%\part{Concluding Remarks}
\chapter{Concluding Remarks}
\label{chp:SummConc}

To conclude, we first present a summary of the motivation of and results obtained in this thesis. Subsequently, we will discuss some open questions and potential next steps to gain even further insight into the efficiency of III-N-based QW emitters.

\section{Summary}

Most modern blue-violet short wavelength visible LEDs incorporate InGaN/GaN QWs, while AlGaN-based QWs have recently attracted significant attention for efficient UV LEDs. While these devices are highly efficient in the short wavelength visible regime, they suffer significant reductions in efficiency at other wavelengths. Moreover, even efficient III-N QW-based devices suffer from efficiency ``droop'' effects: deteriorations of efficiency as drive current density is increased, or when the temperature of the device rises. These drops in efficiency are known as current and thermal droop, respectively. Input from theory is important for understanding the fundamental origins of droop, as well as to guide device design to improve efficiency where possible. However, theoretical results for other semiconductor systems (such as those based on II-VI and other III-V materials) cannot be carried over to III-N systems due to significant differences in the fundamental material properties. In this thesis we sought fundamental understanding of the efficiency bottlenecks affecting III-N QW-based LEDs while accounting particularly for alloy fluctuations, an effect widely neglected in previous theoretical studies in literature. 

One proposed origin of current density dependent droop in $c$-plane InGaN/GaN QWs and connected devices is Auger recombination, a nonradiative process in which an electron recombines with a hole, but instead of a photon being emitted (as in radiative recombination), another carrier is excited, for instance higher into the conduction band. If the rate of Auger recombination grows faster than the rate of radiative recombination as current density increases, overall there will be a negative impact on device performance. The nature of the Auger process underlying droop is still under debate, and a number of recombination mechanisms have been suggested, including a defect-assisted process as well as alloy disorder enhanced Auger recombination. However, the interplay of Auger recombination and alloy disorder in III-N \emph{QW} systems is widely unexplored. Both recent experimental and theoretical studies show that alloy disorder causes carrier localisation in III-N semiconductor alloys, which can significantly alter the electronic and optical properties of III-N heterostructures. Although experimental data indicates the importance of such effects, only very recently have they been accounted for in theoretical studies, as they present a significant modelling challenge. Thus, the aim of this thesis was to address this challenge using an atomistic model.

Our theoretical framework is based on a nearest neighbour $sp^{3}$ tight-binding model, which takes input from a valence force field model to determine the equilibrium lattice positions of the alloy disordered QW structures studied. On top of this, local variations in strain and polarisation fields are accounted for in the framework, along with polarisation field screening effects. The tight-binding energies and wave functions were then used to calculate the radiative and Auger recombination rates.

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We first targeted an investigation of thermal droop in $c$-plane InGaN/GaN QWs. To do so, we disentangled the effects of changing temperature and carrier density by carrying out the analysis at a fixed, low carrier density of $n=3.8\times10^{18}$ cm$^{-3}$. We determined the temperature dependence of the radiative and Auger recombination rates for three different In contents in the well; 10\%, 15\% and 25\%. We found that the radiative rate increases with increasing temperature, in line with experimental studies performed at a similar, fixed carrier density. This behaviour stands in contrast to the expected decrease of radiative rate with increasing temperature predicted from a fundamental physics perspective. We found total radiative recombination coefficients of $B \approx 3\times10^{-11}$ cm$^{3}$/s (10\% In), $B \approx 5\times10^{-12}$ cm$^{3}$/s (15\% In) and $B \approx 4.5\times10^{-13}$ cm$^{3}$/s (25\% In) at a temperature of 300 K and the carrier density above. We found that at low temperatures, when mainly the strongly localised band edge states are populated, the alloy microstructure has a strong impact on the radiative rate, evidenced by the large variations in $B$ coefficient. As temperature increases, the spread in values decreases, due to the population of excited, less localised or even extended states that are less perturbed by the alloy disordered confining potential.

Turning to the temperature dependence of the Auger recombination rates, in order to calculate the total Auger rate we first determined the rates of electron-electron-hole ($eeh$) and hole-hole-electron ($hhe$) Auger recombination. The calculations revealed that the $eeh$ Auger rate shows a weak temperature dependence, or increases slightly, with increasing temperature. The $hhe$ Auger recombination is the dominant contribution, even with increasing In content, and this does not change with temperature. The $hhe$ Auger rate shows a weak temperature dependence, or a slight decrease, with increasing temperature. Our studies revealed that the relative contributions from $eeh$ and $hhe$ Auger rates to the total rate do not change with In content.  We found total Auger coefficients of $C_{\text{tot}} \approx 4\times10^{-30}$ cm$^{6}$/s (10\% In), $C_{\text{tot}} \approx 2\times10^{-30}$ cm$^6$/s (15\% In) and $C_{\text{tot}} \approx 2\times10^{-31}$ cm$^{6}$/s (25\% In) at a temperature of 300 K when averaged over the 10 microscopic alloy configurations investigated. 

With the temperature dependence of the radiative and Auger rates determined, we were in a position to discuss the thermal droop. Our investigations showed that at temperatures greater than 300 K the radiative recombination rate slightly increases while the total Auger recombination rate (slightly) decreases. This finding would indicate a positive impact of increasing temperature on device efficiency when only considering the Auger process as the main nonradiative recombination process. However, this stands in contrast to experimental studies which show that the device efficiency decreases with increasing temperature above 300 K. Therefore, we conclude that the alloy-enhanced Auger recombination process is not responsible for the thermal droop observed in $c$-plane InGaN/GaN quantum wells and other factors, such as carrier transport or defect related processes are more likely to explain the experimentally observed droop effect. 

Having determined the radiative and Auger rates for three different In contents, we also briefly commented on the decrease of efficiency observed in InGaN QW-based emitters when targeting longer wavelength emission (corresponding to higher In contents): the so-called ``green gap''. We found that, when increasing the In content in the well, the radiative recombination rate drops. However, the Auger rate initially drops as In content increases from 10\% to 15\% In in the well, but is composition independent when going to 25\% In. Thus, it is the reduction of the radiative recombination rate and not an increase in the Auger rate that determines the composition dependence of the competition between these processes. As a result, we expect that the reduction of IQE in InGaN-based light emitters operating in the green to yellow spectral range, and thus at higher In contents (i.e. $>15$\%), may not be caused by \emph{alloy-enhanced} Auger recombination effects. This suggests that by reducing defect densities and thus the connected nonradiative recombination in high In content systems, there may be a way forward to close the green gap. Having discussed the thermal droop and green gap in InGaN-based QWs, we turned to an analysis of the carrier density dependent droop.

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To investigate the carrier density dependent droop we targeted a In$_{0.15}$Ga$_{0.85}$N/GaN QW. To determine the radiative and Auger recombination rates as a function of carrier density, we needed to account for the screening of the electrostatic built-in field. This was achieved by self-consistently solving the \sch-Poisson equations in a 1D, VCA-like model. We discussed the impact that the built-in field screening can have on the electronic structure of the QWs, seeing the decrease in the spatial separation of the electron and hole charge densities along the growth direction as carrier density was increased. 

Turning to look at the carrier density dependence of the Auger recombination rates, we again found that the contribution from the $eeh$ Auger process is of secondary importance when compared to the $hhe$ process, and its relative importance increases only slightly with increasing carrier density. The model predicted total Auger coefficients that vary from $C_{\text{tot}}\approx1\times10^{-31}$ cm$^{6}$/s at a carrier density of $n=1\times10^{17}$ cm$^{-3}$ to $C_{\text{tot}}\approx2\times10^{-30}$ cm$^{6}$/s at $n=1\times10^{20}$ cm$^{-3}$. Thus, based on literature expectations, this indicates that our calculated Auger coefficient values should lead to a significant efficiency droop. \blue{These carrier density dependent Auger coefficients, along with the previously determined radiative recombination coefficients, could be used as input to device simulation models in order to include atomistic carrier localisation effects in large scale simulations~\cite{OdFa2024}.}

Equipped with the theoretical data on the radiative and Auger recombination coefficients as functions of carrier density, we targeted a comparison to experimental data. Using samples provided by colleagues in the University of Cambridge, optical studies were carried out in the University of Manchester. Manchester established a novel method to carefully determine the carrier density in InGaN QWs when using pulsed excitation schemes. As literature data recently indicated the presence of defect-assisted Auger recombination processes in InGaN-based devices, special attention was paid to the impact of point defect density and carrier localisation effects on the droop carrier density regime behaviour of the IQE. To do so, three different samples where grown at different growth temperatures to modify the density of point defects in the well. Using these samples, the experimental data revealed that independent of sample (i) droop onset occurs at similar carrier densities in the well and (ii) the high carrier density regime evolution of the integrated PL intensity was basically identical. All this indicated that trap-assisted (defect related) Auger recombination seems to be of secondary importance for the studied samples. 

Furthermore, when using our calculated carrier density dependent radiative and Auger recombination coefficients, excellent agreement was found between theory and experiment for not only the onset of droop, but also the evolution of the IQE in the high carrier density regime. Although we found that the carrier density dependent droop may be driven by Auger recombination intrinsic to the active region, the analysis also found a peak IQE of 70\%, relatively high for a green emitting QW. Thus, our study suggested that factors external to the QW, such as injection efficiency or carrier distribution homogeneity (to avoid regions of locally high carrier density), should be targeted for improving the efficiency of longer wavelength visible III-N emitters. Having established the framework for visible InGaN-based emitters, we turned to UV emitting AlGaN-based QWs.

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Before targeting an analysis of thermal droop in AlGaN-based QWs, we first discussed the impact of alloy disorder on the electronic structure of a $c$-plane 2.85 nm wide Al$_{0.5}$Ga$_{0.5}$N/AlN QWs. We found that alloy disorder leads to strong carrier localisation of not only the hole ground state, but also excited hole states. Electron states may also be impacted by carrier localisation, but to a lesser extent than holes. Turning to the temperature dependence of the radiative recombination, at a fixed carrier density of $n=2.5\times10^{18}$ cm$^{-3}$, the radiative recombination coefficient, $B$, slightly increases with increasing temperature, similar to InGaN/GaN QWs. Also, the calculations showed that the alloy microstructure plays a crucial role at low temperatures as mainly the strongly localised band edge states are occupied. As temperature increases and higher lying, extended states are populated, the spread in $B$ values is strongly reduced, as the details of the alloy microstructure impact the extended states less than the strongly localised states. We found a value of $B \approx 4\times10^{-13}$ cm$^{3}$/s at T = 300 K and $n = 2.5\times10^{18}$ cm$^{-3}$. Experimental studies reporting values of $B$ are sparse and data on $B$ for structurally similar QWs (i.e. with AlN barriers and similar Al contents in the well) is only available for much smaller well widths. That study yielded a higher value for $B$ in comparison to our data. However, this is expected as well width is a major factor that can impact the magnitude of $B$ due to the separation of electron and hole wave functions along the growth direction due to the built-in polarisation field. Thus, we expect that narrower wells will have larger $B$ coefficients. Future studies can now target an investigation of the dependence of $B$ on well width. 

Having discussed the radiative recombination rate via the $B$ coefficient, we then turned to the temperature dependence of the Auger recombination processes. Calculating both the $eeh$ and $hhe$ Auger rates as a function of temperature revealed that both processes are strongly impacted by the alloy microstructure and the localisation of holes. Overall, the $hhe$ Auger coefficient, $C_{hhe}$, was found to be greater than the $eeh$ Auger coefficient, $C_{eeh}$, by four orders of magnitude. The total Auger coefficient at a carrier density of $n=2.5\times10^{18}$ cm$^{-3}$ and a temperature of T = 300 K was $C_{\text{tot}} = 5\times10^{-32}$ cm$^{6}$/s. Experimental studies are sparse, and have been carried out for wells with different structural properties. In comparison to those studies we found smaller Auger coefficients, which is expected as the difference in well width is expected to have a major impact on this quantity due to the presence of the built-in field. We also noted the significant experimental uncertainties in determining the radiative and Auger recombination coefficients more generally. Further studies of the impact of carrier localisation on the recombination processes in AlGaN-based QWs are required to shed light on the role of alloy-enhanced Auger in contrast to, for instance, defect-assisted Auger.

Having determined the temperature evolution of the radiative and Auger recombination rates, we investigated the competition between these two processes as temperature increases. As before, we expect that if the ratio of radiative recombination rate to Auger rate increases, this would indicate an overall improvement in device performance. Experimentally it is found that device performance deteriorates as temperature is increased above 300 K. Our results showed that the ratio of radiative to Auger recombination rate increases with increasing temperature in this range. Thus, our findings stand in contrast to experimental reports, indicating that the here considered alloy-enhanced Auger recombination process is not directly causing thermal droop. However, we note that Auger recombination could have an indirect impact on device performance due to the generation of high energy Auger carriers which may cause defects in the material, thus increasing nonradiative recombination pathways and hampering efficiency.

Overall, the results presented in this thesis show that alloy disorder can have a strong impact on the electronic structure and recombination rates in III-N QWs and thus, the IQE. Although Auger recombination may not represent an intrinsic roadblock to overcoming the green gap or thermal droop in InGaN/GaN QWs, current dependent droop is likely an intrinsic feature of these heterostructures. Thus, designs that aim to reduce current density in the active region may be the best route forwards to mitigate current droop in InGaN-based devices. For AlGaN/AlN QW-based devices, many questions remain. Although our results suggest that Auger recombination again may not represent an intrinsic roadblock to overcoming thermal droop in AlGaN-based devices, we have seen that carrier localisation effects can strongly impact the electronic structure and Auger recombination rate, and may lead to Auger rates large enough to be significant for current droop as well. Thus, we expect that similar strategies employed in InGaN-based emitters to reduce current droop should also be considered for AlGaN-based systems.

\section{Outlook}

As mentioned above, many questions remain with regard to carrier localisation and recombination in AlGaN-based systems. The theoretical framework developed here could be used to investigate the impact of well width, as well as other factors such as Al content in the well. Such an investigation would enable a more direct comparison of expected radiative and Auger coefficients between theory and experiment, along with providing insight into the impact of the built-in field on these processes. It would also be possible to extend the framework to AlGaN/AlGaN QWs (which have also been targeted in experimental studies). In such systems there are open questions over the impact of alloy disorder in the \emph{barriers}. 

Our theory experiment comparison in Chapter~\ref{chp:CarrDensInGaN} suggested that defect-assisted processes appear to be of secondary importance when describing the current dependent droop behaviour at high carrier densities in InGaN QWs, despite recent suggestions that defect-assisted Auger processes are responsible. However, we also observed potential ``trap''-like states in InGaN/GaN QWs that caused enhanced Auger rates. In light of the recent discussions of trap-assisted Auger recombination, an investigation of such states, for instance to determine the rate of their occurrence, may provide important insights into their importance, especially given that most analyses of trap-assisted Auger recombination focus on defect-assisted processes. 

Carrying out a similar experimental comparison for AlGaN/AlN QWs could also provide further insight into the role of alloy-enhanced and defect-assisted Auger in AlGaN systems. To achieve this, the carrier density dependence of the radiative and Auger recombination rates in AlGaN systems would need to be determined, while taking the screening of the built-in field into account. For the InGaN QWs studied here, due to the computational demand of carrying out a fully self-consistent tight-binding \sch-Poisson calculation, we employed a 1D VCA-like model to determine the screening of the built-in field. Based on our theory experiment comparison (and literature results) this approach seems sufficient for InGaN/GaN QWs and so we have used this model for AlGaN/AlN QWs too. However, validating this approximation and investigating the impact of screening effects in a tight-binding framework could provide further valuable insights into carrier localisation effects in AlGaN/AlN QWs. 

For InGaN systems, recent experimental studies have targeted QWs with much larger well widths ($\approx$ 10 nm) than conventionally used ($\approx$ 3 nm) as a potential route to increasing efficiency. In those studies it is argued that populating the energetically lowest band edge states (that are strongly localised and spatially separated at opposite QW interfaces) may be enough to fully screen the built-in polarisation field even at relatively low carrier densities, thus, higher lying states may not be affected by the built-in field. However, our results show that although the radiative recombination rate is indeed increased when the polarisation field is screened, so too is the Auger rate. Thus, an investigation of the competition between the radiative and Auger rates in wider QWs may provide insight into the competition between these processes. 

Another lingering question is the discrepancy between theoretical and experimental estimations of the relative contributions from the electron-electron-hole ($eeh$) and hole-hole-electron ($hhe$) Auger rates. Some experimental studies have found that for InGaN/GaN QWs the $eeh$ rate is larger than the $hhe$ rate by a factor of 1 -- 12, but here we find the $eeh$ rate is approximately four orders of magnitude smaller than the $hhe$ rate, in line with basically all other theoretical studies that comment on this aspect. When discussing this discrepancy above we noted that electron localisation could have an impact on the $eeh$ rate, and more generally the nature and impact of electron localisation remains under debate, for instance, the role of gross well width fluctuations and other mesoscopic structural phenomena in electron localisation. 

The TB framework developed here could be used to guide the development of future modified continuum-based approaches, for instance acting as a benchmark for localisation landscape theory (LLT) calculations. Currently, the main disadvantage of LLT is that it has only been established in a single band framework, whereas the description of some semiconductor materials (such as AlN) and phenomena (such as light polarisation character) require multi-band approaches. Ideally then a multi-band LLT would be developed, but it is not clear if this is possible. In the meantime, our TB framework remains a balanced compromise between a full description of the system and computational feasibility. 

Overall, it is an important tool for investigating a range of phenomena in III-N QWs while accurately taking alloy disorder effects into account. Furthermore, the framework could be extended to novel III-N materials and nanostructures, such as those incorporating boron containing alloys. The introduction of B into III-N alloys is targeted to reduce the lattice mismatch in heterostructures and to control the valence band structure, for instance in AlGaN-based UV emitters. In and B are significantly different in size, so effects such as carrier localisation could also play an important role which would need to be considered in future studies. Indeed, recent \emph{ab initio} theoretical studies of BGaN have shown that the presence of B clusters can lead to carrier localisation effects~\cite{NiSh2023}. As we have seen throughout this thesis, carrier localisation effects can strongly impact the electronic and optical properties of nitride-based devices.

