as carriers populate extended, more delocalised states.At low carrier densities and low temperatures, when mainly localised states near the band edges are being populated, the interplay of builtin field screening and carrier localisation may be more complicated.However, our theory experiment comparison [13] on optical properties in (In,Ga)N QWs at low temperatures and carrier densities revealed that when using unscreened TB wave-functions, a very good agreement between calculated and measured quantities, such as PL peak positions and full width at half maximum (FWHM) are obtained.Moreover, our joint theory experiment investigations [31] on the evolution of the electronic and optical properties of (In,Ga)N QWs as a function of electric field strength in these systems, where, on the theory side, also a 1D field correction to internal electric field was employed, showed very good agreement in terms of the trends observed in experiment and theory.All this supports the conclusion that the approach chosen here should provide a reasonable approximation for capturing polarisation field screening effects with increasing carrier density.While, future studies can target a fully self-consistent 3D approach, our results below already show that (i) we obtain an onset of the built-in potential screening at carrier densities similar to experimental studies and (ii) the expected behaviour of the electron and hole wave-function separation is observed (i.e. that the separation decreases) with increasing carrier density.Thus, we expect that the above detailed method is sufficient to capture the main effects of the built-in field screening on the electronic structure of (In,Ga)N/GaN QWs.

Results and discussion
In the following section we present results for an (In,Ga)N/GaN QW with 15% In and a well width of 3 nm, assuming a random (In,Ga)N alloy in the well.More details, e.g. about the simulation cell, can be found in [15].Overall, the assumed composition and well width are often found in experimental realisations of (In,Ga)N/GaN QWs [17,18,20].
We start our analysis with a brief recap of some of our previous results.We have recently studied the temperature dependence of both radiative and Auger (non-radiative) recombination rate at a fixed carrier density for In 0.15 Ga 0.85 N/GaN QWs [15,24].Figure 1 summarises data on C hhe as a function of temperature (C eeh contributions are of secondary importance at this carrier density, see [15]).The figure reveals that (i) especially at lower temperatures the alloy microstructure can significantly impact the Auger coefficient, C hhe , and (ii) out of the ten studied alloy configurations, eight give very similar coefficients at higher temperatures.As discussed in [15], the two 'outlier' configurations (Cfs.4 and 10) are related to strongly localised (hole) states and may be connected to recent literature discussions on TAAR processes [7,8], as they form localised states in the band gap similar to a defect state [32].
As already discussed above, the computational expense of the Auger rate calculations, given the large number of states and CMEs involved at high carrier densities (⩾1 × 10 19 cm −3 ) is immense.To investigate the carrier density dependence of the Auger rate and thus the Auger coefficients, we have chosen three configurations from the previously studied ten, guided by the following considerations.The experimental data in [20] on (In,Ga)N/GaN QWs with well widths and In contents similar to the systems considered here indicated that for high carrier densities in the wells, the measured optical properties did not vary significantly between different spots on the sample; all the measurements in [20] were carried out at a temperature of 300 K. To take these findings into account, we aimed for two configurations that reflect the 'most commonly' found Auger coefficients in our theoretical analysis.Based on this we selected Cfs. 2 and 7 (also highlighted in figure 1) as they give a very similar Auger coefficient at higher temperatures, similar to 8 of the 10 previously investigated systems.Additionally, the Auger coefficients behave differently with decreasing temperature: for Cf. 2 C hhe increases with decreasing temperature while for Cf. 7 C hhe decreases with decreasing temperature.In addition to having representative configurations at 300 K, these configurations allow us also to shed light on the impact of the alloy microstructure on the results.Finally, we have selected Cf. 4, which belongs to the set of two 'outlier' configurations shown in figure 1, which exhibit significantly larger Auger coefficients when compared to all other configurations studied.Our previous work [15] revealed that these 'outlier' configurations also had noticeably higher hole  ground state energies, thus indicating strong hole localisation effects, e.g.states deeper in the band gap of the QW.This situation may therefore be related to TAAR processes [32] where defect states are formed in the band gap.In our case however, these traps/localised states in the band gap are induced by alloy fluctuations and not defects.If such extreme situations would occur in an (In,Ga)N QW, one could expect that experimental studies observe different recombination characteristics (e.g.much faster non-radiative recombination processes) in different areas of a sample.While these 'outlier' configurations may not be representative of the experiments in [20], where it is concluded that defect-assisted Auger processes are of secondary importance in the samples studied, from a theoretical perspective it is nevertheless interesting to study the carrier density dependence of the Auger recombination in such an 'extreme' configuration.Overall, the three selected configurations furnish a good overview of how Auger coefficients may change with carrier density in (In,Ga)N/GaN QW systems.
Before turning to the carrier density dependence of the Auger coefficients, we first analyse how the screening of the built-in field affects the electronic structure of the QW. Figure 2 depicts the ground-state electron (red) and hole (blue) charge densities for Cf. 2 for varying carrier densities (increasing from left to right).We have selected Cf. 2 here since (i) at low temperatures its Auger coefficient C hhe is very similar to Cf. 4, which gives a significantly larger Auger coefficient over the full temperature range, and (ii) C hhe decreases with increasing temperature for Cf. 2, thus also providing insight into the importance of excited states (further discussion below).Looking first at the 'Top View' (first row, view along caxis), for lower carrier densities the electron charge density is mainly localised by the well width fluctuation (WWF) and with increasing carrier density starts to 'delocalise' in the c-plane.However, the alloy microstructure is of secondary importance when compared to holes (blue charge densities in figure 2).The hole charge density is strongly localised, and remains so even at higher carrier densities.Looking at the 'Side View' (second row), while still being strongly localised by the alloy fluctuations, the hole ground state localisation region changes (when comparing charge densities for built-in fields determined at carrier densities 1 × 10 18 cm −3 and 5 × 10 19 cm −3 ).We attribute this to the following.At low carrier densities, the electrostatic built-in field forces both electron and hole wave-functions to be localised at the lower (holes) and upper (electrons) QW barrier interfaces.Thus, only the alloy microstructure at and near these interfaces plays a role for alloy-induced carrier localisation effects.However, when the built-in field is reduced by screening effects, local potential 'pockets' away from the interfaces now also play a role for carrier localisation effects.This is an important observation as it means that screening of the built-in field not only changes the electron and hole wave-function overlap (both inplane and out-of plane), but it can be accompanied by significant modifications in the electronic structure of the well (e.g. also energetic separation between localised excited states).
Figure 2 reveals, in general, the expected effect that the polarisation field spatially separates electron and hole wavefunctions along the c-axis at lower carrier densities.With increasing carrier density the built-in field is screened and the spatial separation along the growth direction is reduced.As a result, the electron and hole wave-function overlap is increased.Overall, and based on the method described above, we find that the screening of the built-in field becomes noticeable (i.e.reduction of the electrostatic built-in field by, for example, more than 10%) for carrier densities exceeding n = 5 × 10 18 cm −3 ; this finding is in line with recent literature data [6,27].
However, with increasing carrier density and/or temperature, not only ground states, but also excited states are important for Auger recombination, especially for holes where one is left with strong alloy-induced carrier localisation effects.Figure 3 displays the first five hole states for Cf. 2, with the built-in field determined for a carrier density of n = 5 × 10 18 cm −3 .Several features of these states are important.Firstly, figure 3 clearly shows that not only the ground state exhibits strong localisation effects, but also the first few excited states.To characterise carrier localisation effects further, often quantities such as the inverse participation ratio (IPR) are investigated.In [13] a detailed analysis of the IPR values for (In,Ga)N QWs similar to those studied here have been presented, which reveal hole localisation lengths on the order of 1 nm for ground and excited states (e.g. for states in the energy range of 50 meV into the valence 'band').Secondly, from figure 3 one can see that while the ground state, ψ 1 h , and the first excited state, ψ 2 h , are localised by alloy fluctuations, they localise in different spatial in-plane regions.This result is not captured by a calculation where these localisation features are neglected, e.g. a standard one-dimensional continuum-based description (e.g. a virtual crystal approximation (VCA)) of (In,Ga)N/GaN QWs, which is often used in literature to evaluate Auger recombination [9,33,34].Thus, CMEs involving wave-functions localised in different in-plane regions may be strongly reduced in magnitude compared to a VCA description of the same system.However, figure 3 also reveals that the second excited state, ψ 3 h , is strongly localised and localises in the same spatial position as the ground state, ψ 1 h ; this situation could for instance be approximated or described by a 'local' quantum dot-like structure with a ground and an excited state.This also indicates that CMEs involving these states may be strongly enhanced compared to a VCA description.The situation is further modified and complicated by the spatial separation of the third and fourth excited hole states, which have multiple localisation centres, sometimes spatially overlapping with other hole states, and sometimes not.
Table 1 displays the energy separations between the first four excited hole states and the hole ground state depicted in figure 3.As the table shows, the energy separation between these states is only a few meV, so that at room temperature (k b T ≈ 25 meV), and even low carrier densities, the excited, localised hole states are being populated and can contribute to the Auger recombination process.This indicates already that an accurate description of the electronic structure of (In,Ga)N QWs, capturing carrier localisation, is of central importance when targeting Auger recombination processes in these systems.
Overall, the results presented above indicate that while the wave-function overlap along the c-axis increases with increasing carrier density, the wave-function overlap between hole states within the c-plane may strongly depend on the alloy microstructure.This should at least be the case for lower temperatures and lower carrier densities since here mainly the localised hole states near the 'band' edge are important.As a consequence, and even though with increasing carrier density more states are involved in the Auger rate, (1), their contribution may be small due the spatial in-plane separation of the (hole) states.Furthermore, it is important to note that to evaluate the Auger coefficient, the rate is divided by the carrier density cubed, (2).Thus, at lower temperatures, the Auger coefficient may increase or decrease with increasing carrier density, depending on whether n 3 or the number and magnitude of the involved states/CMEs dominate the Auger rate, (1).Finally, the evolution of the Auger coefficient may be further impacted by the fact that the electronic structure changes with increasing carrier density as seen above in figure 2, where, for instance, the hole ground state at lower carrier densities is different from the hole ground state in the high carrier density regime and so are the excited states, including their energetic separation.
In general, at higher temperatures and/or higher carrier densities, where the localised states are saturated, and one is approaching the situation that the wave-functions for electrons and holes are more delocalised, the alloy microstructure is expected to be less important.Thus, more macroscopic effects, e.g. the spatial separation of electrons and holes along the caxis, become important.
Equipped with this insight and taking these considerations as a whole into account, we present in a first step the calculated total Auger coefficient, C tot , as a function of the carrier density, n.We start our analysis with low-temperature (T = 10 K) results, before turning to room-temperature data (T = 300 K).Finally, we investigate in more detail the contribution from electron-electron-hole (C eeh ) and hole-holeelectron (C hhe ) processes to the total Auger coefficient, C tot = C eeh + C hhe .
Figure 4 displays the total Auger coefficient, C tot , for a temperature of T = 10 K as a function of carrier density, n; the data are shown for the three microscopic alloy configurations considered in this study (see discussion above).This figure shows that for carrier densities below n = 1 × 10 19 cm −3 , the alloy microstructure significantly impacts not only the magnitude of C tot , but also how C tot evolves with increasing carrier density n.At low carrier densities, the behaviour seen already in figure 1, is here also visible, namely a large spread in the values of C tot .As carrier density in the well increases up to n = 1 × 10 19 cm −3 , the configurations exhibit different behaviours.In the case of Cf. 7, C tot decreases up to n = 1 × 10 18 cm −3 before starting to increase.Although at low carrier densities Cf. 2 and 4 exhibit Auger coefficients of similar magnitude, their carrier density dependence is quite different: C tot for Cf. 2 decreases when the carrier density increases up to n = 1 × 10 19 cm −3 ; C tot for Cf. 4 has an almost constant (slight decrease) behaviour up to a carrier density of n = 1 × 10 18 cm −3 and a 'kink' in the evolution of C tot at n = 5 × 10 18 cm −3 .Overall, and as discussed above, several factors play an important role (e.g.number of states involved, wave-function localisation effects and associated magnitudes of the CMEs, potential changes in the electronic structure, scaling by 1/n 3 ) in determining how the Auger coefficient evolves with increasing carrier density, which makes it difficult to deconvolute the impact of these different factors.
However, figure 4 also shows that for carrier densities n ⩾ 1 × 10 19 cm −3 , and independent of the alloy configuration, the Auger coefficients, C tot , start to increase with increasing carrier density.Moreover, at the highest carrier density considered here, we find that all three alloy configurations give, to a first approximation, very similar Auger coefficients.Overall, we attribute this to the following two effects.Firstly, with increasing carrier density the electrostatic builtin field is significantly screened (at a carrier density of n = 5 × 10 19 cm −3 the built-in field is reduced by a factor of around 2 when compared to a situation that neglects any screening), which increases the electron and hole wave-function overlap, as indicated in figure 2. As a consequence, one may expect the increase observed in C tot .Furthermore, with increasing carrier density the localised states become saturated and for all configurations more delocalised electron and hole states are populated and thus start to contribute.These more delocalised states are less impacted by the alloy microstructure and thus less prone to in-plane carrier separation.All in all, with increasing carrier density the Auger coefficient increases, and differences between the three different alloy configurations are 'smeared' out, resulting in the observed very similar results for the Auger coefficient at higher carrier densities.
Based on these considerations, one may also expect that at elevated temperatures, for which the carriers are distributed over a larger range of (localised hole) states than at lower temperatures, variations in Auger coefficient for each configuration become less pronounced.This is reflected in figure 5, which depicts C tot as a function of the carrier density at a temperature of T = 300 K.For densities n ⩽ 1 × 10 18 cm −3 all configurations show, in general, a (slight) increase with increasing carrier density.Turning to the magnitude of the Auger coefficient in the lower carrier density regime of n ⩽ 1 × 10 18 cm −3 , we find that Cfs. 4 and 7 are slightly affected by increasing the temperature from T = 10 K (see figure 4) to T = 300 K when compared to Cf. 2. For Cf. 2 we observe a noticeable decrease in the magnitude of C tot when increasing the temperature from T = 10 K (see figure 4) to T = 300 K.This reflects the results already seen in [15], and summarised in figure 1.
In the carrier density range n = 1 × 10 18 cm −3 to n = 5 × 10 18 cm −3 , the Auger coefficient C tot exhibits only a slight carrier density dependence.Turning to the results when increasing the carrier density beyond n = 5 × 10 18 cm −3 , we observe that for all configurations studied, C tot starts to increase.Moreover, and similar to the low temperature results, Cfs. 4 and 7 give very similar Auger coefficients at the high carrier density of n = 1 × 10 20 cm −3 , and Cf. 2 is of the same order of magnitude.Overall, we attribute this to similar effects discussed above for the low temperature results.Firstly, at these higher carrier densities the localised states become saturated.Thus, with increasing carrier density more delocalised states, which are less sensitive to the alloy microstructure, contribute to the Auger rate.Secondly, at high carrier densities the intrinsic built-in field is reduced, and in conjunction with the population of the more delocalised states, the electron and hole wave-function overlap along the growth direction is increased, which results in larger CMEs and consequently an increase in the Auger rate and thus the coefficient C tot .
So far we have only investigated the total Auger coefficient, C tot = C eeh + C hhe , but not the individual contributions from the electron-electron-hole, C eeh , and hole-hole-electron, C hhe , processes.As discussed in our previous study [15] and mentioned above, at lower carrier densities, the C hhe part dominates the magnitude of the total Auger coefficient, C tot , and C eeh is of secondary importance.This is consistent with the DFT studies in [12] on (In,Ga)N bulk systems and also theory work on c-plane (In,Ga)N QWs using a modified continuumbased model and analytic considerations [10,18].However, experimental investigations by Nirschl et al [5] on (In,Ga)N QWs, concluded that the eeh Auger rate is equal to, or larger than, the hhe rate.The work by David et al [18] discusses in general that screening the built-in field increases the relative importance of the C eeh contribution.Jones et al [10], using a modified continuum based model to describe the Auger recombination, conclude when studying non-polar (In,Ga)N QWs, which do not exhibit any macroscopic polarisation field [25], that indeed screening of the polarisation field may lead to the situation found in the experiments by Nirschl et al [5].Thus, to gain insight into this question, figure 6 presents the (percentage) contribution from the electron-electron-hole, ∆C eeh (n) = (C eeh (n)/C tot (n)), and hole-hole-electron, ∆C hhe (n) = (C hhe (n)/C tot (n)), to the total Auger coefficient, C tot , as a function of the carrier density, n.The data are shown for low (T = 10 K) and room temperature (T = 300 K).At low carrier densities (n ⩽ 5 × 10 18 cm −3 ) and independent of the temperature, figure 6 shows that C hhe dominates the total rate, being over 95% of C tot .This finding is consistent with our previous results [15] and literature theory data [10,18] on c-plane (In,Ga)N/GaN QWs.In the higher carrier density regime, i.e. when increasing the carrier density beyond n = 5 × 10 18 cm −3 , C eeh increases slightly, relative to the contribution from C hhe ; the general trend is thus consistent with expectations in, for example, [18].At high carrier densities, C hhe still contributes ⩾90% of C tot which stands in contrast to the experimental results of Nirschl et al [5], and the calculations by Jones et al [10] for non-polar (In,Ga)N QWs.However, it is to note that the electronic structure of a non-polar (In,Ga)N QW system is different to that of a c-plane well [25].Thus Auger rates may behave differently in a non-polar well when compared to c-plane systems with partially screened built-in fields.Also, further studies comparing atomistic and modified continuum-based models are required to shed light on the importance of electronelectron-hole Auger processes in polar c-plane and non-polar (In,Ga)N QWs.
In addition to the above discussed uncertainties, there is also a wide range of Auger coefficient values reported in the literature [15,35]; the C tot values varying by a few orders of magnitude.Often the magnitude of C tot is used to estimate the significance of Auger recombination for the efficiency droop [19].However, it is important to note that several factors must be considered before conclusions about the relationship between Auger recombination and efficiency droop can be drawn.For instance, our calculations show already that the Auger coefficient is not independent of carrier density, as the widely used ABC model may suggest in its original form.We find a total Auger coefficient on the order of 10 −31 cm 6 s −1 in the low carrier density regime while in the high carrier density regime C tot may reach values on the order of 10 −30 cm 6 s −1 , so an order of magnitude larger when compared to the low carrier density result.Thus, when comparing Auger coefficients, this should ideally be done at fixed carrier densities and temperatures.
Secondly, when studying the droop phenomenon not only the magnitude and carrier density dependence of C tot (n) is important but also its competition with the radiative recombination, often described by the so-called B coefficient.In general, the B coefficient itself also carries a carrier density dependence.Therefore, while the magnitude of C tot (n) is important, its evolution with carrier density and how this compares with the evolution of the B coefficient is key.
In our recent theory experiment comparison we have carried out such an analysis [20] for In 0.15 Ga 0.85 N/GaN QWs.Our work indicated that the Auger coefficients obtained here and in conjunction with B(n) coefficients calculated from our TB model, again as a function of carrier density, provided a very good description of the experimental data [20] in the high carrier density regime.We found that in this high carrier density regime, Auger recombination can lead to a significant efficiency droop, even without any defect-assisted processes.Nevertheless, the predicted internal quantum efficiency was only limited to 70% in a green emitting (In,Ga)N QW system, at least when excited resonantly.In conjunction with the experimental data, our results suggest that other factors external to the QW impact the peak external quantum efficiency in green emitting (In,Ga)N-based LEDs.

Conclusion
In this study we presented an atomistic theoretical analysis of non-radiative Auger recombination processes in c-plane (In,Ga)N/GaN QWs as functions of temperature and carrier density.The model accounts for random alloy fluctuations, carrier localisation effects, and screening of the electrostatic built-in field with increasing carrier density.Prior to investigating the evolution of the total Auger coefficient with carrier density, we determined the temperature evolution of the Auger rate.These calculations revealed that at low temperatures and carrier densities, the Auger coefficients are strongly impacted by the alloy microstructure.However, as temperature and carrier density is increased and the localised states are saturated, the different microscopic alloy configurations exhibit very similar Auger coefficients.Moreover, we find that the contribution from the electron-electron-hole Auger process is of secondary importance when compared to the hole-hole-electron process, and its relative importance increases only slightly with increasing carrier density.For the total Auger recombination coefficient, in general, values over 10 −31 cm 6 s −1 are found, which in the high carrier density regime may even reach 10 −30 cm 6 s −1 .Such values are, in general, considered to be sufficient to significantly impact the efficiency of (In,Ga)N-based devices.

Figure 1 .
Figure 1.Hole-hole-electron Auger coefficient, C hhe , as a function of temperature for 10 different microscopic alloy configurations of an In 0.15 Ga 0.85 N/GaN quantum well at a fixed, low carrier density of n = 3.8 × 10 18 cm −3 [15].The configurations used to investigate the carrier density dependence of the Auger recombination rate in the present study are highlighted.

Figure 2 .
Figure 2. Isosurface plots of the ground state electron,|ψ 1 e | 2 , (red) and hole,|ψ 1 h | 2 , (blue) charge densities for alloy configuration 2 with a built-in field determined for varying carrier densities in an In 0.15 Ga 0.85 N/GaN quantum well.The light (dark) isosurfaces correspond to 10% (40%) of the maximum value of the respective charge density.In the 'Top View', the black dashed lines indicate the limits of the simulation cell (black dashed square) and the dimensions of a well width fluctuation (black dashed circle) considered.In the 'Side View', the black dashed lines give an indication of the quantum well interfaces.

Figure 3 .
Figure 3. Isosurface plots of the first five hole charge densities, |ψ 1,2,3,4,5 h | 2 , for alloy configuration 2 of a In 0.15 Ga 0.85 N/GaN quantum well; the built-in potential has been determined for a carrier density of n = 5 × 10 18 cm −3 .The light (dark) isosurfaces correspond to 10% (40%) of the maximum value of the respective charge density.

Figure 4 .
Figure 4. Total Auger coefficient, Ctot, as a function of carrier density, n, at a temperature of T = 10 K for three different microscopic alloy configurations (Cfs.) for an In 0.15 Ga 0.85 N/GaN quantum well.

Figure 5 .
Figure 5.Total Auger recombination coefficient, Ctot, as a function of carrier density, n, at a temperature of T = 300 K for three different microscopic alloy configurations (Cfs.) for an In 0.15 Ga 0.85 N/GaN quantum well.

Figure 6 .
Figure 6.Percentage contribution to the total Auger coefficient, ∆Cα = (Cα/Ctot), from the electron-electron-hole (α = eeh, represented by triangles) and hole-hole-electron (α = hhe, represented by circles) coefficients as a function of carrier density, n, at T = 10 K (filled symbols, solid line) and T = 300 K (open symbols, dashed line) for a In 0.15 Ga 0.85 N/GaN quantum well.The data are averaged over three microscopic configurations.

Table 1 .
Energy separation ∆E = E 1 h − E i h, where E 1 h is the hole ground state energy and E i h is the energy of the first five hole states i ∈ {1, 2, 3, 4, 5} for configuration 2 in the studied In 0.15 Ga 0.85 N/GaN quantum well at a carrier density of n = 5 × 10 18 cm −3 .State numbers correspond to the labels of the charge densities, |ψ i h | 2 , in figure3.