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\begin{abstract}
Most modern blue-violet short wavelength visible light-emitting diodes (LEDs) incorporate group III-nitride (III-N) semiconductor quantum wells (QWs), and ultraviolet (UV) LEDs could be developed using similar materials and heterostructures. \blue{Relative to filament and fluorescent bulb technologies, modern blue-violet short wavelength emitting III-N QW-based devices are significantly more efficient. However, they suffer from efficiency ``droop'' effects, the fundamental causes of which are under debate. For example, the efficiency of III-N LEDs drops with increasing \emph{drive current}, and thus increased carrier density in the well, an effect known as ``current droop''. Furthermore, even at fixed drive current, when the \emph{temperature} of the device rises, there is a corresponding drop in device efficiency, the so-called ``thermal droop''. On top of current and temperature dependent droop, III-N QW-based light emitting devices also suffer significant reductions in efficiency at longer (green to red) and shorter (deep UV) wavelengths. The reduction in efficiency as III-N devices are engineered to emit at longer wavelengths is a major contributor to the so-called ``green gap'' phenomenon, wherein neither III-N nor other III-V heterostructure-based devices (such as those based on phosphides or arsenides emitting in the red to infrared spectral range) can be designed to emit efficiently in the green to yellow spectral range.}

Input from theory is important for understanding the fundamental origins of these droop phenomena, as well as to guide device design to improve the efficiency of not only blue-violet visible wavelength emitters that suffer from droop, but also devices emitting at the shorter and longer wavelengths mentioned above. However, theoretical results obtained for other semiconductor structures (such as those based on II-VI and other III-V heterostructures) cannot be carried over to III-N systems due to significant differences in the fundamental material properties. For instance, alloy disorder causes strong 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. The aim of this thesis is to address this challenge using atomistic modelling.

\blue{Auger recombination has been discussed in the literature as an important nonradiative process in $c$-plane InGaN/GaN QWs in which an electron recombines with a hole, but instead of a photon being emitted (as in radiative recombination) another carrier is excited. If the rate of Auger recombination grows faster than the rate of radiative recombination as temperature or carrier density increases, overall, there will be a negative impact on device performance. For instance, there is much evidence that Auger recombination plays a significant role in carrier density dependent droop, but the exact nature of the Auger process underlying this droop phenomenon is still under debate, and several Auger recombination mechanisms have been suggested, including a defect-assisted process as well as alloy disorder enhanced Auger recombination. Auger recombination has also been explored as a possible cause of thermal droop but there is still much debate over its relevance to this droop phenomenon. Despite experimentally established links between Auger recombination and these droop phenomena, on the theoretical side the impact of alloy disorder on the Auger recombination process in III-N \emph{QW} systems is widely unexplored.}

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 at high carrier densities in the well. The tight-binding energies and wave functions are then used to calculate the radiative and Auger recombination rates.

For InGaN/GaN QW systems, our predicted values for radiative and Auger recombination rates lie within the wide range of experimentally reported values, and confirm that the coefficient of Auger recombination is not as small as one may expect from the expected dependence of Auger coefficient on band gap. \blue{To study the thermal droop, we have evaluated the radiative and Auger recombination rates at a fixed carrier density but as a function of temperature. Our results reflect the unconventional but experimentally observed increase of the radiative recombination rate with increasing temperature. When focusing on the competition between radiative and alloy-enhanced Auger effects, neglecting, e.g., defect-related processes such as Shockley-Read-Hall recombination, our results indicate an improvement of device performance with increasing temperature, in contrast to experimentally determined efficiency data.} Thus, we expect that alloy-enhanced Auger recombination, intrinsic to InGaN-based QWs, is not responsible for thermal droop. As a result, efficiency improvement strategies that target, for instance, factors extrinsic to the well, such as reducing defect densities, should be considered.

\blue{Turning to the carrier density dependent droop (i.e. the current droop), the competition between radiative and Auger rates as a function of carrier density (but at fixed temperatures) was determined. In InGaN/GaN QW systems, already at low carrier densities our model predicts Auger recombination rates large enough to be considered significant contributors to this droop phenomenon (based on expectations from the literature). When investigating the carrier density dependence of the recombination rates, we find that the Auger rate grows faster than the radiative rate as carrier density increases in the $c$-plane InGaN/GaN QWs studied here. Working within a commonly used model of efficiency in LEDs, we compared the theoretically determined carrier density dependent efficiency data to experimental results from collaborators at the Universities of Manchester and Cambridge for samples with varying defect densities. We found a good agreement between theory and experiment at carrier densities where this droop phenomenon is observed; the temperature is kept constant in the experiments and calculations. Furthermore, the drop in efficiency with increasing carrier density was essentially independent of sample, and thus independent of defect density. Overall, these results indicate that defect-assisted Auger processes may be of secondary importance, and alloy-enhanced Auger recombination can be sufficient to explain the current droop. Moreover, despite the large Auger coefficients, the green emitting samples studied in our theory experiment comparison were found to have relatively high internal quantum efficiencies, suggesting that Auger recombination may not be limiting the performance of longer wavelength visible light-emitting devices.}

Lastly, UV LEDs based on AlGaN QW systems have attracted significant attention in recent years for the potential development of more efficient and environmentally friendly UV emitters. Despite this, and despite the experimental observation of both thermal and current droop in AlGaN-based QWs, Auger recombination and carrier localisation have also been widely unexplored in these systems, particularly from a theoretical perspective. Thus, we applied our theoretical framework to determine the radiative and Auger recombination rates at a fixed carrier density but as a function of temperature in $c$-plane AlGaN/AlN QWs, thus providing insight into the thermal droop of such emitters. We found Auger recombination rates on the same order of magnitude as those determined for the InGaN/GaN QWs studied above. Based on our results, we expect that Auger recombination is not the driving cause of thermal droop in AlGaN/AlN QWs. However, based on expectations from the literature, the values calculated here suggest that Auger recombination could be a significant contributor to current droop in AlGaN QWs. 

Overall, our calculations show that alloy-enhanced Auger recombination is indeed strongly contributing to nonradiative recombination processes in III-N heterostructures. However, the thermal droop does not seem to be driven by this intrinsic nonradiative process. Instead, other factors, such as defect-assisted Auger recombination or carrier injection deficiencies, may be responsible. Thus, device optimisation strategies that target these extrinsic effects may still improve efficiency. With regard to the current density dependent droop, our calculations indicate that alloy-enhanced Auger recombination plays a significant role. Efforts to mitigate the impact of alloy-enhanced Auger recombination on the internal quantum efficiency could consider targeting larger active regions to reduce current density, or improved current spreading through the multiple QWs employed in LEDs. \blue{Although alloy-enhanced Auger recombination may present an intrinsic roadblock to reducing current droop, our results suggest that the reduction in efficiency of longer wavelength III-N devices emitting in the visible range (i.e. green) is not driven by this Auger process. Thus, device performance optimisation may still be achievable by targeting extrinsic factors such as carrier injection efficiency and homogeneity.}

\end{abstract}

\begin{dedication}
    To my parents, and my grandparents.
\end{dedication}
\begin{dedication}
    To Pip and Colin.
\end{dedication}
\begin{dedication}
    In loving memory of Niamh Lynch, Conor King and Cian Nugent.\\
    \vspace{0.2in}
    \emph{Ar dheis D\'{e} go raibh a n-anamacha.}
\end{dedication}
\begin{dedication}
    ``Results! Why, man, I have gotten a lot of results! I know several thousand things that won’t work.''\\
    \vspace{0.2in}
    - T. Edison (allegedly) \vspace{0.2in}\\ From ``Edison: His Life and Inventions'' by Frank Lewis Dyer and Thomas Commerford Martin, Volume 2, Chapter 24, p. 616, Harper \& Brothers, New York (1910)
\end{dedication}

\prelim{Publications, presentations and posters}
\subsection*{Journal publications}
\begin{itemize}

    \item \cite{McKi2023} ``Carrier density dependent Auger recombination in $c$-plane (In,Ga)N/GaN quantum wells: Insights from atomistic calculations,'' \underline{J. M. McMahon}, E. Kioupakis, and S. Schulz, \emph{J. Phys. D: Appl. Phys.} vol. 57, p. 125102, 2024
    
    \item \cite{BaMc2023} ``Disentangling the Impact of Point Defect Density and Carrier Localization-Enhanced Auger Recombination on Efficiency Droop in (In,Ga)N/GaN Quantum Wells,'' R. M. Barrett, \underline{J. M. McMahon}, R. Ahumada-Lazo, J. A. Alanis, P. Parkinson, S. Schulz, M. J. Kappers, R. A. Oliver, and D. Binks, \emph{ACS Photonics} vol. 10, no. 8, pp. 2632-2640, 2023
    
    \item \cite{McKi2022} ``Atomistic analysis of Auger recombination in $c$-plane (In,Ga)N/GaN quantum wells: Temperature-dependent competition between radiative and nonradiative recombination,'' \underline{J. M. McMahon}, E. Kioupakis, and S. Schulz, \emph{Physical Review B} vol. 105, p. 195307, 2022

    \item \cite{McTa2020} ``Atomistic analysis of radiative recombination rate, Stokes shift, and density of states in $c$-plane InGaN/GaN quantum wells,'' \underline{J. M. McMahon}, D. S. P. Tanner, E. Kioupakis, and S. Schulz, \emph{Applied Physics Letters} vol. 116, p. 181104, 2020
    
    \item \cite{TaMc2018} ``Interface roughness, carrier localization, and wave function overlap in $c$-plane (In,Ga)N/GaN quantum wells: interplay of well width, alloy microstructure, structural inhomogeneities, and Coulomb effects,'' D. S. P. Tanner, \underline{J. M. McMahon}, and S. Schulz, \emph{Physical Review Applied} vol. 10, 034027, 2018
\end{itemize}

\subsection*{Conference talks}
\begin{itemize}

    \item ``Atomistic study of radiative and Auger recombination rates in $c$-plane InGaN/GaN quantum wells``, \underline{J. McMahon}, E. Kioupakis, and S. Schulz, at the \textit{International Workshop on Nitride Semiconductors}, Berlin, October 2022
    
    \item ``Theoretical study of the carrier density dependence of Auger recombination in polar (In,Ga)N/GaN quantum wells'', \underline{J. M. McMahon}, E. Kioupakis, and S. Schulz, at the \emph{UK Nitride Consortium Winter Meeting}, Online, January 2022
    
    \item ``Temperature dependence of radiative and non-radiative recombination rates in $c$-plane InGaN/GaN quantum wells'', \underline{J. M. McMahon}, D. S. P. Tanner, E. Kioupakis, and S. Schulz, at the \emph{UK Nitride Consortium Winter Meeting}, Online, January 2021
    
    \item ``Carrier localization in polar InGaN QWs: Consequences for the temperature dependence of the radiative recombination'', \underline{J. M. McMahon}, D. S. P. Tanner, E. Kioupakis, and S. Schulz, at the \emph{UK Nitride Consortium Winter Meeting}, Cardiff, January 2020
    
\end{itemize}
\subsection*{Conference posters}
\begin{itemize}

    \item ``Influence of temperature on the radiative and non-radiative recombination rates in c-plane InGaN/GaN quantum wells'', \underline{J. M. McMahon}, E. Kioupakis, and S. Schulz, at \emph{Photonics Ireland}, Online, June 2021
    
    \item ``Atomistic analysis of radiative and nonradiative recombination in InGaN quantum wells'', \underline{Joshua McMahon} and Stefan Schulz, at the \emph{IPIC Industry Workshop}, March 2020
    
\end{itemize}
% \prelim{Contributions to invited talks}
\subsection*{Contributions to invited talks}
 \begin{itemize}

    \item ``Carrier transport, radiative and non-radiative recombination in (In,Ga)N heterostructures: Insights from atomistic and multi-scale simulations,'' Stefan Schulz, at \emph{Applied Mathematics and Simulation for Semiconductors and Electrochemical Systems}, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, September, 2021
    
\end{itemize}

\begin{acknowledgements}
This research would not have been possible without the support of Science Foundation Ireland and the Sustainable Energy Authority of Ireland.

This research would also not have been possible without the inexhaustible, always helpful and supportive guidance and encouragement of Dr Stefan Schulz. My sincerest thanks must go to him first and foremost for believing in me and giving me this opportunity all those years ago. And doubly so for continuing to support and guide me even through the tumultuous times since then. 

The unwavering support of Prof. Eoin O'Reilly as well, from my first time stepping foot in Tyndall in 2012 up to now, has been a consistent source of comfort and inspiration. It is an honour to be able to say that I worked with both Dr. Schulz and Prof. O'Reilly.

To complete the excellent supervision team for my time in Tyndall, I am so grateful to Dr. Farzan Gity in his role as monitor. Despite the stress of preparing for yearly reviews I can safely say I looked forward to them knowing that Dr. Gity's presence would only have a positive impact, along with his willingness to help and curiosity for the work we were doing.

A huge thank you to our collaborators, Prof. Manos Kioupakis in the US and Dr. Dave Binks and Dr. Rachel Barrett in the UK for the consistently incredibly insightful conversations. It was a pleasure to work with ye. 

I do not think I can convey how thankful I am to my parents, Chris and Miriam, for supporting me every step of the way, from the first time I said I wanted to study theoretical physics to today. I owe them everything. This work is as much the fruit of their labour, as it is mine. As they say; ``all that is good in me, began with you''.

To my grandparents too, I don't think I can properly convey how thankful I am to them. I will always remember the times I stayed far too late drinking tea and chatting and venting about PhD stress. I will also always remember discussing the mystifying nature of how magnets work with Grandad driving home from school one afternoon. Trying to get to the bottom of that question played an important role in my burgeoning interest in physics at the time.

To Denis, Dave and Hank, yere company got me through some of the hardest times in the last five years and I am so grateful for yere friendship. I am looking forward to our adventures in the future.

To Sin\'{e}ad, Ciaran, Kate, Niall, Jamesy and Clara, I am so glad our friendships continue despite the distance between us. I am so looking forward to seeing ye more often. Yere support has always been such a massive comfort, especially through COVID but, really, every day.

Thank you to everyone I've crossed paths with in the office too. I know I didn't spend much time in there after COVID but it was always a nice treat to see ye. To Daniel and Rob, it was a pleasure working with ye, I wish ye all the best in the future. To Noor and Saroj, thank you for yere company on those late evenings, it would've been a very lonely office otherwise. And particularly, thank you to Michael. I sometimes forget how wild it is that we've known each other for 23 of the last 27 years, going from playschool to PhD. I am also so glad that despite the distance between us as well, our friendship is one that we can always pick up from where we left off. 

A big, general thank you to everyone in UCC Ultimate, Rebel Ultimate and XVI Ultimate. Frisbee has always been a wonderful escape. I also want to briefly look further back, and acknowledge the support of everyone I was close with when I started this journey. To the theoretical physics class of 2018, although it was intimidating at times, ultimately yere intelligence and, more importantly, hard work was always an inspiration. To Jess and Danny, Kevin B and Kevin Q, Sean B and Ruair\'{i} (this is all your fault), a huge thank you for being wonderful friends. I have such fond memories of our times together. I would not have walked this path without yere support, as well as the support of Rory, Chloe, Ailne and Seamus, and especially, for your constant encouragement and belief in me, Christina.

Lastly, I cannot forget the unwavering support of Pip and Colin, and Inca now too, reminding me to relax, and teaching me how as well.

\emph{Ar sc\'ath a ch\'eile a mhaireann na daoine.}
\end{acknowledgements}
