%\part{Introduction}

%\setcounter{chapter}{-1}
\chapter*{Prologue}
\addcontentsline{toc}{chapter}{Prologue}

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.8\columnwidth]{Figures/elecGenMix.eps}
    \caption{Breakdown of fuel types used to generate electricity between 1971 and 2019. Data available under Creative Commons license CC BY 4.0 from Ref.~\cite{IEAWebsite}.}
    \label{fig:WorldEnergyUse}
\end{figure}

From the use of primitive animal fat candles to light caves for painting by prehistoric humans~\cite{Weisbuch2018ComptesRendu}, to the mass production of extremely precisely engineered nanoscale light emitters in modern times, human civilisations have always sought innovative ways to dispel darkness. Nowhere is this innovation more evident than in the vast efforts of the past century to develop solid state lighting (SSL) technologies. 

SSL technologies hold many advantages over conventional filament and fluorescent bulb-based technologies. They are generally less fragile, more environmentally friendly and more efficient~\cite{Hump2008}. Indeed, efficiency has always been one of the most important metrics by which the development of not only SSL, but all lighting, has been measured~\cite{Weisbuch2020Review}. 

Now, more than ever, developing more efficient lighting technologies is crucial. Energy consumption due to lighting demands continues to grow, and is expected to continue further. Global electricity consumption due to lighting grew to over 1700 TWh in the years leading up to 2020, and has grown year on year since then~\cite{IEAWebsiteElecUse}. This increasing usage is directly linked to significant greenhouse gas emissions~\cite{Hump2008}. Figure~\ref{fig:WorldEnergyUse} shows the fuels used to generate electricity from 1971 to 2019. Although the use of renewable energy sources has increased, still, nearly two thirds of electricity generation is facilitated by fossil fuel-based technologies~\cite{IEAWebsite}. In order to accelerate the crucial reduction of carbon emissions to avoid global warming of more than 2$^{\circ}$C, all feasible paths to mitigating energy consumption must be explored.

Although improvements to the efficiencies of filament and fluorescent bulbs continue, in 2010, Humphreys~\cite{Hump2008} reported that ``incandescent light bulbs convert about 5\% of the electricity they use into visible light'' and ``energy saving compact fluorescent lamps ... are only about 20\% efficient.'' SSL technologies, in the form of light-emitting diodes (LEDs), can achieve electricity to light conversion efficiencies of over 90\% in certain applications and at certain wavelengths~\cite{SaDo2013} and are generally more efficient than filament or fluorescent bulbs due to the direct emission of light from the material~\cite{Hump2008}. Furthermore, both filament and fluorescent technologies suffer from short lifetimes relative to SSL technologies. Filament bulbs generally have lifetimes of approximately 1000 hours, and although fluorescent bulbs can last longer, poor usage practises can significantly shorten their lifetime~\cite{Hump2008}. LEDs can have lifetimes on the order of 100 times longer than filament bulbs, and are generally far more robust. 

\blue{Despite their greater operational efficiencies, the potential for larger energy expenditure during the manufacturing, transportation and disposal of an LED lamp, in comparison to filament of fluorescent technologies, has led to recent ``life-cycle'' analyses \cite{ScDi2012P1,ScDi2012P2}. For instance, in the analysis of Ref.~\cite{ScDi2012P1}, it was reported that, in 2011, fluorescent technologies and LED-based lamps had similar average life-cycle energy consumption values, which were both approximately a quarter of the value for filament-based lamps. However, the report also concluded that ``energy-in-use'', i.e., the energy used when the lamp is switched on, is the dominant contribution to the average life-cycle energy consumption for both technologies. Thus, although fluorescent lamps are slightly less energy intensive in the manufacturing, transport and disposal phases than LEDs, these processes contribute less than 10\% of the overall average life-cycle energy consumption for both technologies. As a result, improvements to LED operational efficiencies (i.e. that reduce the ``energy-in-use'') would significantly reduce the overall average life-cycle energy consumption of LEDs. It was estimated that these improvements would halve the overall average life-cycle energy consumption of LEDs by 2015~\cite{ScDi2012P2}. On top of this, improvements to manufacturing and disposal/recycling techniques could also be realised, reducing the overall energy consumption further. Although beyond the scope of this thesis, we also note that there is much debate over the potential for a so-called ``rebound effect''. The rebound effect refers to the increased adoption and usage of LED-based lamps due to their lower initial purchasing costs and, in particular, due to their lower related energy-in-use costs. As a result, in the short term, the introduction of a more efficient LED technology may lead to an overall reduction in energy consumption for lighting, but there may be a subsequent increase to a level of energy consumption larger than the value before the efficiency was improved. Thus, any overall energy saving due to improved efficiency may be negated~\cite{ThCh2008}. Regardless of whether such a rebound effect will occur, reducing the rate of increase of energy usage and connected carbon output remains crucial, with SSL technologies providing the best path forward.}

\blue{Given the advantages SSL technologies hold over conventional lighting technologies, in terms of pathways to improve efficiency and overall life-cycle energy consumption, along with commercial interests and interest in the fundamental physics of these devices, huge research efforts have gone into understanding the materials and heterostructures that underlie LEDs. Furthermore, optimisations of the designs of LED-based light emitting devices continue to attract huge interest.} In this chapter we will briefly recap the major milestones in the development of LED technologies before discussing some of the current bottlenecks to increasing the efficiency of these devices further and theoretical efforts to understand and predict solutions to these bottlenecks. Finally, we will lay out the structure of this thesis.

\section*{A brief history of solid state lighting}

The first documented observation of the direct emission of light from a material, called electroluminescence (EL), is credited to H. J. Round when he observed a weak form of EL from SiC in 1907~\cite{Weisbuch2020Review}. However, the significance of this discovery was essentially lost until the early 1920s when O. Losev independently rediscovered the phenomenon in the semiconductor materials SiC and ZnO. Losev suggested that the phenomenon may be an inverse of the photoelectric effect, as he was able to relate the emission wavelength to the voltage he applied to the material. Interest in EL continued but efficiency did not significantly improve despite extensive research efforts. Losev had originally identified two forms of EL, which depended on the bias of the applied voltage (types I and II). One of these Losev identified as occurring under high reverse bias (type I), and the other required a smaller forward bias (type II). It was suggested by Destriau in 1955 that only type II EL would be able to achieve efficiencies required for lighting applications. Indeed, type II EL forms the basis of light emission from almost all modern LEDs, as we will see~\cite{Weisbuch2020Review}.

Nowadays we understand that the structures investigated by Round, Losev and Destriau were examples of Schottky barrier diodes, in which there is a contact (also called a junction) between a metal and a semiconductor. The invention of the $p$-$n$ junction and so-called double heterostructures (DHs), such as quantum wells (QWs), heralded the era of modern LED devices in subsequent decades~\cite{Weisbuch2020Review}. In the 1970s and 1980s much work was carried out to develop DHs based on alloys of AlAs and GaAs, in part due to the relative ease in creating high quality DHs made of those materials. These devices mainly emitted in the red-infrared spectral range. 

AlAs and GaAs are examples of III-V semiconductors, i.e., they are compounds of group III and group V elements of the periodic table. In the search for materials that could be used in devices that emit over a wider visible spectral range, many compounds and alloys were investigated including II-VI and other III-V semiconductors. III-V semiconductors based on phosphorus containing compounds (phosphides, e.g., InP, GaP) also found significant use in red-infrared wavelength devices. \blue{However, when targeting shorter wavelength emission (i.e. from red to green), the efficiency of these devices deteriorated significantly. This was in part due to the incorporation of Al containing alloys such as AlAs or AlP to increase the band gap. Both AlAs and AlP, as well as GaP, are indirect gap semiconductors~\cite{TsQw2023}, which exhibit inherently reduced radiative efficiencies. In ternary and quaternary alloys of the indirect semiconductors mentioned above with direct gap materials, such as GaAs, InAs or InP, the band gap of the resulting alloy will switch from direct to indirect once the concentration of indirect gap material reaches a critical level. Thus, when targeting larger band gaps (and smaller wavelengths) by including larger AlAs, AlP or GaP contents, the band gap becomes indirect and the radiative efficiency of the materials is reduced.} 

III-V materials based on nitrogen containing compounds (nitrides, e.g., InN, GaN, AlN) had been explored as early as the 1960s. \blue{III-N alloys differ significantly from arsenide and phosphide-based semiconductors. For instance, III-N alloys remain direct gap semiconductors over the entire visible wavelength range.} However, III-N systems suffered from issues with crystal quality and difficulties in doping (a technique used to increase conductivity), thus the direct gap II-VI semiconductor ZnSe became the front-runner for short wavelength, blue spectral range applications~\cite{Weisbuch2020Review}. III-N semiconductors overtook ZnSe as the material of choice when Amano, Akasaki and Nakamura overcame the poor crystal qualities and issues with $p$-doping GaN~\cite{Weisbuch2020Review}. These breakthroughs paved the way for Amano, Akasaki and Nakamura to develop the first efficient blue LEDs, for which they were awarded the Nobel prize in 2014~\cite{AmMa1989,NaTa1992}. III-N systems are now the material of choice for blue LEDs, having outperformed ZnSe~\cite{Hump2008}.

\subsection*{III-nitrides}

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{Figures/nitrideBandgaps_fullColour.eps}
    \caption{Band gap, $E_{\text{g}}$, as a function of the in-plane lattice constant, $a$, for the commonly used wurtzite III-N semiconductors, InN, GaN and AlN. The solid lines represent the range of emission energies from alloys of the III-N semiconductors, which are represented by black filled circles. Photon energies corresponding to the visible spectrum are represented appropriately. Binary III-N band gap values are taken from Ref.~\cite{WuWa2003} (InN and GaN) and Ref.~\cite{TsCh2018} (AlN) with band gap bowing parameters to describe the variation of the band gap as a function of the alloy composition from Ref.~\cite{VuMe2003}. Lattice constants are taken from Ref.~\cite{CaSc2011}.}
    \label{fig:nitrideBandgaps}
\end{figure}

III-N semiconductors have attracted massive amounts of research attention for a variety of reasons, one of which is the possibility of creating LEDs emitting across a much wider range than the blue section of the visible spectrum~\cite{Hump2008}. Figure~\ref{fig:nitrideBandgaps} shows the range of theoretically possible emission energies of III-N alloys. \blue{The variation in the band gap between InN, AlN and GaN essentially arises from the differences in the interatomic spacings of these materials, which lead to different interactions between atomic orbitals localised on the constituent cation and anion sites}. When semiconductors of differing band gaps are alloyed, the band gap of the resulting material lies, in general, somewhere between the two constituent materials. With the band gaps of GaN ($E_{\text{g}} \approx 3.51$ eV~\cite{WuWa2003}) and InN ($E_{\text{g}} \approx 0.69$ eV~\cite{WuWa2003}) lying on either side of the visible spectrum, in principle, alloys of these semiconductors (InGaN) could emit across that entire wavelength range. Furthermore, the band gap of AlN ($E_{\text{g}} \approx 6.0$ eV~\cite{TsCh2018}) lies in the deep ultraviolet (UV) spectral window, thus allowing for the potential creation of LEDs with wavelengths spanning from the infrared to the deep UV. Although work on AlGaN-based UV LEDs has picked up in recent years, current state-of-the-art deep UV LEDs still suffer from relatively poor efficiencies~\cite{AmCo2020}. 

The realisation of efficient blue LEDs allowed not only for the implementation of efficient blue light emitters, but also for their use as the blue component of red-green-blue (RGB) displays, as well as for their use in white light sources~\cite{Hump2008}. RGB LEDs can be used to generate white light but there are a number of issues that arise. For instance, in practice, realising efficient InGaN-based LEDs with wavelengths in the red-green spectral range has proven extremely challenging~\cite{TaDa2020}. \blue{Recalling from above that arsenide and phosphide-based emitters also suffer from poor efficiencies when targeting green wavelengths, the generally much lower efficiency of green LEDs when compared to red and blue emitters is referred to as the so-called ``green gap''. Generally, arsenide-based LED technologies are used to produce efficient red light emitters, but integrating different material systems on the same chip raises further challenges~\cite{MoUe2022}.} 

InGaN-based LEDs operating at the blue end of the visible spectrum have found many applications, aided by their relatively high efficiencies, and make up a significant portion of the lighting market~\cite{ZiBe2021}. Thus, given the large historical investment in the methods, expertise and equipment needed to mass produce InGaN-based LEDs, developing established III-N LED technology is more desirable for manufacturers in the short to medium term. Although, work on semiconductor LED alternatives such as organic and polymer-based LEDs continues~\cite{Hump2008,ZiBe2021}.

Initially, ZnSe was the front-runner for visible blue wavelength range LEDs due to the much higher dislocation densities found in GaN and InGaN. Although the crystal growth methods developed by Amano \emph{et al.} and Nakamura \emph{et al.} allowed the growth of much higher quality GaN crystals, the density of dislocation defects remained much higher ($\approx 10^{6}$ cm$^{-2}$) than in other III-V semiconductors such as arsenides and phosphides ($\approx 10^{3}$--$10^{5}$ cm$^{-2}$), and was even higher still in InGaN ($\approx 10^{9}$ cm$^{-2}$). Even though growth techniques have improved since the initial breakthroughs of Amano \emph{et al.} and Nakamura \emph{et al.}, modern GaN and InGaN QW-based LEDs still exhibit higher dislocation densities than other III-V materials, and also, counter-intuitively, higher efficiencies. 

The prevailing theory used to explain this is that charge carriers become localised, and as such are prevented from diffusing to defects, where they might recombine nonradiatively (i.e. without the emission of a photon). Carrier localisation has been shown to have a significant impact on the optical and electronic properties of InGaN-based LEDs and we will discuss it in more detail later, but the exact origin of this effect and the length-scale over which it occurs is still debated~\cite{OlBe2010}. \blue{As mentioned above, III-N semiconductors differ significantly from ``conventional'' semiconductor systems, such as those based on arsenides or phosphides. For example, in semiconductor materials, band structures are often characterised by their so-called effective mass. The effective mass is a key ingredient in understanding the electronic properties of a semiconductor and is usually given in units of the free electron mass, $m_{0}$. When electrons are excited from the valence band to the conduction band, the ``missing'' electron in the valence band can also be described by a positively charged quasi-particle, a so-called hole. As in the case of the electrons, an effective mass can also be assigned to a hole. The effective mass of holes in III-N semiconductors ($m_{\text{InN}}^{*} = 1.61$, $m_{\text{GaN}}^{*} = 1.87$ and $m_{\text{InN}}^{*} = 2.68$) is significantly larger than in III-As ($m_{\text{InAs}}^{*} = 0.26$, $m_{\text{GaAs}}^{*} = 0.45$ and $m_{\text{AlAs}}^{*} = 0.51$) or III-Ps ($m_{\text{InP}}^{*} = 0.46$ and $m_{\text{GaP}}^{*} = 0.79$). Due to these higher effective masses in III-N systems, small variations in the confining energy landscape of an alloy disordered crystal or heterostructure may already lead to carrier localisation effects. On top of this, III-Ns tend to crystallise in the wurtzite phase, in contrast to III-As and III-P systems, which tend to crystallise in the zinc blende phase. Due to the nature of the wurtzite crystal structure, which will be discussed in more detail in Sec.~\ref{sec:LEDsIIINMaterials}, semiconductor materials in this phase exhibit stronger electrostatic built-in fields. The result of this is that in III-N alloys, where there is a varying local environment of In, Ga or Al ions, the energy landscape is further locally affected when compared to III-V alloys crystallising in the zinc-blende phase. Coupled with the larger hole effective masses, the local polarisation field fluctuations can further enhance carrier (hole) localisation in III-N alloys.} Thus, it is an important aspect of the physics of III-N emitters and requires careful consideration. Furthermore, a gain in our understanding of carrier localisation and its impact on optical and electronic properties may also provide insight into how it affects efficiency. Significant efforts have been undertaken to understand the properties of III-N systems generally, and especially the factors that limit the efficiency of such devices. Efforts to further understand and improve the efficiency of III-N LEDs continue, as several bottlenecks have arisen.

\section*{Efficiency bottlenecks}

Previously we mentioned the green gap, partly caused by the reduction in the efficiency of InGaN-based LEDs as the band gap of the active (light emitting) region of the device is tailored to lower values by increasing the In content in InGaN, in order to emit longer wavelengths in the yellow-green visible spectral range. The causes of the green gap, from the perspective of InGaN-based emitters, are still under debate, but it has been attributed to, for instance, the formation of defects~\cite{UsMu2021,MaDa2014} or the increase in the strength of the built-in piezoelectric polarisation fields found in III-N QWs as In content increases, which tends to separate electrons and holes, hampering radiative recombination~\cite{AuPe2016}. The latter aspect is also known as the quantum confined Stark effect (QCSE). Indeed, the theoretical work in Ref.~\cite{AuPe2016} suggests that carrier localisation effects lead to a larger reduction in efficiency as In content increases.

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{Figures/thermalDroop.eps}
    \caption{Experimentally determined normalised optical output power as a function of temperature for an InGaN/GaN QW. Adapted from data in Ref.~\cite{DeMe2016}. Note: data is normalised to the output power at low temperatures, hence, it reflects efficiency.}
    \label{fig:ThermalDroop}
\end{figure}

Although the green gap has been attributed to material/active region specific issues, generally there are a number of factors that affect overall device efficiency, such as the efficiency of ``injecting'' electrons into the active region or extracting light out of the device. There are two main efficiency bottlenecks affecting overall device performance that will be focus of this thesis, thermal ``droop'' and current ``droop''~\cite{Weisbuch2020Review,MeDe2020,DeMe2017,Pipr2010}. Thermal droop is the reduction in the efficiency of both InGaN and AlGaN QW-based LEDs at elevated temperatures~\cite{MeDe2020,DeMe2017}. Figure~\ref{fig:ThermalDroop} shows experimental data adapted from Ref.~\cite{DeMe2016} on the (normalised) optical output power as a function of temperature in an InGaN/GaN QW-based LED at a fixed current. Generally, for LED devices, at room temperatures, $\sim$ 25$^{\circ}$C ($\sim$ 300 K), the efficiency of the device will have already dropped, but there will further heating of the device from electrical energy that is not converted to light, leading to common device temperatures around 80$^{\circ}$C ($\sim$ 350 K), with commercial devices rated up to even higher maximum junction temperatures of 175$^{\circ}$C ($\sim$ 450 K)~\cite{DeMe2016,DeMe2017}. In general, in semiconductor materials, Auger recombination is an important (nonradiative) process that can negatively affect device performance. It has been found experimentally in InGaN/GaN QWs that the rate of Auger recombination may increase with increasing temperature~\cite{NiKa2016} and thus the Auger process could be connected to thermal droop. However, other factors such as defect related processes have also been suggested, along with inefficiencies in injecting current into the QW~\cite{MeDe2020}. 

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{Figures/SampleCurrentDroop.eps}
    \caption{Theoretical IQE as a function of carrier density in a typical droop scenario and in an illustrative ideal scenario in which there is no Auger recombination. For the typical scenario, IQE coefficients of $A=1\times10^{6}$ s$^{-1}$~\cite{BaMc2023}, $B = 1\times10^{-11}$ cm$^{3}$/s~\cite{McTa2020} and $C=1\times10^{-30}$ cm$^{6}$/s~\cite{McKi2022} have been used in the ABC model. For the ideal scenario the $C$ coefficient has been set to zero.}
    \label{fig:CurrentDroop}
\end{figure}

Current droop (often simply, but confusingly referred to as efficiency droop) is the reduction in the efficiency of both InGaN and AlGaN QW-based LEDs as the current density through the LED is increased~\cite{Pipr2010}. Specifically, it is the reduction in the internal quantum efficiency (IQE), essentially the efficiency with which electrons are converted to photons, in the light emitting region of the structure (active region). Ideally, when the current passed through the LED is increased it will lead to an increase in the amount of light emitted. However, it is found that above a threshold current density the efficiency of light conversion decreases. These two scenarios (ideal and ``drooped'') are depicted in Figure~\ref{fig:CurrentDroop}, which shows the IQE as a function of carrier density (which is proportional to current density). To combat this, most commercial LEDs are operated at current densities lower than required for high brightness applications~\cite{Weisbuch2020Review}. This has limited the use of InGaN-based visible LEDs in applications such as vehicle headlights and may become a significant issue for the use of AlGaN-based UV LEDs in future applications such as large scale water disinfection treatments~\cite{AmCo2020}.

Much effort has been directed towards understanding the origin of current droop both experimentally and theoretically. Commonly, the carrier density dependence of the IQE is investigated in terms of the so-called ABC model, in which there are three recombination pathways governed by three coefficients, $A$, $B$ and $C$: the $A$ coefficient describes the Shockley-Read-Hall recombination, $B$, the radiative recombination and $C$ is the Auger recombination coefficient. However, experimental studies that target the determination of these coefficients are required to make significant assumptions about the coefficients and the IQE in order to disentangle the different pathways. For instance, it is often assumed that the IQE is perfect at low temperatures, but this has been questioned in III-N systems~\cite{NiTo2018}. Furthermore, there is often much uncertainty in the extracted values, and fitting the data for $C$, for example, may require the assumption of a value for $B$. It has also been found that parameters obtained via fitting procedures for one sample may not be transferable to other samples. Significant uncertainty remains, and although the role of Auger recombination has received much attention, other causes such as carrier leakage have also been suggested~\cite{Weisbuch2020Review,WeNa2020}.

To support the experimental studies, theoretical work has also targeted the determination of the recombination parameters entering the ABC model. To this end, theoretical frameworks that had been successfully applied to other III-V materials were employed to analyse III-N systems, but provided conflicting results~\cite{HodmasElectronicBands,HarrisonQWWD,MultibandEffMassIntro}. However, in those analyses alloy disorder and carrier localisation effects were largely neglected, despite the fact that these phenomena were thought to be the primary cause of the high efficiencies of III-N emitters despite their high defect densities. The impact of alloy disorder was largely neglected as it presents a significant modelling challenge. Carrier localisation can occur in InGaN QWs, for example, on the length scale of 2 -- 3 atoms~\cite{OlBe2010}. Thus, ideally, an electronic structure theory with an atomistic resolution of the system is required. Simultaneously, while some carriers may be localised on atomistic length scales, others may be more extended, on the length scale of 2 -- 3 nm~\cite{OlBe2010}. To account for this, again ideally, a simulation that can be scaled up to these length scales is required. On top of this, including realistic QW widths and the barrier material, along with the polarisation fields mentioned above, presents a unique modelling challenge in nitride-based systems, which may not be present for III-As or III-P systems.

Although there have been some ``state-of-the-art'' \emph{ab initio} studies of Auger recombination in bulk InGaN~\cite{KiSt2015}, modelling heterostructures, such as QWs, is generally beyond the scope of first principles frameworks~\cite{HodmasElectronicBands}. Instead, continuum-based models based on, for instance, \kp theories, or virtual crystal approximations are often used~\cite{AuPe2016,HodmasElectronicBands,HarrisonQWWD,MultibandEffMassIntro,JoTe2017,RuKa2020,BeGo2010,BeZh2013}. Many of these continuum-based models provide a good description of the electronic structure in a restricted energy range and region of the first Brillouin zone (FBZ), but as we will see, calculating the Auger recombination rate requires an accurate description of the electronic structure in a wider energy range, and potentially over the entire FBZ. Continuum-based models can be extended to accurately describe the full FBZ and larger energy ranges but doing so increases the computational cost~\cite{BeZh2013}. Furthermore, issues can arise in the application of \kp-based models to systems with interfaces, such as QWs~\cite{MultibandEffMassIntro}. Perhaps most importantly though, all of the approaches mentioned above require some spatial averaging of the alloy composition, which reduces the resolution of the simulation and may thus underestimate the impact of carrier localisation. For example, recent results have suggested that the coefficients of radiative and nonradiative Auger recombination in InGaN QWs are not carrier density independent, as is often assumed~\cite{BaMc2023}, and experimental reports of the temperature dependence of the radiative recombination rate stand in stark contrast to expectations based on fundamental theory and analysis in modified continuum-based frameworks~\cite{McTa2020}.

Due to the fundamentally atomistic nature of alloy disorder, the intrinsically atomistic tight-binding model represents a natural approach to tackling this problem~\cite{AufderMaurReview}. However, including effects such as the polarisation fields discussed above or deformations of the semiconductor crystal due to the presence of a heterostructure, present further challenges. So far, there have been very few theoretical studies of carrier recombination in III-N QWs (including InGaN-based QWs emitting in the visible spectral window, despite their technological importance) in an atomistic framework, with the work in Refs.~\cite{AuPe2016},~\cite{DiPe2019} and~\cite{DiPe2020} being the exceptions. For instance, the focus of the studies in Refs.~\cite{AuPe2016} and~\cite{DiPe2019} is the impact of carrier localisation on the radiative recombination rate in InGaN QWs. In this thesis we aim to develop a model that accounts for alloy fluctuations and carrier localisation effects to gain insight into the radiative and Auger recombination rates of realistic InGaN and AlGaN QW structures.

\section*{Thesis overview}

In Chapter~\ref{chp:LEDs} we will discuss some preliminary information on LEDs and how their efficiency is quantified, as well as the III-N heterostructures that form the light-emitting structures at the heart of LEDs. The wurtzite crystal structure of the relevant III-N materials will be introduced as well as some of the properties unique to III-N systems, such as the previously mentioned polarisation fields and carrier localisation effects. 

In order to capture these unique features, in Chapter~\ref{chp:ElecStruc}, we will discuss electronic structure theories such as the \emph{ab initio} and continuum-based models mentioned above, as well as the advantages and disadvantages of these methods. Subsequently we will introduce the tight-binding model used in the rest of this thesis to investigate the electronic and optical properties of III-N QWs. 

Building on this electronic structure theory, in Chapter~\ref{chp:RecombinationRates} we will discuss the determination of the radiative and Auger recombination rates in III-N QWs in general, and how these calculations are carried out within the established tight-binding framework. Subsequently, we will introduce modifications to the electronic structure to account for changes in carrier density.

With the theoretical framework established, in Chapter~\ref{chp:TempInGaN} we will primarily focus on the phenomenon of thermal droop in InGaN/GaN QWs by investigating the temperature dependence of the radiative and Auger recombination rates. This analysis is carried out for three different In contents in the well, allowing us to also comment on the impact of Auger recombination on the green gap.

In Chapter~\ref{chp:CarrDensInGaN} the carrier density dependence of the radiative and Auger recombination rates will be analysed in order to gain insight into the current droop in InGaN/GaN QWs. Here, we will also compare our theoretical results to experimental data extracted by collaborators at the University of Manchester on samples grown at the University of Cambridge.

Lastly, we will target an investigation of thermal droop in UV emitting AlGaN-based systems in Chapter~\ref{chp:TempAlGaN}. We will discuss modifications to the theoretical framework developed here to determine the temperature dependence of the radiative and Auger rates in AlGaN/AlN QWs, giving insight into the role of Auger recombination in thermal droop.

To conclude, we will summarise the main results of the thesis and discuss open questions and potential future research topics in the outlook in Chapter~\ref{chp:SummConc}.


\chapter{Light-Emitting Diodes}
\label{chp:LEDs}
\setcounter{page}{14}
\vspace{-0.5in}

\section{Introduction}
\label{sec:LEDsIntro}

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.9\columnwidth]{Figures/fig_2_1_MetalInsulatorsSemiconductors.eps}
    \caption{Schematic band diagram of a) a metal, b) a $p$-doped, intrinsic and $n$-doped semiconductor and c) an insulator. The range of energies that form the conduction (valence) band in a semiconductor and insulator are indicated by red (blue) boxes. Red (blue) energy ranges also represent unoccupied (occupied) electron states at 0 K. The Fermi level, $\text{E}_{\text{F}}$, is represented by a black dashed line. The band gap, $\text{E}_{\text{g}}$, is the difference in energy between the conduction band and valence band edges.}
    \label{fig:bandsElectronsHoles}
\end{figure}

In the Prologue we mentioned that the advent of $p$-$n$ junctions paved the way for efficient modern LEDs. All $p$-$n$ junctions are formed from semiconducting materials. A semiconductor is a crystalline material with an electrical conductivity between that of a metal (which freely conducts charge) and an insulator (which does not freely conduct charge). To understand the categorisation of materials into metals, semiconductors and insulators, we must first discuss the idea of energy ``bands'' and the Fermi level. 

From fundamental solid state physics, electrons in a crystal may only occupy certain allowed energy states, but these states can be closely spaced in certain energy ranges called ``bands'', depicted in Figs.~\ref{fig:bandsElectronsHoles} a), b) and c)~\cite{Kittel}. For a system of allowed electron energy states, the Fermi level, $\text{E}_{\text{F}}$, marks the midway point between occupied and unoccupied states (blue and red energy ranges in Fig.~\ref{fig:bandsElectronsHoles}, respectively). In other words, it is the energy at which there is a probability of 1/2 of occupation, even if there is no allowed state at that energy. In a metal, the Fermi level lies within a band, for example in Fig.~\ref{fig:bandsElectronsHoles} a), thus, electrons are easily excited into unoccupied states by the thermal energy of the system, allowing charge to freely move in the material. In an insulator, the Fermi level lies in the ``forbidden energy gap'' or ``band gap'', and there is a large energy separation between bands, for example in Fig.~\ref{fig:bandsElectronsHoles} c), thus charge does not easily flow. In a semiconductor, the Fermi level also lies in the band gap, but the energy separation between conduction and valence bands is smaller than in an insulator. In this case, the thermal energy of the system is sufficient to excite a small number (i.e. a smaller number than in a metal) of electrons across the band gap into unoccupied states, for example in Fig.~\ref{fig:bandsElectronsHoles} b). 

When an electron is excited into an unoccupied state, it leaves behind a vacant state. Electron vacancies can be thought of as positively charged quasi-particles called \emph{holes}. The unoccupied states above the band gap in a semiconductor are usually referred to as the conduction band, in contrast to the occupied valence band states below the band gap. If an electron that has been excited to the conduction band loses its energy and recombines with a hole in the valence band, a photon of light with energy equal to (or greater than) the band gap of the material can be emitted. In a ``direct'' band gap material, electron hole pairs recombine and emit a photon as described. In an ``indirect'' band gap material, the electron must also interact with the underlying crystal lattice in order for the recombination to occur. This reduces the probability of light emission in indirect gap materials compared to direct gap materials.

A $p$-$n$ junction consists of two regions in a semiconductor with a metallurgic interface between them, an $n$-doped region and a $p$-doped region. In an $n$-doped material, impurities (called dopants) have been added that introduce excess electrons. In this case they are called ``donor'' atoms as they ``donate'' extra electrons. In a $p$-doped material, the dopants have a deficit of electrons and so introduce vacant sites for electrons to occupy, as such, they are called ``acceptor'' dopants. In other words, $p$-doped materials have an excess of holes. 

The introduction of dopants to the semiconductor also affects the Fermi level in the system. In intrinsic semiconductors the Fermi level usually sits near the middle of the band gap (see ``Intrinsic'' in Fig.~\ref{fig:bandsElectronsHoles} b)). Introducing donor atoms into the system moves the Fermi level towards the conduction band edge (CBE) energy (see ``$n$-doped'' in Fig.~\ref{fig:bandsElectronsHoles} b)) and introducing acceptor atoms moves the Fermi level towards the valence band edge (VBE) energy (see ``$p$-doped'' in Fig.~\ref{fig:bandsElectronsHoles} b)). The energy levels of the $n$-dopants ($\text{E}_{\text{D}}$ for donors) or $p$-dopants ($\text{E}_{\text{A}}$ for acceptors) ideally are near the conduction or valence band edge energies, respectively. These energy levels are indicated by red ($\text{E}_{\text{D}}$) and blue ($\text{E}_{\text{A}}$) dashed lines in Fig.~\ref{fig:DepletionBandBending}. The closer these levels are to the relevant band edge energies, the easier it is to excite the excess electrons or holes from the dopants to the band. If the $n$ and $p$-doped regions consist of the same semiconductor material, and thus have the same band gap, the junction is called a homojunction. In a heterojunction, different semiconductors (which can have different band gaps) are used. When multiple heterojunctions are combined, a heterostructure is formed. The inclusion of heterostructures was a major step in the development of modern efficient LEDs~\cite{Weisbuch2020Review}. We will return to the discussion of heterostructures below.

In this chapter we will discuss the general underlying principle of operation of most modern LEDs, i.e., the injection of charge carriers across a $p$-$n$ junction. The quantum well (QW) double heterostructures which are crucial to the efficiency of LED devices will also be introduced. Subsequently, we will turn to the parameters and ABC model of efficiency often employed in the analysis of LED devices. Having discussed some key concepts pertaining to LEDs in general, lastly, the peculiarities of III-N systems, including their crystal structure, the impact of strain in III-N heterostructures, related polarisation fields, and carrier localisation effects will be discussed.

\section{Principles of operation}
\label{sec:LEDsPrinciplesOfOperation}

As mentioned, before the introduction of double heterostructures (DHs), the $p$-$n$ junctions used in LEDs were homojunctions. For the purpose of discussion, if we assume that there is a discrete, step-like change in the concentration of dopants across the interface between the $n$- and $p$-doped regions, and no external voltage applied to the junction, then there will be a region with a high concentration of electrons (the $n$-doped region) next to a region with a low concentration of electrons (the $p$-doped region), and vice versa for holes. Furthermore, due to the alteration of the Fermi level of the system with the introduction of $n$- and $p$-dopants, there will a discontinuity of the Fermi level across the interface. Charge will flow from a region of higher Fermi level to lower Fermi level~\cite{SzeLee2012}.

Fick's law describes the tendency of particles to diffuse from regions of high to low concentration, thus electrons from the $n$-doped region will diffuse into the $p$-doped region and holes into the $n$-doped region. Electrons diffusing into the $p$-doped region will recombine with the excess of holes, and similarly for holes diffusing into the $n$-doped region. On either side of the interface a region of low electron and hole concentration will form called the depletion region.

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{Figures/fig2_2.eps}
    \caption{Conduction (CBE) and valence band edge (VBE) energies as a function of position along the growth direction of a $p$-$n$ junction. Electrons in the $n$-doped region see a potential barrier due to the bending of bands by the built-in potential, similarly for holes in the $p$-doped region. The potential barrier is indicated by qV$_{\text{BI}}$. The depletion region forms around the interface between the $n$- and $p$-doped regions. Donor impurities in the $n$-doped ($p$-doped) region move the Fermi level, $\text{E}_{\text{F}}$ towards the CBE (VBE). Dopant energies are indicated by red (blue) dashed lines for donors (acceptors). The built-in potential ensures that there is no discontinuity in the Fermi level.}
    \label{fig:DepletionBandBending}
\end{figure}

Furthermore, as electrons diffuse from the $n$-doped region, positively charged donors remain. Similarly for holes, negatively charged acceptors remain in the $p$-doped region. The resulting net positive and negative charges in the $n$- and $p$-doped regions induce an electric field that opposes the diffusion process. Associated with this field is a potential drop, V$_{\text{BI}}$, that will align the Fermi levels in the $n$- and $p$-doped regions, bringing the system into equilibrium, and resulting in a ``band bending'' effect, shown above in Fig.~\ref{fig:DepletionBandBending}. Thus, for instance, electrons in the $n$-doped region see an energy barrier preventing their movement to the $p$-side of the junction. 

If we apply a voltage across the junction that pulls carriers away from the depletion region (i.e. so that electrons are pulled to the right and holes to the left in Fig.~\ref{fig:DepletionBandBending}, called reverse bias) the width of the depletion region and the magnitude of the potential energy barrier will increase. Conversely, applying a forward bias will reduce the width of the depletion region and magnitude of the potential barrier which will allow current to flow. 

When current flows, electrons and holes cross the depletion region and may then recombine radiatively. This is depicted schematically below in Fig.~\ref{fig:pnHJvsDH} a). As we will see in more detail in the next section, the rate of radiative recombination depends on the concentration of charge carriers, thus increasing the carrier concentration would ideally lead to an increased rate of radiative recombination. This can be achieved by the inclusion of a double heterostructure.

\subsection{Double heterostructures and quantum wells}

Double heterostuctures (DHs) offer a number of advantages. The DHs targeted during the development of modern LEDs consisted of a material with a band gap that was lower than the band gap of the surrounding (barrier) material. One major advantage of this is that the light emitted by recombination in the lower band gap material (the so-called ``active layer''/``active region'') would not be absorbed by the barrier material, improving efficiency~\cite{Weisbuch2020Review}. Furthermore, due to the lower energy in the active layer depicted in Fig.~\ref{fig:pnHJvsDH} b), charge carriers become confined in the DH. A schematic of the impact of the inclusion of a DH on the electrons and holes in a $p$-$n$ junction is shown in Fig.~\ref{fig:pnHJvsDH} b). The confinement of electrons and holes in the active layer leads to a much higher rate of light emission from this region.

\begin{figure}[t!]
\centering
    \includegraphics[width=\columnwidth]{Figures/fig2_3ab.eps}
    \caption{a) Under forward bias, the built-in potential and depletion region width in the $p$-$n$ junction reduce, allowing carriers to enter. Carriers can undergo recombination, potentially resulting in the emission of a photon. b) Introduction of a double heterostructure confines carriers in the same spatial region, increasing the likelihood of recombination occurring.}
    \label{fig:pnHJvsDH}
\end{figure}

When first developed, DHs consisted of active layers with widths of $\sim$ 10 -- 100 nm (e.g. the first nitride DHs developed by Nakamura \emph{et al.} included an InGaN active layer 45 nm thick~\cite{Nakamura2015}). These improved device performance significantly, increasing light output power from 42 $\mu$W for a GaN homojunction (i.e. a homojunction consisting simply of $n$- and $p$-doped GaN) to 1.5 mW when including an InGaN DH~\cite{Nakamura2015}. Further improvements to growth techniques allowed the realisation of DHs with much narrower widths, increasing the concentration of charge carriers and thus the amount of light emitted~\cite{Nakamura2015}. 

Semiconductor QWs are DHs that are so thin ($\sim$ a few nanometres) that quantum confinement effects begin to manifest, bringing further advantages, such as the possibility to tailor the emission wavelength. By changing the band gap of the material in the active layer (using, for instance, alloys such as InGaN) and also the width of the active layer~\cite{BoLo2000}, QWs can be engineered to emit light at a specific wavelength. Modern blue LEDs often include QWs with widths of around 2 -- 3 nm. The variation of the conduction and valence bands form a confining potential. A simple approximation of this situation is as a 1D particle in a box problem, schematic solutions of which are shown in Fig.\ref{fig:BandAlign}.

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{Figures/fig2_4.eps}
    \caption{Type I band alignment of a typical QW showing variation of band edges as a function of position along the direction of quantum confinement. Carriers are confined in the region of lower band gap. Schematic solutions to a 1D particle in a box problem are represented by the carrier wave functions, $\Psi$, and lowest energy levels, E.}
    \label{fig:BandAlign}
\end{figure}

Figure~\ref{fig:BandAlign} depicts the change in conduction band edge energy (CBE) and valence band edge energy (VBE) as a function of position along the direction of quantum confinement, when a material of lower band gap is embedded in a material of higher band gap. The QW shown in Fig.~\ref{fig:BandAlign} is an example of a type I heterostructure (or type I band alignment), in which both electrons and holes are confined in the active layer (i.e. the CBE of the well material is lower in energy than the barrier material, and the VBE of the well material is higher in energy than the barrier material). In type II heterostructures, only one type of carrier is confined, for instance, the CBE of the well may be lower in energy than the CBE of the barrier, but the VBE of the well may also be lower in energy than the VBE of the barrier, in which case only electrons would be confined in the QW, and vice versa. 

As discussed above, confining both electrons and holes in the active layer leads to increased carrier concentration in the same spatial region and thus an increased rate of radiative recombination. For this reason, type I band alignments are desirable in optoelectronic applications. Further details on band alignments in type I and II heterostructures can be found in Ref.~\cite{RoKu2015}. From a fundamental physics perspective the band alignment is determined by the band gap and the valence band offset, which can be determined theoretically or experimentally~\cite{VuMe2003,MoVa2010,MoMi2011}. For example, for InN and GaN, the valence band offset (i.e. the difference in energy between the top of the valence band edges) has been determined to be approximately 0.62 eV~\cite{MoVa2010,MoMi2011} (with the VBE of InN siting at higher energies than the VBE of GaN), and, as we saw above, the band gap of InN (and InGaN alloys) is smaller than that of GaN.

So far, we have discussed radiative recombination a number of times, as well as the efficiency of LEDs. We return now to the common measures of efficiency in LEDs in more detail, as well as quantifying the relationship between radiative (and nonradiative) recombination and efficiency. 

\section{Quantifying and modelling efficiency in LED devices}
\label{sec:LEDsModellingEfficiency}

Throughout the history of the development of light-emitting diodes (LEDs) efficiency has been a key parameter since the optimisation of energy conversion is desirable for many reasons (e.g. commercial interests, pure scientific curiosity and, more recently, carbon output mitigation)~\cite{OkBi2015}. However, in an LED device there are a number of parameters that are used to characterise the efficiency, the product of which gives the overall LED efficiency, known as the wall-plug efficiency (WPE)~\cite{Weisbuch2020Review}. The following definitions are presented in Ref.~\cite{ShSh2018} and only summarised here. A very thorough discussion of measures of efficiency, as well as factors that can affect efficiency in LEDs, can also be found in Ref.~\cite{Weisbuch2020Review}. The WPE ($\eta_{\text{WPE}}$, also known as power efficiency) is the overarching measure of the total light output of an LED compared to the total electrical input to the LED, and accounts for all energy losses in the structure. WPEs can then be broken down into two contributions, the voltage efficiency (VE, $\eta_{\text{VE}}$, also known as feeding efficiency or electrical efficiency) and the external quantum efficiency (EQE, $\eta_{\text{EQE}}$) such that 
\begin{equation*}
    \eta_{\text{WPE}} = \eta_{\text{VE}} \times \eta_{\text{EQE}} \,\, .
\end{equation*}
VE is defined as the fraction of electrical power supplied to injected electron hole pairs in the active region (e.g. in the QW). Losses measured by this parameter are mainly attributed to Ohmic heating of the contacts between the device power supply and the $p$-$n$ junction. Another contributing factor to the VE is the injection efficiency (IE, $\eta_{\text{IE}}$). The IE is the ratio of the number of electrons injected into the active layer (shown in Fig.~\ref{fig:pnHJvsDH} b) above) to the number of electrons injected into the overall LED structure. IE can then take into account losses due to, for instance, carrier overshoot, where an electron or hole does not become confined in the active region.

EQE is defined as the ratio of the number of photons emitted from the device to the number of electrons injected into the active region. \blue{However, some photons produced in the active region of the device will not be emitted, for instance, they may be reabsorbed by free carriers in the barriers or in the metal contacts, or may be emitted in the wrong direction, e.g., towards the underlying substrate on which the LED is grown. There, in the substrate, they may also undergo absorption~\cite{Weisbuch2020Review}.} Thus, EQE can be further broken down into two contributions, the light extraction efficiency (LEE, $\eta_{\text{LEE}}$) and the internal quantum efficiency (IQE, $\eta_{\text{IQE}}$) such that
\begin{equation*}
    \eta_{\text{EQE}} = \eta_{\text{LEE}} \times \eta_{\text{IQE}} \,\, .
\end{equation*}
LEE is a measure of the ability of photons to be extracted from the device and is a significant limiting factor in, for instance, the efficiency of deep UV emitting LEDs~\cite{AmCo2020,FiSc2022,ScCa2013}.

Furthermore, in III-N-based LEDs, as mentioned above in the Prologue, the IQE has been shown to ``droop'' with increasing carrier density. The IQE is defined as the ratio of the rate of photon generation to the total rate of recombination (including radiative and nonradiative processes) in the active region. The IQE allows us to quantify the efficiency of the active region in converting recombining electrons to photons. However, experimentally determining the IQE can prove challenging, due to the challenge of disentangling the measurements of the parameters described above~\cite{Weisbuch2020Review}. Regardless, efforts have been made that rely on the so-called ABC model, that we will discuss in the following section, to shed light onto the droop effect and the IQE in general.

\subsection{ABC model}

In the ABC model~\cite{Weisbuch2020Review,Pipr2010,PiprekReview2020}, three major recombination pathways are usually considered:
\begin{itemize}
    \item A nonradiative Shockley-Read-Hall (SRH)~\cite{ShRe1952,Hall1952,NeamenSemiconductors} recombination process that scales linearly with electron carrier density, $n$ ($R_{\text{SRH}} \propto n$),
    \item a radiative recombination process proportional to the product of electron, $n$, and hole, $p$, carrier densities ($R_{\text{rad}} \propto np$),
    \item a nonradiative Auger recombination process involving three carriers ($R_{\text{Auger}} \propto n^{2}p$ or $np^{2}$).
\end{itemize}
The coefficients of proportionality are then the $A$, $B$ and $C$ recombination coefficients such that $R_{\text{SRH}} = An$, $R_{\text{rad}} = Bnp$ and $R_{\text{Auger}} = C_{nnp}n^{2}p + C_{npp}np^{2}$. The $C_{nnp}$ and $C_{npp}$ Auger coefficients are also often referred to as $C_{eeh}$ and $C_{hhe}$, respectively. One of the main assumptions of the ABC model is that these proportionality coefficients are constants and thus independent of carrier density or temperature. \blue{Furthermore, in experimental reports of recombination coefficients, the coefficients can often be considered as ``effective'' parameters that include contributions from many processes. For instance, values of $C$ reported in the literature may include contributions from different Auger-type processes, such as those enhanced by phonons or defects, or even other nonradiative processes. On top of this, when extracting coefficient values experimentally, often some assumptions about the value of, for instance, the $B$ coefficient is required in order to determine $C$. Also, the coefficients extracted for a given experimental sample might not be transferable to other samples, given the empirical nature of the fitting. Thus, targeting a direct calculation of the coefficients may allow us to gain insight into the relative importance of different recombination processes, both radiative and nonradiative.}

\begin{figure}[t!]
\centering
    \includegraphics[width=\columnwidth]{Figures/fig2_5abc.eps}
    \caption{Schematic representations of the three recombination pathways in conventional ABC models. Changes in electron energy are represented by red arrows, changes in hole energy are represented by blue arrows. a) Shockley-Read-Hall recombination at a defect (trap) state is nonradiative. b) Radiative recombination results in the emission of a photon. c) Auger recombination is a three particle process. An electron hole pair recombine and interact with either another electron ($eeh$ Auger) or another hole ($hhe$ Auger) as shown.}
    \label{fig:ABCschema}
\end{figure}

Figure~\ref{fig:ABCschema} shows schematic depictions of the a) SRH, b) radiative and c) Auger recombination processes. The SRH model of recombination in semiconductors involves the recombination of an electron hole pair via an intermediate state (called a ``trap'' state), with an energy that lies in the band gap~\cite{ShRe1952,Hall1952,NeamenSemiconductors}. The intermediate state is generally associated with defects in the material, such as impurities or threading dislocations~\cite{OlBe2010}. Radiative recombination refers to a recombination event in which a photon is emitted. Auger recombination, also referred to as Auger-Meitner recombination~\cite{ZhTu2022,MaCo2019}, is a three carrier process independently discovered by Pierre Auger and Lise Meitner in the mid-1920s. In an Auger recombination event, an electron hole pair recombines and interacts with either an electron or a hole via the Coulomb interaction. Thus, there are two distinct Auger processes, namely electron-electron-hole ($eeh$, sometimes called $nnp$) and hole-hole-electron ($hhe$, also called $npp$), depicted in Fig.~\ref{fig:ABCschema} c). The total Auger rate is the sum of the $eeh$ and $hhe$ rates, and here the coefficient of total Auger recombination, $C_{\text{tot}}$, is the sum of $C_{eeh}$ and $C_{hhe}$. Auger recombination is a nonradiative recombination process as it produces ``hot'' Auger carriers instead of photons.

In general, from the definition of the IQE; 
\begin{align}
    \text{IQE} &= \frac{\#\left(\text{radiative recombinations}\right)}{\#\left(\text{recombinations in total}\right)} \nonumber \\
    &= \frac{R_{\text{radiative}}}{R_{\text{radiative}}+R_{\text{nonradiative}}} \nonumber\\
    &= \frac{Bnp}{Bnp + An + (C_{eeh}n^{2}p+C_{hhe}np^{2})} \,\, .
    \label{eq:IQEExpression}
\end{align}

Here we have made the assumption that every electron injected into the active region undergoes one of the above recombination processes. Different approaches to determining $A$, $B$ and $C$ experimentally have been discussed in the literature but often require significant assumptions. For instance, as mentioned, it is often assumed that these coefficients do not vary with temperature or carrier density~\cite{WePi2015}, or these approaches require some \emph{a priori} knowledge of the $B$ coefficient in order to determine $A$ and $C$~\cite{WePi2015}. Accurately determining the carrier density in the active layer can also present another challenge.

The ABC model was, in the context of III-Ns, first invoked by Shen \emph{et al.} to explain the observation of current droop. They reported that the droop was related to an internal loss process, and not, for instance, injection efficiency~\cite{ShMu2007}. Although debate over the applicability of the ABC model to nitrides continues~\cite{PiprekReview2020}, this approach remains (sometimes with modifications) one of the primary methods used to quantify the IQE, as well as the radiative and nonradiative recombination rates both experimentally and theoretically~\cite{NiBi2016,BiNi2013,GaDr2012,TiMc2014}. Often, experimental determinations of IQE take a ``top-down'' approach, by fitting the expression for IQE in Eq.~\eqref{eq:IQEExpression} to experimental data with $A$, $B$ and $C$ being free parameters. 

Theoretically, however, it is possible to attempt a ``bottom-up'' analysis, determining the recombination coefficients (and thus the IQE) from the electronic structure. In order to gain this information about the electronic structure, we must account for material specific aspects, for instance, the underlying crystal structure and QW properties (e.g. well width, alloy composition, etc.). Although the ABC model is generally applicable to semiconductor LEDs~\cite{Weisbuch2020Review}, as discussed in the Prologue, III-N materials differ from ``conventional'' semiconductor materials in several ways, with these peculiarities, to a large extent, stemming from their crystal structure, material properties and growth direction. To obtain a detailed understanding of the IQE, we must gain insight into the electronic structure, which is tightly linked to the underlying crystal structure and resulting and connected properties of III-N materials. In the next section we discuss these features as well as the resulting impact on heterostructures consisting of these materials.

\section{III-nitride materials and derived heterostructures}
\label{sec:LEDsIIINMaterials}

In general, when referring to III-N semiconductors, we are referring to the binary alloys InN, GaN and AlN, along with ternary alloys derived from these such as InGaN and AlGaN. One feature of the binary III-Ns that sets them apart from conventional III-V semiconductors is that they are thermodynamically stable in the wurtzite crystal structure, instead of the zinc-blende crystal structure~\cite{SarojPatraThesis,MiguelCaroThesis,ChengIIIVSemiconductors}. So, we begin by examining features of a wurtzite crystal structure. In this analysis we assume the material is perfectly crystalline and does not contain defects.

\subsection{Wurtzite crystal structure}

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.5\columnwidth]{Figures/Wurtzite.eps}
    \caption{Representation of the wurtzite crystal structure (not to scale). The ions of the unit cell are labeled \textbf{A} to \textbf{D} and represented by the filled coloured circles, red circles representing cations (positively charged) such as In, Ga and Al, and blue representing anions (negatively charged), in this case N. The hatched circles represent periodic replicas of the ions of the unit cell. Cartesian $x$, $y$ and $z$ axes are presented relative to the lattice. The in-plane ($x$-$y$-plane), $a$, and out-of-plane ($z$-direction), $c$, lattice constants are also shown. Adapted from Ref.~\cite{MiguelCaroThesis}.}
    \label{fig:WZStruct}
\end{figure}

Figure~\ref{fig:WZStruct} shows an overview of the wurtzite unit cell. Features of this crystal structure are well documented and here we mainly summarise the discussion in Refs.~\cite{SarojPatraThesis} and~\cite{KoSa1983}. The wurtzite crystal structure has $C_{6v}$ symmetry~\cite{StefanSchulzThesis}. In contrast to cubic systems, this hexagonal system is characterised by two lattice constants, $a$ and $c$ (see Fig.~\ref{fig:WZStruct}). If one makes an observation of the layout of atoms in the wurtzite lattice at a certain point, and then translates by a distance $c$ along the $\hat{z}$-direction, the new atomic configuration will be indistinguishable. This direction is also called the $c$-axis/$c$-direction. The ideal wurtzite structure is formed by two interpenetrating hexagonal close-packed sub-lattices, offset by $(5/8)c$ in the $z$-direction. Similar translational invariances hold in the plane defined by $x$ and $y$, which is called the $c$-plane, as it is normal to the $c$-axis. 

Along with $c$, along the base of the structure there is the in-plane lattice constant $a$. Often, we will refer to in-plane ($x$-$y$ plane, $a$ lattice constant) and out-of-plane ($z$-direction, $c$ lattice constant) directions, which correspond to directions in and perpendicular to the $c$-plane, respectively. Translations of the lattice that result in an indistinguishable lattice are defined by the primitive lattice vectors: 
\begin{align*}
    \mathbf{a} &= (a,0,0) \,\, , \\
    \mathbf{b} &= ((1/2)a,(\sqrt{3}/2)a,0) \,\, , \\
    \mathbf{c} &= (0,0,c) \,\, .
\end{align*}
In Fig.~\ref{fig:WZStruct}, the ions of the unit cell are denoted by the coloured circles, red circles denoting cations (positively charged) such as In, Ga and Al, and blue denoting anions (negatively charged), in this case N. The anions and cations form alternating layers. The filled circles represent atoms of the primitive unit cell, i.e., the group of ions that generate the entire crystal when translated by the primitive lattice vectors. Shaded circles represent atoms in neighbouring periodic replicas of the unit cell. Often, the positions of ions in the lattice are given in terms of $a$, $c$ and the internal parameter $u$. The internal parameter is defined to be the length of the cation-anion bond divided by $c$. In an ideal wurtzite structure $u=3/8$~\cite{Morkoc2008}. Within the primitive unit cell, atomic sites are located at the following positions~\cite{Morkoc2008}:
\begin{align*}
    \mathbf{r}_{\textbf{A}} &= (0,0,0) \,\, , \\
    \mathbf{r}_{\textbf{B}} &= (0,0,uc) \,\, , \\
    \mathbf{r}_{\textbf{C}} &= ((1/2)a,(1/(2\sqrt{3}))a,c/2) \,\, , \\
    \mathbf{r}_{\textbf{D}} &= ((1/2)a,(1/(2\sqrt{3}))a,(u+1/2)c) \,\, .
\end{align*}

Above, we defined the $c$-plane as the plane normal to the $c$-axis. In general, crystallographic directions and planes can be defined by Miller indices; $h$, $k$ and $l$~\cite{Kittel}. These indices are the reciprocal of the point of intersection of the plane of interest with the axes defined by the primitive lattice vectors, and negative values are identified by a bar above the index, e.g., $\bar{h}$. In general, a crystallographic \emph{plane} is labelled with round brackets, $(hkl)$, thus, in our example above, the $c$-plane would be denoted by (001). Square brackets are used to denote a \emph{direction}, thus, above, the associated $c$-direction would be [001]. However, in hexagonal lattices a modified version called Miller-Bravais indices are used where the index $i=-h-k$ is introduced. The modified label is then $(hkil)$. This allows for quick identification of equivalent directions in hexagonal lattices. In Miller-Bravais notation, the wurtzite $c$-plane above becomes the (0001)-plane, and the $c$-direction is the [0001]-direction. 

Although the definition of the Miller and Miller-Bravais indices may seem unusual, it allows for easier identification of crystallographic planes during experimental investigations of these crystals via measurements of the diffraction of, for instance, X-rays through the material~\cite{PeCo2001}. In fact, the concept of the reciprocal of a real space length is key to investigating not only crystal structures, but also the nature of the quantum mechanical states of charge carriers. We will return to this idea of a ``reciprocal space'' below when we discuss the electronic structure. In general, the $c$-direction is the most common growth direction of III-N semiconductors, with layers deposited throughout the growth process on the $c$-plane. As we will be primarily discussing $c$-plane III-N QWs, we will often refer to the $c$-plane as the growth plane, and the $c$-direction as the growth direction. It is possible to grow III-Ns on other planes such as the (1$\bar{\text{1}}$00)-plane ($m$-plane) and the (11$\bar{\text{2}}$0)-plane ($a$-plane), however, it is more difficult to grow high quality crystals in these directions~\cite{FeSc2009}.

Furthermore, as mentioned previously, the atomic positions in a wurtzite crystal structure are often expressed in terms of the internal parameter, $u$. In an \emph{ideal} structure, $u=3/8$. However, in \emph{realistic} wurtzite III-Ns $u$ deviates from this value~\cite{AmMa2002,ZoBe2001,PaKu2014,CaSc2013local}. The consequence of this deviation is that wurtzite systems exhibit a built-in electric polarisation, $\mathbf{P}$. From electrostatics we know that any discontinuity in $\mathbf{P}$ will lead to bound polarisation induced charges and built-in electrostatic fields~\cite{Zangwill,Jackson1999}. Although growth along other directions may be targeted to avoid the formation of polarisation fields, $c$-plane growth is the most common. Thus, the electric polarisation is the subject of our next discussion, as it can have a strong impact on the optical and electronic properties of $c$-plane III-N heterostructures.

\subsection{Polarisation}

Wurtzite III-N heterostructures have been shown experimentally to contain strong built-in electrostatic fields, stronger than those found in conventional III-V and II-VI semiconductors~\cite{BeFi1997,BeFi1999}. As mentioned above, due to the deviation of the internal parameter from ideal in real wurtzite III-Ns, there will be a built-in polarisation. This deviation gives rise to a net dipole moment in each primitive unit cell along the vector from ion \textbf{D} to \textbf{C} in Fig.~\ref{fig:WZStruct} above, $\mathbf{d}_{\textbf{DC}}$, i.e., the $-z$ direction. The net dipole moment per volume is defined to be the spontaneous polarisation vector field, $\mathbf{P}_{\text{SP}}$. In unstrained systems, spontaneous polarisation is not found in zinc-blende materials due to reasons of symmetry~\cite{HodmasPolInIIINSemi,MultibandEffMassElecPropsZBvsWurt}. Furthermore, it has been observed that the magnitude of the spontaneous polarisation in wurtzite III-Ns is much larger than in wurtzite II-VI materials such as ZnO or BeO~\cite{BeFi1997}. Further discussion can be found in Refs.~\cite{StefanSchulzThesis} and~\cite{HodmasPolInIIINSemi}. 

In III-N heterostructures, on top of the spontaneous polarisation, $\mathbf{P}_{\text{SP}}$, there is also the possibility of a strain induced piezoelectric polarisation vector field, $\mathbf{P}_{\text{PZ}}$. It has been found experimentally that III-N materials exhibit significantly higher piezoelectric responses compared to other semiconductors given their high bond ionicity, and can be up to ten times larger in wurtzite III-N materials when compared to, for instance, zinc-blende arsenides~\cite{BeFi1997,BeFi1999}. We now turn to discuss the piezoelectric polarisation in more detail.

\subsubsection{Piezoelectric polarisation and strain}

Deformations of a material cause strain to arise. In semiconductor crystals that lack inversion symmetry, strain can result in the formation of a net dipole moment~\cite{DanielTannerThesis}. This phenomenon, known as the piezoelectric effect, is strongly linked to the symmetry of the underlying crystal structure, the applied strain (hydrostatic, biaxial compressive, shear strain) and the growth direction. In general, we can relate the piezoelectric polarisation field, $\mathbf{P}_{\text{PZ}}$, in a material to an applied strain, $\varepsilon$, via the components of the strain tensor, $\varepsilon_{jk}$, and the components of the piezoelectric tensor, $e_{ijk}$, such that
\begin{equation*}
    P_{\text{PZ},i} = \sum_{j=1}^{3}\sum_{k=1}^{3}e_{ijk}\varepsilon_{jk} \,\, ,
\end{equation*}
where the indices $i$, $j$ and $k$ ($\in \{1,2,3\}$) run over the three Cartesian directions $x$, $y$ and $z$ of 3D space. This formalism, and the discussion below, broadly follow the approach set out in Ref.~\cite{HodmasPolInIIINSemi}.  

In linear elasticity theory, the strain tensor, $\varepsilon$, is of the form~\cite{HodmasPolInIIINSemi}
\begin{equation*}
    \varepsilon = \begin{pmatrix}
        \varepsilon_{xx} & \varepsilon_{yx} & \varepsilon_{zx} \\
        \varepsilon_{xy} & \varepsilon_{yy} & \varepsilon_{zy} \\
        \varepsilon_{xz} & \varepsilon_{yz} & \varepsilon_{zz}
    \end{pmatrix} \,\, ,
\end{equation*}
however, it can be shown~\cite{HodmasPolInIIINSemi} that three of the nine entries are redundant as the $j$ and $k$ indices are interchangeable due to symmetries of $\varepsilon_{jk}$~\cite{DanielTannerThesis}. This simplification, due to Voigt~\cite{HodmasPolInIIINSemi}, is often employed. If we describe pairs of indices as $xx = 1$, $yy = 2$, $zz = 3$, $yz = 4$, $xz = 5$, $xy = 6$, the strain tensor becomes
\begin{equation*}
    \varepsilon = \begin{pmatrix} 
    \varepsilon_{1} & \varepsilon_{2} & \varepsilon_{3} & \varepsilon_{4} & \varepsilon_{5} & \varepsilon_{6} 
    \end{pmatrix}^{T} \, \, .
\end{equation*}
It is also noted in Ref.~\cite{HodmasPolInIIINSemi} that in this convention $\varepsilon_{\alpha} = \varepsilon_{jk} + \varepsilon_{kj} = 2\varepsilon_{jk}$ with $j\neq k$, and the inclusion or exclusion of the factor of two has been inconsistent in the literature. Regardless, in Voigt notation the piezoelectric tensor becomes
\begin{equation*}
    e = \begin{pmatrix}
        e_{11} & e_{12} & e_{13} & e_{14} & e_{15} & e_{16} \\
        e_{21} & e_{22} & e_{23} & e_{24} & e_{25} & e_{26} \\
        e_{31} & e_{32} & e_{33} & e_{34} & e_{35} & e_{36} 
    \end{pmatrix} \,\, .
\end{equation*}
However, some of these components can be shown to be either zero, or equal to other components, depending on the symmetry of the crystal structure of the material being described~\cite{HodmasPolInIIINSemi}. In a system with wurtzite symmetry, the piezoelectric tensor is
\begin{equation}
    \label{eq:PiezoCoeffs}
    e = \begin{pmatrix}
        0      & 0      & 0      & 0      & e_{15} & 0 \\
        0      & 0      & 0      & e_{15} & 0      & 0 \\
        e_{31} & e_{31} & e_{33} & 0      & 0      & 0 
    \end{pmatrix} \,\, .
\end{equation}
Overall then, for instance, in Voigt notation, the component of the piezoelectric polarisation vector field in the $z$-direction is given by
\begin{equation}
    \label{eq:zComponentStrain}
    P_{\text{PZ},3} = \sum_{\alpha=1}^{6}e_{3\alpha}\varepsilon_{\alpha} \,\, .
\end{equation}
The expression in Eq.~\eqref{eq:zComponentStrain} illustrates that strain in the growth plane of $c$-plane III-Ns will lead to a piezoelectric response. In III-N QWs, in-plane strain arises when III-N alloys with different in-plane lattice constants are grown on top of each other. We now look at the consequences this has for $c$-plane III-N QWs.

\subsection{Impact of polarisation on III-N quantum wells}

Most modern short-wavelength visible spectral range LEDs incorporate QWs consisting of thin (2 -- 3 nm) $c$-plane InGaN layers in the active region, surrounded by, for instance, $c$-plane GaN barriers~\cite{NiBi2016,BiNi2013,GaDr2012,TiMc2014}. Thus, already we expect that the discontinuity in spontaneous polarisation will lead to the formation of a polarisation field across the well. Furthermore, the in-plane lattice constant mismatch between, for example, InN and GaN is approximately 11\% ($a(\text{InN}) = 3.545$ \r{A} and $a(\text{GaN}) = 3.189$ \r{A}~\cite{CaSc2011}) and so will lead to a significant strain in the system. As discussed above, this strain will lead to a piezoelectric contribution to the polarisation field in the well. In order to discuss the impact of the polarisation field on the electronic and optical properties of III-N QWs, as an illustration, we will focus on a simplified, 1D description of an InGaN/GaN QW.

In order to investigate the built-in polarisation field we require some knowledge of the strain that will arise from the lattice mismatch between GaN and the InGaN alloy in the well. To do this, we need to determine the lattice constants of the bulk (i.e. not in a QW) InGaN alloy. However, due to the distribution of In in the alloy it is not immediately clear how to do so, as the value will vary between areas of higher and lower In content. One approximation commonly used in this situation is the virtual crystal approximation (VCA). In a VCA various physical parameters of the bulk alloy are approximated as being weighted averages of the parameters of the constituent materials (such as the lattice constants, elastic constants, band gaps, etc.~\cite{CaSc2011,AmMa2002}). This approach works remarkably well when describing, for example, the lattice constants of bulk InGaN alloys~\cite{LiLi2005}. We denote InGaN alloys with specific InN contents by In$_{x}$Ga$_{1-x}$N, where $x$ is the fractional molar InN content in the alloy. We can apply this approach to determine the $a$ lattice constant of an In$_{x}$Ga$_{1-x}$N alloy by using the following interpolation due to V{\'e}gard (called V{\'e}gard's law)~\cite{CaSc2011}; 
\begin{equation}
    \label{eq:vegard}
     a(\text{In$_{x}$Ga$_{1-x}$N}) = xa(\text{InN}) + (1-x)a(\text{GaN})\,\, .
\end{equation}
For instance, using parameters from Ref.~\cite{CaSc2011}, for a typical InN content of 10\%, i.e., $x=0.1$, with the in-plane lattice constants for InN and GaN given above, the resulting in-plane lattice constant for a In$_{0.1}$Ga$_{0.9}$N alloy is $a(\text{In$_{0.1}$Ga$_{0.9}$N}) = 3.225$ \r{A}. For the lattice parameters, \EQ{eq:vegard} is often employed~\cite{CaSc2011,AmMa2002}, but for other material parameters, for instance band gaps, \EQ{eq:vegard} needs to be modified to include a quadratic dependence of parameter on composition. The quadratic dependence is characterised by the so-called bowing parameter, which describes the deviation from a linear dependence. The bowing parameter can be determined by fitting experimental or theoretical data~\cite{VuMe2003,MoVa2010,CaSc2013local}.

When such an In$_{x}$Ga$_{1-x}$N alloy is grown on a GaN substrate, the mismatch in the lattice constants will lead to a compressive strain in the In$_{x}$Ga$_{1-x}$N alloy in the growth plane if pseudomorphic growth is assumed. In such a case, the strain is constant in the QW region, but to minimise the elastic energy of the system, the InGaN ``expands'' along the growth direction. This assumption will generally be reasonable as long as the well width is below the critical thickness of an InGaN layer~\cite{JaAl1998,HoCo2007,LeSt2008}. Thus, the in-plane lattice constants of the In$_{x}$Ga$_{1-x}$N layer will be the same as the in-plane lattice constants of the GaN substrate and the in-plane strain will be isotropic, i.e., $\varepsilon_{1} = \varepsilon_{2}$. In terms of the $a$ lattice constants of the well ($a_{\text{W}}$) and barrier ($a_{\text{B}}$) materials, $\varepsilon_{1} = (a_{\text{B}} - a_{\text{W}})/a_{\text{W}}$. With the structure relaxed in the growth direction, there will be no stress in the $c$-direction, and we can relate the growth direction strain to the in-plane strain via~\cite{AmMa2002}
\begin{equation*}
    \varepsilon_{3} = -2\frac{C_{13}}{C_{33}}\varepsilon_{1} \,\, ,
\end{equation*}
where the factor $2C_{13}/C_{33}$ is proportional to the Poisson ratio for the material~\cite{HarrisonQWWD}. The $C_{ij}$ values are the elastic constants of the well material, which may be estimated by a linear interpolation using V{\'e}gard's law, in a first approximation~\cite{CaSc2011}. 

Given our assumption of pseudomorphic growth of the QW structure, we expect that shear stresses are absent~\cite{HodmasPolInIIINSemi}. Expanding the sum in Eq.~\eqref{eq:zComponentStrain} for the $z$ component of the piezoelectric polarisation vector field, the magnitude of the resulting polarisation in the well in the $z$/$c$-direction is~\cite{AmMa2002}
\begin{align*}
    P_{\text{PZ},3} &= \varepsilon_{1}e_{31} + \varepsilon_{2}e_{32} + \varepsilon_{3}e_{33} \\
    &= 2\varepsilon_{1}e_{31} + \varepsilon_{3}e_{33} \\
    &= 2\varepsilon_{1}\left(e_{31} - e_{33}\frac{C_{13}}{C_{33}}\right) \,\, ,
\end{align*}
where $e_{i\alpha}$ are the piezoelectric coefficients (see Eq.~\eqref{eq:PiezoCoeffs} above). 

\begin{figure}[t!]
\centering
    \includegraphics[width=\columnwidth]{Figures/FigBoundCharges.eps}
    \caption{a) Formation of bound charges on the interfaces of a QW consisting of a material with polarisation \textbf{P}$_{\text{B}}$, embedded in a material of polarisation \textbf{P}$_{\text{A}}$. Negative (positive) charge is represented by red circles with negative symbols (blue circles with positive symbols). b) Resulting potential, $\varphi_{\text{pol}}$, across the well. Position along the growth direction is denoted by z.}
    \label{fig:BoundCharges}
\end{figure}

In order to see the consequences of the presence of this polarisation, we note that in general, in a dielectric material, there will be a charge density (called the bound charge, $\rho_{\text{bound}}$) associated with a polarisation field, $\mathbf{P}$, such that~\cite{HarrisonQWWD,Zangwill,Jackson1999}
\begin{equation}
    \label{eq:ChargeDensityPolField}
    \rho_{\text{bound}}(\mathbf{r}) = -\mathbf{\nabla} \cdot \mathbf{P}(\mathbf{r}) \,\, .
\end{equation}
Thus, if there is no change in $\mathbf{P}(\mathbf{r})$, there will be no bound charge. For $c$-plane InGaN/GaN QWs, there will be a discontinuity in the polarisation field, $\mathbf{P} = (P_{\text{SP}} + P_{\text{PZ}})\mathbf{\hat{z}}$, at the interfaces, which will give rise to bound surface charges, depicted in Fig.~\ref{fig:BoundCharges} a). These bound charges generate an electric field, with an associated potential across the well of a similar form to the potential between two capacitor plates, depicted in Fig.~\ref{fig:BoundCharges} b). It is possible to determine this ``built-in'' potential, $\varphi_{\text{pol}}$, by combining Gauss's law;
\begin{equation*}
    \nabla\cdot\mathbf{D}(\mathbf{r}) = \rho_{\text{total}}(\mathbf{r}) = \rho_{\text{free}}(\mathbf{r}) + \rho_{\text{bound}}(\mathbf{r}) \,\, ,
\end{equation*}
with the constitutive relation between the displacement, $\mathbf{D}$, and electric, $\mathbf{E}$, fields~\cite{Zangwill,Jackson1999};
\begin{equation*}
    \mathbf{D}(\mathbf{r}) = \epsilon(\mathbf{r})\mathbf{E}(\mathbf{r}) \,\, .
\end{equation*}
Assuming there is only the polarisation induced bound charge, i.e., that no free charge exists in the system ($\rho_{\text{free}}(\mathbf{r}) = 0$)~\cite{HarrisonQWWD,Jackson1999}, and bearing in mind the relationship between electric field and potential,
\begin{equation*}
    \mathbf{E}(\mathbf{r}) = -\nabla \varphi_{\text{pol}}(\mathbf{r}) \,\, ,
\end{equation*}
we find that
\begin{equation}
    \label{eq:Poisson}
    -\nabla\left(\epsilon(\mathbf{r})\nabla\varphi_{\text{pol}}(\mathbf{r})\right) = \rho_{\text{bound}}(\mathbf{r}) = -\nabla\cdot\mathbf{P}(\mathbf{r}) \,\, ,
\end{equation}
where $\epsilon(\mathbf{r})$ is the position dependent permittivity of the system. Returning to a 1D model as an illustration, we must solve
\begin{equation*}
    -\frac{d}{dz}\left(\epsilon(z)\frac{d}{dz}\varphi_{\text{pol}}(z)\right) = \rho_{\text{pol}}(z) = -\frac{d}{dz}P(z) \,\, .
\end{equation*}
The resulting potential for a QW embedded in an unstrained barrier material of spontaneous polarisation $P_{\text{SP}}^{\text{B}}$ is~\cite{CaSc2011}
\begin{equation*}
    \varphi_{\text{pol}}(z) = \left(\frac{P_{\text{SP}}^{\text{w}} - P_{\text{SP}}^{\text{B}} + P_{\text{PZ}}^{\text{w}}}{2\epsilon_{0}\epsilon_{r}^{\text{W}}}\right)\left(|z|-|z-h|\right) \,\, .
\end{equation*}
Here, $h$ is the QW width, $\epsilon_{r}^{\text{W}}$ is the relative permittivity in the well, $P_{\text{SP}}^{\text{W}}$ is the spontaneous polarisation in the well and $P_{\text{PZ}}^{\text{W}}$ is the piezoelectric polarisation in the well~\cite{BeFi1999}. This potential will modify the electronic and optical properties of the QW, causing the tilting of bands shown above in Fig.~\ref{fig:BandsFlatTilted}. This effect can be quite large in III-N QWs with electric field strengths on the order of MV/cm reported in InGaN/GaN QWs~\cite{BeFi1999,BeFi1998}.

\begin{figure}[t!]
\centering
    \includegraphics[width=\columnwidth]{Figures/fig2_7ab.eps}
    \caption{Conduction (CBE) and valence band edge (VBE) energies as a function of position along the growth direction, z, through a $c$-plane III-N QW a) without and b) with the presence of the built-in polarisation field. When the polarisation field is present, the carriers are separated along the growth direction. The red-shift of the emission energy is a signature of the quantum confined Stark effect~\cite{RyYo2009}.}
    \label{fig:BandsFlatTilted}
\end{figure}

Figures~\ref{fig:BandsFlatTilted} a) and b) show schematics of the conduction and valence band edge energy profiles along with the electron and hole ground states in a $c$-plane III-N QW a) in the absence of and b) in the presence of the built-in polarisation field. Due to the tilting of the bands the difference in energy between the lowest conduction band state and highest valence band state decreases. As electrons and holes occupy their respective ground states, the energy of photons emitted when electron hole pairs recombine radiatively will reduce, causing a red-shift of the emission~\cite{RyYo2009}. Furthermore, the charge carriers will become separated along the growth direction. In Fig.~\ref{fig:BandsFlatTilted} b), the holes are confined at the left side of the well, and electrons on the right. The spatial separation of charge carriers and associated red-shift of emission wavelength are referred to as the quantum confined Stark effect (QCSE)~\cite{RyYo2009}. This spatial separation may also impact the recombination rates in III-N QWs. Larger separation between carriers leads to smaller interactions, as the wave function overlap will also be reduced. The result of this will be smaller radiative and Auger rates, which, as we will see below in Chapter~\ref{chp:RecombinationRates}, depend on the overlap of carrier wave functions.

In this section we saw that the piezoelectric polarisation is strain dependent, thus large fluctuations in strain can lead to large fluctuations in piezoelectric polarisation field. In a realistic III-N alloy there is a random distribution of cation species throughout the well~\cite{SmKa2003}. Due to the differences in lattice constant/bond length between III-N materials, this will lead to fluctuations in strain even within a given unit cell, when there are different cation species present. In the case of InN and GaN the difference in lattice constant is quite large (approximately 10\%) and so the deformations of the four bonds surrounding a given atom can lead to significant ``local'' fluctuations in strain. Consequently, not only is the macroscopic piezoelectric field (described above in a 1D model) present, but also large 3D local fluctuations in strain and piezoelectric field must be accounted for as they will have a significant impact on the carrier wave functions, potentially enhancing carrier localisation. In fact, experimental studies suggest that InGaN and AlGaN systems exhibit strong carrier localisation effects~\cite{Weisbuch2020Review,FrNi2020LocalisationWellWidth}. 

However, simulating alloy disorder effects has proven to be a unique challenge. In order to fully capture these effects, as discussed in the Prologue, we require an approach that is fully three dimensional, and is ideally atomistic in nature, i.e., an approach that takes into account atom by atom variations in material parameters. In the following section we discuss some general consequences of alloy disorder in III-N semiconductors, particularly InGaN systems. As has been mentioned previously, InGaN systems have received far more attention than AlGaN systems, but recent work suggests similar effects may be present in both.

\subsection{Alloy disorder and carrier localisation in III-nitrides}
\label{sec:IntroAlloysDisorder}

Above we discussed that alloy disorder can lead, for instance, to local fluctuations in strain and polarisation fields~\cite{BaSp2007}. Moreover, given the relatively large difference in the band gaps of, for instance, InN and GaN ($\text{E}_{\text{g}}(\text{InN}) \approx 0.69$ eV compared to $\text{E}_{\text{g}}(\text{GaN}) \approx 3.51$ eV~\cite{WuWa2003}), variations in alloy content will also lead to significant local fluctuations in confining potential and thus regions of lower (In rich) and higher (Ga rich) energy~\cite{WeNa2020}. A similar situation arises for AlGaN-based QWs ($\text{E}_{\text{g}}(\text{AlN}) \approx 6.0$ eV~\cite{TsCh2018}). These local variations in confining potential, coupled with the local variations in polarisation field, can lead to strong carrier localisation effects. As already mentioned above, experimental studies give clear indications of such carrier localisation effects. 

\begin{figure}[t!]
\centering
    \includegraphics[width=\columnwidth]{Figures/fig2_8ab.eps}
    \caption{Conduction (CBE) and valence band edge (VBE) energies as a function of position in the growth plane in a) a perfect semiconductor (homogeneous emission energy) and b) a disordered semiconductor (inhomogeneous emission energies).}
    \label{fig:BandsFlatDisordered}
\end{figure}

A strong indication of carrier localisation is the broadening of low temperature photoluminescence (PL) peaks observed experimentally in InGaN and AlGaN QWs~\cite{GrSo2005}. During PL studies a sample is illuminated by light with energy above the band gap of the well material which leads to photoexcitation of carriers, i.e., electrons are excited from the valence to the conduction band. The carriers then relax to the lowest energy state in the region they were excited in, and then recombine, emitting light. 

If the energy potential across the well was similar to that in Fig.~\ref{fig:BandsFlatDisordered} a) above, at low temperatures, due to the locally flat band edge profile, radiative recombinations would result in emitted photons of a single energy. Such a PL spectrum would then show a single, narrow peak at that energy. For example, ``conventional'' III-V InGaAs/GaAs QWs have been shown to have very narrow peaks, characterised by full width at half maximum (FWHM) values of $<$ 10 meV~\cite{SiBa1985,Devine1988,BeHs1988}. However, alloy disorder can lead to very strong fluctuations in the potential energy landscape, as depicted schematically in Fig.~\ref{fig:BandsFlatDisordered} b). Thus, at low temperatures the carriers will be confined in local ``pockets'', or ``valleys'', and when recombining, will emit photons in a range of energies. For example, typical $c$-plane InGaN/GaN QW FWHM are found to be $\approx$ 20 -- 80 meV~\cite{WatsonParrisThesis}.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{Figures/DOSQWBROAD.eps}
    \caption{Energy dependence of the density of states (DOS) of a QW structure with (red) and without (blue) carrier localisation effects. The ``step-function''-like DOS of an ideal QW structure with two degrees of freedom from fundamental quantum mechanical considerations is given by the blue solid line. In a QW system experiencing carrier localisation effects, the DOS is broadened by ``tail states'' with energies in the band gap, characterised by the exponential DOS tail (red solid line). The Stokes shift is the difference in energy between 1) Absorption and 3) Emission, due to 2) Capture, indicated by the different arrow lengths, see main text for details. After Ref.~\cite{OuYi1998}.}
    \label{fig:DOSQWBROAD}
\end{figure}

Another experimentally observable result of carrier localisation is the relatively large Stokes shift, $\text{E}_{\text{S}}$, which is defined as the difference in PL peak energy and absorption edge energy, in III-N QWs~\cite{Weisbuch2020Review}. The Stokes shift arises from the broadening of the density of states (DOS) in an alloy disordered QW. The DOS in a homogeneous QW system, in contrast, will correspond to that of a system with two degrees of freedom. From fundamental quantum mechanics, it can be shown that the DOS of such a system is ``step-function''-like in character, a schematic of which is shown by the blue line in Fig.~\ref{fig:DOSQWBROAD}. However, the alloy disordered confinement potential in InGaN systems, coupled with local fluctuations in strain and polarisation field, leads to the confinement of states with a range of energies near the ``band'' edge and into the band gap, introducing the exponential ``tail'' represented by the red line in Fig.~\ref{fig:DOSQWBROAD}. 

The presence of tail states due to alloy disorder can contribute to the Stokes shift for the following reasons. Absorption processes are generally dominated by the excitation of carriers into unoccupied states higher into the band (see step 1 in Fig.~\ref{fig:DOSQWBROAD}). In contrast, during PL studies, after absorption, carriers are ``captured'' and relax down in energy to localised band edge states (see step 2 in Fig.~\ref{fig:DOSQWBROAD}). Finally, emission is dominated by carriers recombining from localised band edge states, emitting photons with lower energies than were initially absorbed (see step 3 in Fig.~\ref{fig:DOSQWBROAD}). In a homogeneous QW the there would be no difference between the absorption and emission energies. Although the Stokes shift can also be affected by structural properties such as the built-in field and fluctuations in the width of the well, generally it is found to be significantly larger in InGaN/GaN QWs (up to $\text{E}_{\text{S}} \approx$ 500 -- 800 meV~\cite{WatsonParrisThesis,DaGr2000,LeMo2001}) compared to, for instance, arsenide QWs ($\text{E}_{\text{S}}$ < 30 meV \cite{PoPa1996,DeMe1985,HeDa2000,CoGu1990}) or CdSe/CdS systems ($\text{E}_{\text{S}}$ < 100 meV~\cite{TaKo2003}).

\begin{figure}[t!]
\centering
    \includegraphics[width=\columnwidth]{Figures/fig2_9ab.eps}
    \caption{a) Schematic of the evolution of PL peak energy, $\text{E}_{\text{PL}}$, with temperature, T, in $c$-plane InGaN/GaN QWs, see, for example Ref.~\cite{ChGa1998}. b) Potential energy landscape in a disordered QW at low temperatures. A carrier, represented by a red circle, is localised in a local energy minimum (``pocket''). c) As temperature increases the carrier gains energy and can ``hop'' to lower energy pockets.}
    \label{fig:SShape}
\end{figure}

The description of the locally fluctuating potential landscape has also been applied to the explanation of the temperature dependence of the PL peak energy, which exhibits a characteristic ``S-shape''. A schematic of the evolution of PL peak energy with temperature for a typical $c$-plane InGaN QW is depicted in Fig.~\ref{fig:SShape} a). At low temperatures, as temperature initially increases, the PL peak red-shifts to lower energies, before blue-shifting as temperature continues to increase, and finally begins to red-shift again. An example of this can be seen in the experimental work of Ref.~\cite{ChGa1998}. 

Figures~\ref{fig:SShape} b) and c) present a schematic representation of how carriers distribute in a fluctuating energy landscape at low temperature, and as temperature initially increases. The initial decrease in PL peak energy is attributed to the population of states in regions of higher local energy minima (maxima for holes), depicted in Fig.~\ref{fig:SShape} b). These carriers may ``hop'' between local potential minima and reach lower energy valleys, indicated in Fig.~\ref{fig:SShape} c). As temperature continues to increase the carriers can occupy states with higher energies, up to and above the fluctuations, thus the PL peak will increase. Subsequently, the band gap of the system begins to decrease, which is the so-called Varshni shift~\cite{Varshni1967,Eliseev2003}, hence the PL peak energy also decreases~\cite{TeCo1998}.

A further consequence of alloy disorder is that localised carriers trapped in regions of higher In content are less likely to diffuse through the well and interact with nonradiative (defect-related) recombination centres. Overall, this may lead to an increase in the LED efficiency~\cite{OlBe2010}. Thus, as mentioned in the Prologue, carrier localisation is often assumed to be a key ingredient in the remarkably high efficiencies of InGaN QW-based devices despite their large defect densities~\cite{Weisbuch2020Review,OlBe2010}. However, there is still some uncertainty in the exact origin and length scale of carrier localisation in III-N QWs~\cite{OlBe2010}. For example, on the nanometre scale, the suggestion of non-random indium ``clustering'' was made by Narukawa \emph{et al.}~\cite{NaKa1997}, when they reported In rich regions of the well, which would facilitate hole localisation. However, it was found subsequently that the In rich regions were an artefact of the experimental transmission electron microscopy (TEM) method used to investigate the In distribution, and subsequent studies using atom probe tomography (APT) to avoid similar artefacts found a random distribution of In atoms throughout the well~\cite{OlBe2010,SmKa2003,GaOl2007}. Indeed, theoretical studies suggested that single In-N-In chain are sufficient to localise carriers, particularly holes~\cite{BeMa1999,Wang2001}, with experimental results supporting this~\cite{ChUe2006}. 

\blue{The method widely adopted in the literature for studying the impact of alloy disorder on the electronic structure is to assume randomly distributed Ga and In atoms in a given supercell~\cite{ChLi2010,UsOR2014,SmWa1989,Farkas2021}. This is often done by generating a special quasi-random structure that mimics the correlation function of a random alloy, in the case of density functional theory studies~\cite{WeFe1990}, or considering a large number of different alloy configurations in the case of empirical atomistic models. The benefit of the latter approach is that it allows for the impact of the microstructure on the results to be studied, such as the potential broadening of the transition energies, which can then be compared to experimental data on the broadening of the PL structure~\cite{UsOR2014}. Given the broad PL linewidth of these structures, this indicates that already the local alloy configuration plays an important role for optical properties of InGaN heterostructures. We follow this widely adopted method of assuming a random alloy distribution as input to our atomistic theoretical model. To do so, when distributing cation sites, no statistical preference is given to assigning In atoms to sites that already neighbour In sites. This approach is then consistent with, for instance, atom probe tomography measurements that indicate a random distribution of In atoms in InGaN QWs~\cite{SmKa2003}.}

However, explanations of experimental data generally expected that not only holes, but also electrons are localised, and prevented from diffusing to nonradiative recombination sites~\cite{OlBe2010}. It was expected that electrons would form bound (``excitonic'') states with the holes that were localised on In-N-In chains, and thus be localised in a similar manner~\cite{BeMa1999,Wang2001,ChUe2006}. Excitonic effects occur due to the interaction of the electron and hole via the Coulomb force. The related quasi-particle (exciton) can, for example, have different electronic energy levels. However, Kalliakos \emph{et al.}~\cite{KaZh2002} found that the excitonic model could not explain their experimental data on low temperature PL spectra, and suggested that a so-called ``donor-acceptor'' pair (DAP) model (similar to the picture of dopants above in Sec.~\ref{sec:LEDsIntro}) was more suitable~\cite{KaZh2002}. Morel \emph{et al.} expanded on this suggestion with their pseudo-DAP model, which takes into account the differences between actual dopant atoms and the so-called ``isoelectronic traps'' that arise in regions of higher In content~\cite{MoLe2003}. 

In the pseudo-DAP picture, not only are the electrons and holes spatially separated in the growth direction due to the built-in polarisation field, but the carriers are also separated in the growth plane due to fluctuations in In content and the associated isoelectronic trap. However, this model only explained the low temperature PL spectra in terms of the spatial separation of the carriers, and did not aim to provide insight into the microstructural nature of the ``pseudo-donor'' and ``pseudo-acceptor'' traps, i.e., how they are localised and on what length scale. Thus, although hole localisation on In-N-In chains may explain the origin of pseudo-acceptors, the nature of pseudo-donor-like electron localisation was not clear.

Another suggested cause of electron localisation on the nanometre scale is the presence of approximately monolayer (i.e. one atomic layer) fluctuations in well width (called well width fluctuations (WWFs)), which have been observed experimentally~\cite{GrSo2005,GaOl2008}. Theoretically, WWFs have been shown to impact electron localisation in III-N QWs~\cite{WatsonParrisThesis,WaGo2011,TaCa2016}, and experimentally have been shown to impact carrier localisation in, for example, GaAs-based QWs~\cite{PhSt2003}. However, other explanations have also been put forward, such as electrons becoming trapped on extended defects with energies near the band edges, but not recombining nonradiatively, or the formation of V-shaped pits around threading dislocations during growth. Both of these phenomenon could prevent electrons from reaching nonradiative recombination centres. For further discussion see Ref.~\cite{Weisbuch2020Review}.

In this section we have seen that the localisation of carriers due to alloy disorder can lead to many non-trivial results. However, the ABC model described in Sec.~\ref{sec:LEDsModellingEfficiency} usually neglects these effects. Recently, efforts have been made to gain insight into the impact of carrier localisation on the efficiency of III-N QWs, particularly InGaN QWs. For example, carrier localisation effects have been shown to theoretically increase the radiative and Auger recombination rates in those systems~\cite{AuPe2016,KiSt2015}. However, ideally, a framework that can determine the radiative and Auger recombination rates to gain insight into the IQE would include a number of features. Such a framework would be fully 3D, in order to take growth direction and in-plane localisation into account. It would include the effects of local fluctuations in strain and polarisation field as well as being atomistic in nature to account for the localisation of holes on In-N-In chains. Finally, it would also take into account length scales large enough to capture the more extended nature of localised electrons. With an understanding of the impact of carrier localisation on the radiative and Auger rates in such a framework, we could gain insight into the IQE via the ABC model described above. Developing such a framework to investigate the impact of alloy disorder on the IQE poses a significant modelling challenge, as to calculate the recombination rates we must first determine the electronic structure of the system. Thus, in the next chapter we discuss how to calculate the electronic structure of semiconductors and semiconductor heterostructures, while accounting for alloy disorder.
