%\part{Results}

\chapter{Temperature dependent recombination in InGaN QWs}
\label{chp:TempInGaN}

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

In the preceding chapters we have discussed the ingredients of our theoretical framework that can be used to calculate the radiative and Auger recombination rates in $c$-plane III-N QWs. This framework accounts for effects unique to III-N systems, such as the strong built-in polarisation field, and accurately includes carrier localisation effects. With this framework established we seek to gain insight into the experimentally observed thermal droop in InGaN/GaN QWs~\cite{MeDe2020}. To do so we need to understand the temperature dependence of the recombination processes, thus, our primary aim in this chapter is to investigate the temperature dependence of the radiative and Auger recombination rates in such systems. This will allow us to discuss the competition between these rates, and the impact that Auger recombination may have on the IQE of InGaN/GaN QWs, which are at the heart of modern short wavelength visible LEDs. Before doing so, in a first step, we will discuss the specifics of the QW model systems studied here. In a second step, and before looking at radiative and Auger recombination, we will investigate the impact of alloy disorder on the electronic and optical properties in general. Here we will look, for instance, at the density of states (DOS) and Stokes shift. Finally, we will turn to the temperature dependence of the radiative and Auger recombination rates.

\subsection{QW model systems}

In this and the following chapter discussing InGaN/GaN QW systems, we have targeted structures similar to those grown, and analysed experimentally, in Refs.~\cite{TaDa2020} and~\cite{GrSo2005}. Details of the model systems and experimental information can be found in Ref.~\cite{TaDa2020}, and here we only summarise the main points. The results for electronic and optical properties obtained from our TB model discussed above and in detail in Ref.~\cite{TaDa2020} are in good agreement with low temperature experimental data, such as the photoluminescence (PL) peak positions and full width at half maximum (FWHM) values, on $c$-plane InGaN/GaN QWs with In contents ranging from 5\% up to 25\%~\cite{TaDa2020}. All calculations have been carried out on simulation cells with a size of ($10\times9\times10$) nm$^{3}$, with periodic boundary conditions; the system corresponds to $32\times32\times20$ wurtzite unit cells. For all of these studies the assumption of a random alloy has been made, in agreement with general atom probe tomography results on $c$-plane InGaN/GaN QWs~\cite{RiBo2018}, and as discussed above in more detail in Chapter~\ref{chp:LEDs}. For the study of DOS, absorption, Stokes shift and radiative recombination rate, we have generated 125 microscopic In alloy distributions, in order to investigate the impact of alloy microstructure on the results.

Subsequently, in the study of Auger recombination, due to computational expense we have carried out the analysis for 10 microscopic configurations. In our model we also allow for the ``bleeding'' of In atoms into the GaN barrier. However, we restrict this penetration of In atoms into the barrier material to a disc-like region with a height of 2 monolayers (ML), and a width of approximately 5 nm, following the experimental and theoretical work of Refs.~\cite{WatsonParrisThesis},~\cite{WaGo2011}, and~\cite{ScCa2015}. Given that we assume a random alloy distribution of the In atoms in the QW region, the actual ``shape'' of the structural inhomogeneities (well width fluctuations) vary between different microscopic alloy configurations. For the following study of temperature dependent radiative and Auger recombination rates, along with absorption coefficients, DOS and Stokes shift, we have calculated 100 electron and 100 hole states closest to the respective band edges (i.e. the energetically lowest lying states). \blue{We note that the temperature dependence in the calculations arises from the population of states via Fermi-Dirac statistics described above in Sec.~\ref{sec:RecombinationRadiative}; the TB parameters are kept at their values determined by fitting to DFT band structures which are effectively calculated at zero temperature. Future studies could also target temperature dependent TB parameters. In the calculation of Auger coefficients, a further temperature dependence is introduced via the temperature and carrier density dependent screening length described as above in Sec.~\ref{sec:RecombinationAuger}.}

As discussed in the preceding chapter, in general, the introduction of charge carriers into the QW will tend to screen the polarisation field, however in the following we have targeted a relatively low carrier density in line with the experimental study in Ref.~\cite{NiKa2016}, which has also been investigated theoretically in Ref.~\cite{DiPe2019}. Our calculations have been carried out at a fixed carrier density of $n_{3D}=3.8\times10^{18}$ cm$^{-3}$ (corresponding to a sheet carrier density of $n_{2D}=1.2\times10^{12}$ cm$^{-2}$). For this relatively low carrier density we have performed initial, self-consistent, one-dimensional Schr{\"o}dinger-Poisson calculations which show that the screening of the built-in field (i.e. its reduction) is of secondary importance. This approximation is supported by the experimental and theoretical works of Refs.~\cite{DaYo2020review}, \cite{HaCh2022}, and \cite{MuHa2022}. Furthermore, we have tested several configurations with such a reduced (i.e. screened) field and found that the screening of the intrinsic (spontaneous and piezoelectric) built-in field only affected the ground and excited electron and hole states by less than 24 meV (most often by only $\approx$ 3 -- 5 meV). 

However, the screening of the built-in field may affect states near the band edges (e.g. localised hole states) differently to energetically higher lying states, which may exhibit a more delocalised character (see Ref.~\cite{TaCa2016} for more details on the distribution of localised, ``semi-localised'' and delocalised states). To analyse this situation in more detail, while taking into account the numerical expense of these calculations, we have evaluated the Auger coefficient at a temperature of 300 K for one configuration with and without the built-in field screening. In such a higher temperature calculation not only energetically lower lying states but also excited states will become relevant, which provides insight into the impact of the screening on the energetically higher lying states (both valence and conduction states). This study revealed that the results differed by less than 8\%. Overall, this difference is regarded as not only smaller than the uncertainties in some of the material parameters entering the calculations (e.g. the dielectric constants), but also smaller than variations introduced by the alloy microstructure. Furthermore, the calculations by David \emph{et al.}~\cite{DaYo2019manybody,DaYo2019} show that the Coulomb enhancement of the radiative recombination rate is of secondary importance for a carrier density of $n_{3D}=3\times10^{18}$ cm$^{-3}$. Thus, we directly use the TB wave functions (without Coulomb enhancement or built-in field screening effects) when calculating the $C$ and $B$ coefficients.

\section{Results}
\label{sec:TempInGaNResults}

Before discussing the temperature dependence of the radiative and Auger recombination rates in InGaN/GaN QWs, we first discuss the impact of carrier localisation on general features which allows us to connect our theory results to experimentally measurable quantities in these systems. In particular, we look at the density of hole states (h-DOS), as this provides insight into the energy range over which localisation effects are important. In a second step we discuss the broadening of the absorption edge, also due to the presence of localised states, and the Stokes shift, which is the difference in PL peak energy and absorption edge energy.

\subsection{Impact of localisation on optical properties}

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{DOS_25pcIn_deltaE30meV_verF.eps}
    \caption{Hole density of states (h-DOS) for an In$_{0.25}$Ga$_{0.75}$N/GaN well. Black circles: tight-binding data; green solid line: fit to the data based on an Urbach-type model (see main text for details). Inset: natural logarithm of the data from the main figure, indicating the region used for the fitting of the Urbach-type model. Zero of energy is the unstrained GaN valence band edge.}
    \label{DOS15}
\end{figure}

In Sec.~\ref{sec:LEDsIIINMaterials} we discussed that the presence of localised states in a QW system introduces an exponential tail to the ideally ``step-function''-like DOS (in a homogeneous QW, see Fig.~\ref{fig:DOSQWBROAD}). With this in mind, we now turn to the results from our theoretical framework that accounts for carrier localisation. Figure~\ref{DOS15} shows, as an example, the calculated h-DOS for an InGaN/GaN QW with 25\% In. To calculate the h-DOS, hole state energies from all 125 configurations are collected and sorted, before being counted into energy ``bins'' to find $N(E)$ where $N$ is the number of hole states in the energy bin from $E$ to $E+\Delta E$, where $\Delta E$ is the size of the bin. For the 5\% and 10\% In systems, a bin size of $\Delta E = 20$ meV is used, while for 15\% and 25\% In, the bin size is $\Delta E = 30$ meV. Lastly, we calculate $dN/dE$ to find the DOS. The TB data is denoted by the solid black circles and indicates the departure from an ideally "step-function"-like DOS of a homogeneous QW; the presence of tail states is observed for energies larger than approximately 750 meV. In all calculations, the zero of energy is the valence band edge of unstrained GaN. Random alloy fluctuations and the resulting fluctuations in strain and (macroscopic and local) built-in potentials lead to the situation that the valence ``band'' edge varies with position across the QW structure; the h-DOS then includes a contribution from all these factors. For our analysis primarily the evolution of the h-DOS (shape of the curve) with energy is important. An Urbach-like exponential decay of the h-DOS of the form
\begin{equation}
    \label{eq:hDOSExp}
    \text{h-DOS}(E) = A\exp{\left(\frac{E_{0}-E}{E_{\text{tail}}}\right)} \,\, ,
\end{equation}
which is assumed in many studies~\cite{WiTo2008}, is confirmed through the plot of the natural logarithm of the data, as shown in the inset of Fig.~\ref{DOS15}. In Eq.~\eqref{eq:hDOSExp}, $A$, $E_{\text{tail}}$ and $E_{0}$ are free fitting parameters. The TB data is again given by solid black circles and the green solid line is a fit to the region in which the natural logarithm of the TB data changes linearly with energy (above 750 meV). From the slope of the solid green line we extract the characteristic tail energy, $E_\text{tail}$, from
\begin{equation*}
    \ln{\left(\text{h-DOS}(E)\right)} = \left(\frac{1}{E_{\text{tail}}}\right)(E_{0}-E) + \ln\left(A\right) \,\, .
\end{equation*}
Using this information, the solid green line in the main part of Fig.~\ref{DOS15} yields a very good description of the h-DOS calculated from our atomistic model. The same analysis has been carried out for all considered In contents (5\%, 10\%, 15\%, 25\% In) and provides an equally good description (not shown).

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{EUrbachVpcIn_quadfit_verF.eps}
    \caption{Tail state energies, $E_\text{tail}$, as a function of the In content, $x$, in In$_{x}$Ga$_{1-x}$N/GaN quantum wells. The black diamonds are the tight-binding data and  the black solid line gives a fit, $E^\text{f}_\text{tail}=ax-bx^2$, with the coefficients $a$ and $b$ given in figure.}
    \label{E_tailVsindiumcontent}
\end{figure}

Figure~\ref{E_tailVsindiumcontent} displays the extracted tail state energies, $E_\text{tail}$, as a function of the In content, $x$, and reveals an increase in $E_\text{tail}$ with increasing $x$. For the 25\% In case, $E_\text{tail}$ reaches a value of over 30 meV. However, even for the low 5\% In content well, a tail state energy of around 10 meV is found. We also note that the rate of increase in $E_\text{tail}$ in the 5\% to 15\% In content range is larger than that in the 15\% to 25\% In content range, despite the In content difference being equal ($\Delta x$ = 0.1). In order to estimate tail state energies for other In contents we use $E^\text{f}_\text{tail}(x)=ax-bx^2$ to fit the TB results (solid black line), which gives a good description of the data.

While this analysis provides insight into the broadening of the h-DOS, it cannot be directly measured or validated against experimental data. However, the presence of tail states in the DOS leads to a broadening of the absorption spectrum, which is widely observed in experiment, and characterised by the so-called Urbach energy (which is distinct from the tail energy $E_{\text{tail}}$ above)~\cite{DaGr2000,PiLi2017LLT2,ODoMa1999}. Thus, building on our calculations of the DOS, we evaluate the absorption spectra of our InGaN/GaN QWs via $\alpha'(h\nu)\propto \sum_{i,j} |d^{e,h}_{ij}|^{2} \Theta \left( h\nu - E_{e,i} - E_{h,j}\right)$~\cite{WatsonParrisThesis}, where $E_{e,i}$ ($E_{h,j}$) is the electron (hole) eigenenergy of state $i$ ($j$) and \mbox{$d^{e,h}_{ij}=e\langle\psi^e_i|\mathbf{a}\cdot\mathbf{r}|\psi^{h}_j\rangle$} are the dipole matrix elements; $e$ is the elementary charge, $\mathbf{r}$ is the position operator, and $\mathbf{a}=1/\sqrt{3}(1,1,1)$ is the light polarisation vector. As discussed in Sec.~\ref{sec:RecombinationRadiative}, the dipole matrix elements have been directly calculated employing the TB wave functions~\cite{ScSc2006}. The absorption spectrum, $\alpha'(h\nu)$, is evaluated for each of the 125 different microscopic configurations per In content (5\%, 10\%, 15\%, 25\%) using 100 electron and 100 hole TB states.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{absorb_Es_inset_vFb.eps}
    \caption{Calculated absorption spectra for In$_{x}$Ga$_{1-x}$N/GaN quantum wells with 5\% (violet line), 10\% (blue line), 15\% (green line) and 25\% (red line) In content. Inset: Stokes-shift energies, $E_{S}$, as a function of the PL peak energies, $E_\text{PL}$.}
    \label{absorptionedges}
\end{figure}

Figure~\ref{absorptionedges} displays the final spectra, which are obtained by averaging over the 125 configurations per In content $x$. These spectra reflect the experimentally well known broadening, which is often described by a sigmoidal function~\cite{DaGr2000,ODoMa1999}: 
\begin{equation}
    \label{eq:AbsorbSig}
    \alpha'(h\nu) = \frac{\alpha_{0}}{1+\exp{\left(\frac{h\nu-E_\text{a}}{E_\text{u}}\right)}} \,\, .
\end{equation}
In Eq.~\eqref{eq:AbsorbSig}, $E_\text{u}$ is the characteristic, or Urbach, tail energy, and $E_\text{a}$ is the absorption edge energy. Using the above expression to fit the TB data displayed in Fig.~\ref{absorptionedges} enables the extraction of $E_\text{u}$ and $E_\text{a}$ as functions of the In content, $x$. The obtained $E_\text{u}$ and $E_\text{a}$ values are summarised in Table~\ref{tab:stokes_table}. $E_\text{u}$ values on the order of 90 - 100 meV are found, which do not show much variation as In content, $x$, increases. The extracted absorption edge energies, $E_{a}$, decrease with increasing $x$, given that InN has a lower band gap than GaN. Equipped with this information and having previously calculated (low T) PL peak energies, $E_\text{PL}$~\cite{TaDa2020}, which are summarised in Table~\ref{tab:stokes_table}, we can calculate the Stokes shift, $E_{S}$, as a function of In content, $x$, via $E_{S}(x) = E_{\text{a}}(x) - E_{\text{PL}}(x)$~\cite{WatsonParrisThesis}; these results are presented in Table~\ref{tab:stokes_table}. In the experimental study of Ref.~\cite{DaGr2000}, a Stokes shift energy of $E_{S}$ = 830 meV is reported for a well with a PL peak energy of $E_{\text{PL}}$ = 1.96 eV. The In$_{0.25}$Ga$_{0.75}$N QW studied here has a PL peak energy of $E_{\text{PL}}$ = 1.99 eV and a Stokes shift of $E_{S}$ = 850 meV, thus, is in good agreement with the experiment~\cite{DaGr2000}. The experimentally observed linear increase of $E_{S}$ with decreasing $E_{\text{PL}}$ is reflected in our data (cf. inset of Fig.~\ref{absorptionedges}).

\begin{table}
    \centering
    \begin{tabular}{|c|c|c|c|c|}
        \hline In content & $E_{a}$ (eV) & $E_{\text{u}}$ (meV)& $E_\text{PL}$ (eV) & $E_{S}$ (meV) \\
        \hline 5 \% & 3.72 & 95.5 & 3.23 & 490 \\
        \hline 10 \% & 3.54 & 103.6 &  2.96 & 580 \\
        \hline 15 \% & 3.31 & 98.3 & 2.62 & 690 \\
        \hline 25 \% & 2.84 & 94.2 & 1.99 & 850 \\
        \hline
    \end{tabular}
    \caption{Absorption edge energy ($E_\text{a}$), low temperature photoluminescence peak energy ($E_\text{PL}$), Urbach tail energy ($E_\text{u}$), and Stokes shift ($E_{S}$) for In$_{x}$Ga$_{1-x}$N/GaN QWs with varying In contents extracted from a sigmoidal model. $E_\text{PL}$ have been taken from
    Ref.~\cite{TaDa2020}.}
    \label{tab:stokes_table}
\end{table}

\subsection{Radiative recombination}

Having discussed the h-DOS, absorption spectra, and Stokes shifts of InGaN/GaN QWs for a wide In content range, and thus emission wavelength range, we now turn to the discussion of the temperature dependence of the radiative recombination rate. Figure~\ref{BversusT_lowest} displays our calculated $B_{2D}$(T) for QW systems with 10\%, 15\%, and 25\% In, averaged over 125 microscopic configurations, along with the experimental data from Ref.~\cite{NiKa2016}. Several factors are important to note when looking at the experimental data in comparison to our theoretical results. Firstly, it should be noted that in the experimental study of Ref.~\cite{NiKa2016}, $B_{2D}$ values are extracted by fitting $A$, $B$ and $C$ coefficients simultaneously. Thus, different combinations of $A$ and $C$ values may affect the extracted $B_{2D}$ values. On the theoretical side we have not targeted the specific QW structures analysed in Ref.~\cite{NiKa2016} in terms of In content or well width. Therefore, a one-to-one comparison is not possible and not targeted. However, in general the comparison between our theoretical results and the experimental data from Ref.~\cite{NiKa2016} provides a good initial reference frame to investigate \emph{trends} in the evolution of $B_{2D}$ with increasing temperature T.

\begin{figure}
\centering
    \includegraphics[width=0.75\columnwidth]{B2d_w_exp_vF.eps}
    \caption{Radiative recombination rate $B_{2D}$ in In$_{x}$Ga$_{1-x}$N/GaN wells as a function of temperature and alloy composition at a carrier density $n = p = 1.2\times10^{12}$ cm$^{-2}$. 10\% In (purple filled circles), 15\% In (blue filled triangles), 25\% In (orange filled squares), and experimental data from Ref.~\cite{NiKa2016} (black open squares). The data are averaged over 125 microscopic configurations.}
    \label{BversusT_lowest}
\end{figure}

Overall, we find the expected trend that for a given temperature, $B_{2D}$ decreases with increasing In content, $x$. This stems from two factors. Firstly, as our assumed QW model systems were targeting the InGaN QW systems studied in Refs.~\cite{TaDa2020} and~\cite{GrSo2005}, the well width increases slightly with increasing $x$. Secondly, a higher In content leads to a stronger piezoelectric field, since this quantity is strain dependent, as discussed in Sec.~\ref{sec:LEDsIIINMaterials}. The combination of these two factors leads to a stronger quantum confined Stark effect, and thus a stronger spatial separation of electron and hole wave functions, which reduces the spontaneous emission rate and thus $B_{2D}$.

Furthermore, Fig.~\ref{BversusT_lowest} shows that for all three In contents, $B_{2D}$ (slightly) increases with increasing T. The extremely good agreement between theory and experiment should be regarded as a coincidence, following our discussion of, for instance, the structural properties above. Nevertheless, given that independent of the In content considered in the calculations the same trend is reflected, our theoretical model captures the main effects observed in the experiment. This result is achieved \emph{without} assuming In atom clustering in the QW and so is in contrast to Ref.~\cite{DiPe2019}, where In atom clustering is \emph{essential} to model an increasing $B_{2D}$ with increasing T; in the random alloy case, the data in Ref.~\cite{DiPe2019} shows a \emph{decrease} in $B$ with increasing T. Our presented results are fully consistent with the experimental observation of a random In atom distribution in InGaN/GaN QWs while still predicting the experimentally observed trend of increasing $B_{2D}$ with T.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{eres_dipmatels_vF.eps}
    \caption{Energy resolved dipole matrix elements, $|d^{e,h}_{ij}|^{2}$, for an In$_{0.1}$Ga$_{0.9}$N/GaN well. Energies are given with respect to the ``band'' edges. More details are given in the main text.}
    \label{fig:eresdipole}
\end{figure}

The cause of the above discussed trend in $B_{2D}$ with T can be traced back to the changes in the magnitude of the dipole matrix elements between electron and hole states when populating energy levels further away from the ``band'' edges. Figure~\ref{fig:eresdipole} displays $|d^{e,h}_{ij}|^2$ as a function of electron and hole state energies relative to the conduction and valence ``band'' edges; the In content in the well is 10\%. The results from the 125 different configurations are collated and grouped into energy bins of width 40 meV and averaged over the number of elements in each bin. The bottom left corner of Fig.~\ref{fig:eresdipole} corresponds to dipole matrix elements between states near the ``band'' edges, that are strongly localised states and which contribute mainly at low temperatures and carrier densities. Energetically higher lying states, exhibiting larger dipole matrix elements, are populated as T increases thus resulting in the observed increase of $B_{2D}$ with T in Fig.~\ref{BversusT_lowest}.

The above analysis shows that the unusual temperature dependence of $B_{2D}$ stems from carrier localisation, and thus indicates that an accurate description of these effects is key. The description of alloy fluctuations and connected carrier localisation effects may noticeably vary between different models, even within the same theoretical framework. In TB models, one factor is for instance how local built-in potential fluctuations are treated. TB models which include or exclude local field fluctuations may predict different carrier localisation characteristics; these differences will be further compounded by differences in TB parameters~\cite{TaDa2020}. Similar but also different aspects come into play in modified continuum-based models, in which, for example, the local In content is obtained by an averaging procedure~\cite{JoTe2017}. This leads to a coarser spatial resolution in comparison to an atomistic model and may lead to different spatial localisation characteristics and deviations in energies of the electron and hole states when compared to an atomistic model.

We also note that, for instance, the observed slight increase in $B_{2D}$ with temperature may be dependent on the carrier density. With increasing carrier density the localised states may then be occupied so that the $B_{2D}$ could become temperature independent. Furthermore, given that in the experiment $B_{2D}$ is extracted by fitting $A$ and $C$ coefficients simultaneously, further investigations of, for instance, how $C$ changes with temperature and carrier density are important. To answer this question from a theory perspective we discuss in the next section the temperature dependence of the $C$ coefficient. Equipped with information on the temperature dependence of $B$ and $C$ we then target the competition between these processes to gain insight into the thermal droop.

\subsection{Auger recombination}

As discussed above in Sec.~\ref{sec:RecombinationAuger}, the theoretical framework above allows us to investigate the \emph{alloy-enhanced} electron-electron-hole, $C_{eeh}$, and hole-hole-electron, $C_{hhe}$, Auger coefficients; the total Auger coefficient, $C_{\text{tot}}$, is then obtained via $C_{\text{tot}}=C_{eeh}+C_{hhe}$. Phonon-assisted Auger processes are not considered since the DFT calculations in Ref.~\cite{KiSt2015} show that (i) Auger processes can be significantly enhanced by alloy fluctuations and that (ii) in the band gap range of approximately 1.9 - 3.0 eV, which corresponds to the ground state transition energies of the QW structures studied here, contributions from phonon-assisted Auger processes are smaller compared to the alloy-enhanced scattering effects. Thus, it can be expected that alloy-enhanced Auger recombination is the dominant contribution in the InGaN-based QWs investigated here. As mentioned above, due to the large computational expense in calculating Auger coefficients (compared to, for instance, radiative recombination coefficients), we target now 10 randomly selected microscopic alloy configurations, in contrast to the 125 investigated for the calculation of the $B$ coefficient above.

As already mentioned above, we are interested in the temperature dependent competition between the radiative and Auger recombination processes, thus we consider a relatively low carrier density of $n_{3D}=3.8\times10^{18}$ cm$^{-3}$. As such it is justified to take the TB wave functions directly as input for the Auger recombination rate calculation framework detailed in Chapter~\ref{chp:RecombinationRates}. However, we note that while the wave functions are directly taken from the TB model, a carrier density dependent Debye screening length for the Coulomb interaction between the carriers has been taken into account (see Eq.~\eqref{eq:CME} in Sec.~\ref{sec:RecombinationAuger} above). Moreover, previous studies indicate that a carrier density of $n_{3D}=3.8\times10^{18}$ cm$^{-3}$ is not large enough to saturate the localised hole states in InGaN-based QWs with 10\%, 15\%, and 25\% In~\cite{ChSc2018}; electrons are far less strongly affected by alloy fluctuations and exhibit a more delocalised wave function character~\cite{TaCa2016}. Therefore, one can still expect that carrier localisation effects will impact the Auger recombination processes in the systems studied here.

\begin{figure}[t!]
\centering
    \begin{tabular}{c}
        \includegraphics[width=0.59\columnwidth]{Figures/CeehVT_10_15_25.eps} \\
        \includegraphics[width=0.59\columnwidth]{Figures/ChheVT_10_15_25.eps} \\
        \includegraphics[width=0.59\columnwidth]{Figures/PercentCtotVT_10_15_25.eps}
    \end{tabular}
    \caption{a) Electron-electron-hole, $C_{eeh}$, and b) hole-hole-electron, $C_{hhe}$, Auger coefficients as a function of temperature for $c$-plane InGaN/GaN quantum wells with different In contents (10\%, 15\% and 25\%). The data, averaged over 10 microscopic alloy configurations, are given by the filled symbols; error bars indicate maximum and minimum values of $C_{eeh}$ at a given temperature. The carrier density is fixed at $n_{3D}=3.8\times10^{18}$ cm$^{-3}$. c) Percentage break-down of the contribution from $C_{eeh}$ and $C_{hhe}$ to the total Auger coefficient, $C_{\text{tot}}$.}
    \label{fig:CEEHCHHEPERCENTVT}
\end{figure}

Equipped with all this information, Figs.~\ref{fig:CEEHCHHEPERCENTVT} a) and~\ref{fig:CEEHCHHEPERCENTVT} b) display the evolution of the alloy-enhanced $C_{eeh}$ and $C_{hhe}$ Auger coefficients, respectively, as a function of temperature, T, for the different In contents considered; as already highlighted above, the carrier density is fixed at $n_{3D}=3.8\times10^{18}$ cm$^{-3}$. The data are averaged over the 10 different microscopic configurations per In content; error bars indicate the \emph{maximum/minimum} value of $C_{\alpha}$ ($\alpha = \{eeh,hhe\}$) at a given T. The reason for using the min/max values is to highlight that the magnitude of the predicted coefficients may strongly depend on the alloy microstructure. Therefore, using just a single alloy configuration to evaluate Auger coefficients may over- or underestimate these coefficients. Overall, we observe that the $C_{eeh}$ coefficients increase with increasing T, Fig.~\ref{fig:CEEHCHHEPERCENTVT} a), while the $C_{hhe}$ coefficients show a weak T dependence or (slight) decrease, Fig.~\ref{fig:CEEHCHHEPERCENTVT} b). Furthermore, Figs.~\ref{fig:CEEHCHHEPERCENTVT} a) and b) reveal that, independent of temperature and In content, $C_{hhe}$ is greater than $C_{eeh}$ (note the different scales in Figs.~\ref{fig:CEEHCHHEPERCENTVT} a) and b)). 

To gain further insight into the relative contributions from $C_{eeh}$ and $C_{hhe}$ to $C_{\text{tot}}$, a detailed break-down is presented in Fig.~\ref{fig:CEEHCHHEPERCENTVT} c). As already discussed above, in general, the contribution from $C_{eeh}$ to $C_{\text{tot}}$ is smaller relative to the contribution from $C_{hhe}$. For all considered In contents, $C_{hhe}$ makes up $\gtrapprox 99$\% of $C_{\text{tot}}$, independent of T. This finding is consistent with previous \emph{theoretical} studies~\cite{KiSt2015,JoTe2017}. There are very few \emph{experimental} studies that aim to determine the relative contributions from $C_{eeh}$ and $C_{hhe}$ to $C_{\text{tot}}$ however the studies of Ref.~\cite{NiBi2016} and~\cite{GaLu2013} suggest that $C_{eeh}$ should be larger than $C_{hhe}$ by a factor of 1 -- 12. However, the approach used in both experimental studies does not take into account the carrier density dependence of the Auger coefficients.

Having discussed the relative contributions from $C_{eeh}$ and $C_{hhe}$ to the total Auger coefficient, $C_{tot}$, in the next step we focus on the interplay of the alloy microstructure, temperature and In content. Starting with $C_{hhe}$, Fig.~\ref{fig:CEEHCHHEPERCENTVT} b) shows that the Auger coefficient decreases with increasing In content as one may expect due to the increasing (piezoelectric) built-in field. A stronger built-in field will result in stronger spatial separation of the electron and hole wave functions, which thus will affect the Coulomb matrix elements determining the Auger rate. We note, however, that there is a large ``spread'' in the Auger coefficients due to the alloy microstructure; we will come back to this aspect below. Turning to $C_{eeh}$, Fig.~\ref{fig:CEEHCHHEPERCENTVT} a), we find a similar trend: with increasing In content $C_{eeh}$ decreases, also due to the increasing built-in field.

We now turn to the impact of the alloy disorder and temperature on the results. The error bars in Figs.~\ref{fig:CEEHCHHEPERCENTVT} a) and b) give insight into the importance of the alloy microstructure for the Auger coefficients. We observe that with increasing T the ``spread'' in both $C_{eeh}$ and $C_{hhe}$ coefficients, in general, decreases. We attribute this to the effect that at low temperatures only states near the conduction and valence ``band'' edges are populated by the carriers. These are strongly localised states~\cite{TaCa2016} and the wave function overlap between them may strongly depend on the alloy microstructure. As the temperature is increased, the carriers start to populate states where the corresponding wave functions are more delocalised compared to the ``band'' edge states. This has three consequences: (i) in general, the wave function overlap between electron and hole wave functions along the growth direction is (slightly) increased, (ii) in contrast, potential contributions from locally strongly overlapping hole wave functions due to carrier localisation effects are reduced  and (iii) for these more delocalised states the alloy fluctuations present a weaker perturbation and the wave function overlap is less dependent on the alloy disorder. A similar effect is seen for the $B$ coefficient above and illustrated by the dipole matrix elements shown in Fig.~\ref{fig:eresdipole}. As shown in Figs.~\ref{fig:CEEHCHHEPERCENTVT} a) and b), consequences (i) and (ii) lead to opposite effects, with $C_{eeh}$ slightly increasing with increasing temperature while $C_{hhe}$ (slightly) decreases with increasing temperature. Furthermore, consequence (iii) results in the situation that the importance of the alloy microstructure is reduced with increasing temperature and thus the ``spread'' in both the $C_{eeh}$ and $C_{hhe}$ coefficients decreases. 

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{Figures/Fig4_7_Corr.eps}%{ChheVT_15pc_10Cfs_fullAlpha.eps}
    \caption{Hole-hole-electron Auger coefficient, $C_{hhe}$, as a function of temperature for 10 different microscopic alloy configurations of an In$_{0.15}$Ga$_{0.85}$N/GaN quantum well at a fixed, low carrier density of $n=3.8\times10^{18}$ cm$^{-3}$.}
    \label{fig:CHHEVT15pcBreakdown}
\end{figure}

However, for the 15\% In QWs we find that even at higher temperatures there is still a noticeable spread in the Auger coefficient values. As previously shown, for instance in Ref.~\cite{TaCa2016}, the interplay of the built-in field, structural inhomogeneities and the alloy microstructure can affect the wave function localisation. When breaking the 15\% In QW average $C_{hhe}$ coefficient down into the contributions from individual configurations, we find 8 configurations which exhibit very similar $C_{hhe}$ values while two configurations result in much higher values. Figure~\ref{fig:CHHEVT15pcBreakdown} summarises data on $C_{hhe}$ as a function of temperature for the QW system with 15\% In studied here. 

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{Figures/Fig4_8_Corr.eps}%{ChheVHoleGroundStateEnergy.eps}
    \caption{\blue{Auger coefficient, $C_{hhe}$, as a function of the valence ``band'' edge energy (hole ground state energy, GS, filled symbols) and the first excited state (1$^{\text{st}}$, open symbols) for the 10 different alloy configurations of the 15\% In quantum well system.}} 
    \label{fig:VBedge}
\end{figure}

\blue{To gain further insight into how the Auger coefficient value, $C_{hhe}$, is related to the electronic structure, Fig.~\ref{fig:VBedge} shows the $C_{hhe}$ coefficients for the different alloy configurations with respect to their valence ``band'' edge, i.e., the lowest hole ground state (``GS'', filled symbols) and first excited hole state (``$1^{\text{st}}$'', open symbols) energy at 10 K. As one can see from this plot, the two configurations (Cfs. 4 and 10) with the highest Auger coefficient values have the highest valence ``band'' edge (lowest hole ground state) energy. Such a high valence band energy value is indicative of strong carrier (hole) localisation effects~\cite{TaCa2016}. We also find that the next valence states (first excited hole state), for both Cfs. 4 and 10, have energies larger than the valence ``band'' edge energy of most of the remaining 8 configurations. This indicates that not only the hole ground state/valence ``band'' edge state but also excited states exhibit strong localisation effects. So, even at elevated temperatures, contributions from strongly localised states to the Auger coefficients can arise. Given the elevated nature of the Auger coefficients associated with Configs. 4 and 10, along with the fact that the hole ground state and first excited hole state sit energetically above the effective valence ``band'' edge of most of the other 8 configurations, these strongly localised (hole) states may be connected to recent literature discussions of trap-assisted Auger recombination (TAAR) processes~\cite{DaYo2019TAAR,MyEs2020}.}

\blue{TAAR processes require trap-like (localised) states that sit in the band gap similar to a defect state~\cite{ZhTu2022}. However, the nature of the trap-like states underlying TAAR is not yet clear. A number of possible defect/impurity states exist in InGaN systems such as nitrogen interstitials, unintentionally incorporated Ca or Mg dopants~\cite{HaCa2018,ShWi2017,VaNe2004}. Although defect states have been suggested as significant contributors to TAAR~\cite{ZhTu2022}, the strongly localised states found here also sit in the band gap and are linked to elevated Auger coefficients, thus they may also play a role in TAAR. We stress that 10 configurations are, of course, not sufficient to evaluate the statistical significance of the above observed ``extreme'' configurations. However, if such configurations played an important role in InGaN QWs, experimental studies should give indications about their importance: given the large Auger coefficient these configurations should dominate the nonradiative recombination rates. A comparison with experimental studies could therefore provide insight into this question, which we target in Sec.~\ref{sec:CarrDensInGaNResults}.}
 
\begin{figure}[t!] 
\centering 
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{\Large{15\% In, Configuration 10}} \\  
\hline \rotatebox[origin=lb]{90}{\hspace{0.8in}\huge{Top View}} & \includegraphics[scale=0.4]{Figures/Cf10_top_GS_123_AxesLabels.eps} \\ 
\hline \rotatebox[origin=lb]{90}{\hspace{0.2in}\huge{Side View}} & \includegraphics[scale=0.4]{Figures/Cf10_side_GS_123_AxesLabels.eps} \\ 
\hline 
\end{tabular}
\caption{\blue{Isosurface plots of the charge densities of the electron ground state ($|\psi_{e}^{\text{1}}|^{2}$, red), hole ground state ($|\psi_{h}^{\text{1}}|^{2}$, blue), first ($|\psi_{h}^{\text{2}}|^{2}$, green) and second excited hole state ($|\psi_{h}^{\text{3}}|^{2}$, purple) for microscopic Configuration 10 of a 3 nm wide $c$-plane In$_{0.15}$Ga$_{0.85}$N/GaN QW. The light (dark) isosurfaces correspond to 10\% (40\%) of the respective maximum charge density. The top view is along the growth direction, while the side view is along an axis perpendicular to the growth direction.}}
\label{fig:config10ChargeDensities}
\end{figure}

\blue{Nevertheless, to shed light onto the electronic structure of the outlier/extreme configurations, Fig.~\ref{fig:config10ChargeDensities} shows isosurface plots of the charge densities of the electron ground state ($|\psi_{e}^{\text{1}}|^{2}$, red), hole ground state ($|\psi_{h}^{\text{1}}|^{2}$, blue), first ($|\psi_{h}^{\text{2}}|^{2}$, green) and second excited hole states ($|\psi_{h}^{\text{3}}|^{2}$, purple) for Cf. 10, as an example. The charge densities of these different states show that indeed all states are localised by the random alloy fluctuations. A further important observation is that the hole ground and first excited state localise in the same spatial position. Therefore, this particular configuration behaves almost like a quantum dot with two bound energy levels. We note as well that the presence of local quantum dots inside InGaN QWs was put forward to explain experimental observations, e.g., the S-shaped temperature dependence of the PL peak energy, which requires localised states~\cite{FlRa2004}. This co-localisation leads to an enhanced hole-hole wave function overlap entering the calculation of the $hhe$ Auger coefficient even at elevated temperatures where excited states are being populated. Such a situation may thus arise naturally in a random alloy context, or may even depend on the growth conditions in experiments. This co-localisation of the ground and first excited hole states is not found in the 8 non-``extreme'' configurations, while for Config. 4 these states are located in close proximity to each other. Hence, the much larger magnitude of the Auger coefficients in Configs. 4 and 10 are driven by the co-localisation of these states.}
 
\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{CtotVT_10_15_25.eps}
    \caption{Total Auger coefficient, $C_{\text{tot}}$, as a function of the temperature, T, for $c$-plane InGaN/GaN quantum wells with different In contents (10\%, 15\% and 25\%). The data, averaged over 10 microscopic configurations, are given by the filled symbols; error bars indicate maximum and minimum values of $C_{tot}$ at a given temperature, across the 10 configurations. The calculations have been carried out at a fixed carrier density of $n_{3D}=3.8\times10^{18}$ cm$^{-3}$.}
    \label{fig:CTOT}
\end{figure}

Having discussed the temperature dependence of, and the impact of the alloy microstructure on, $C_{eeh}$ and $C_{hhe}$ in detail, we now turn to analysing the \emph{total} Auger coefficient, $C_{\text{tot}}=C_{eeh}+C_{hhe}$. Figure~\ref{fig:CTOT} displays $C_{\text{tot}}$ as a function of temperature for the three different In contents considered. In general, $C_{\text{tot}}$ reflects the features of $C_{hhe}$ in that (i) $C_{\text{tot}}$ decreases with increasing In content, independent of the temperature, (ii) the impact of the alloy microstructure and thus the spread in the $C_{\text{tot}}$ values, in general decreases with increasing T and that (iii) the Auger coefficient shows a weak T dependence, or a (slight) decrease with increasing T.

As already highlighted in the Sec.~\ref{sec:ElecStrucSemiEmp}, there have been multiple theoretical studies focusing on Auger coefficients in InGaN/GaN QWs. While these studies come in different flavours (e.g. effective-mass models plus scaling of the bulk Auger coefficients, 8-band $\mathbf{k}\cdot\mathbf{p}$ models, full-zone $\mathbf{k}\cdot\mathbf{p}$ models) they basically all rely on continuum-based electronic structure models. Thus, most of these models neglect alloy fluctuations. Looking at the results from some of these theoretical models/approaches; Bertazzi \emph{et al.}~\cite{BeZh2013} find an upper bound estimate for the Auger coefficient in an In$_{0.25}$Ga$_{0.75}$N/GaN QW of $C_{\text{tot}} \approx 5\times10^{-31}$ cm$^6$ s$^{-1}$ using a full-zone $\mathbf{k}\cdot\mathbf{p}$ model. While this number is similar to our calculated values, the approach in Ref.~\cite{BeZh2013} neglects not only polarisation fields but carrier localisation effects too. The polarisation fields may reduce the Auger coefficients, while alloy fluctuations should increase the Auger coefficients, since $k$-selection rules are removed. This was already highlighted by Jones \emph{et al.}~\cite{JoTe2017} where, by using a 3D effective mass model in conjunction with scaled bulk Auger coefficients, they showed that including alloy fluctuations led to an enhancement of the Auger coefficients compared to systems in which alloy fluctuations are neglected in the simulations. For a $c$-plane In$_{0.25}$Ga$_{0.75}$N/GaN QW, Jones \emph{et al}. report total Auger coefficients of $C_{\text{tot}}\approx7\times10^{-33}$ cm$^{6}$ s$^{-1}$, at 300 K and a carrier density of $n_{3D}=1\times10^{19}$ cm$^{-3}$, when alloy fluctuations are taken into account. This value is smaller than the value reported here, but as we have recently shown, effective mass models can significantly underestimate carrier localisation effects. Thus, if alloy fluctuations lead to an increase in the Auger coefficient, stronger carrier localisation may increase the values reported by Jones \emph{et al.} significantly.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{CtotVEgap_10_15_25.eps}
    \caption{Comparison of calculated and measured Auger coefficients ($C_{\text{tot}}$) as a function of (effective) band gap and emission wavelength. Data from several literature sources are included. Ref.~\textbf{a} = Ref.~\cite{PiprekReview2020}, Ref.~\textbf{b} = Ref.~\cite{GaDr2012}, Ref.~\textbf{c} = Ref.~\cite{DaYo2019scaling}.} 
    \label{fig:LITC}
\end{figure}

A direct comparison of our calculated Auger coefficients with experiment, in terms of the In content evolution is difficult since, for instance, in Refs.~\cite{NiKa2016} and~\cite{ScBi2013} only emission wavelengths, or classifications such as ``green'' or ``blue'' emitting wells, are reported and these different wavelength regimes may be realised by different alloy and well width combinations. However, to gain some first insights into how our calculated values compare with literature data, Fig.~\ref{fig:LITC} compares our calculated total Auger coefficients at a temperature of 300 K with literature data, both from experiment and theory, as a function of the (effective) band gap/emission wavelength. To make this comparison we have used the expression from Ref.~\cite{Eliseev2003},
\begin{eqnarray*}
    E_{g}(T)  = E_{g}(0) - \frac{\alpha T^{2}}{\beta + T} - \frac{\sigma^{2}}{k_{B}T^{2}}\,\,,
\end{eqnarray*} 
to obtain the (effective) band gap for our QW structures at 300 K. The low temperature band gap, $E_{g}(0)$, is found by calculating the TB single-particle electron-hole ground state transition energy averaged over the 10 microscopic configurations considered. The values used for $\alpha$, $\beta$ and $\sigma$ are taken from Ref.~\cite{TaDa2020}. As already seen in other work~\cite{PiprekReview2020}, Fig.~\ref{fig:LITC} indicates the unconventional plateau/increase of $C_{\text{tot}}$ with increasing (effective) band gap in the InGaN systems, when compared to the sharp decline in $C_{\text{tot}}$ for other material systems with increasing band gap; furthermore, our calculated values are within the large spread of literature values. 

\blue{The above analysis cannot give insight into the statistical significance of the extreme configurations, therefore it is important to study, in general, how the results may change if such extreme/outlier configurations are present in an alloy.} When including all 10 microscopic configurations in the averaging procedure (blue filled circle), our calculations indicate that the Auger coefficient increases with increasing band gap. Although the magnitudes are different, a similar dependence of the Auger coefficient on the (effective) band gap value is observed in the experimental studies from Ref.~\cite{DaYo2019scaling}. On the other hand, when excluding the two ``outlier'' configurations in the 15\% In results (averaging over 8 alloy configurations), we find that the Auger coefficient is, in terms of magnitude, very similar to the 25\% In case. As a consequence, our calculated Auger coefficients only vary weakly in the energy range of $\approx 1.9$ eV ($\approx 644$ nm at 25\% In) to $\approx2.6$ eV ($\approx482$ nm at 15\% In); a similar behaviour was found in the experimental studies of, for instance, Schiavon~\emph{et al.}~\cite{ScBi2013}. However, we find that when going to larger effective band gaps (i.e. $\approx2.9$ eV/424 nm at 10\% In) the Auger coefficient increases, while in Ref.~\cite{ScBi2013} the coefficient is largely constant/may only be slightly increasing beyond 2.6 eV. Overall, our analysis highlights that the alloy microstructure (for instance ``trap states'') can play an important role in the magnitude of the Auger coefficient and thus further studies are required to gain insight into this question. Continuing our analysis of the temperature dependence of the recombination rates and to provide further insight into the thermal droop phenomenon in InGaN-based QW systems, we briefly recap the previously studied temperature dependence of the radiative recombination (expressed by the $B$ coefficient) at high temperatures, alongside the nonradiative Auger recombination rate; special emphasis will be placed on the competition between these two rates to gain insight into the thermal droop.

\subsection{Competition between radiative and Auger recombination}

\begin{figure}[t!]
\centering
    \begin{tabular}{c}
        \includegraphics[width=0.63\columnwidth]{Figures/BVhighT_10_15_25.eps} \\
        \includegraphics[width=0.63\columnwidth]{Figures/CtotVhighT_10_15_25.eps} \\
        \includegraphics[width=0.63\columnwidth]{Figures/BCRatioVhighT_10_15_25.eps}
    \end{tabular}
    \caption{a) Radiative coefficient, $B$, b) total nonradiative Auger coefficient, $C_{\text{tot}}$ and c) ratio of these coefficients, $B/C_{\text{tot}}$, as a function of temperature (above 300 K) for 10\%, 15\% and 25\% indium in the well, averaged over 125 configurations. The carrier density is fixed at $n_{3D}=3.8\times10^{18}$ cm$^{-3}$. The $B/C_{\text{tot}}$ ratio is normalised to its value at 300 K.}
    \label{fig:BCBCRATVTHIGH}
\end{figure}

The calculation of $B$ is discussed in more detail above, and in Ref.~\cite{McTa2020}. We highlight two important aspects here. Firstly, the strong decrease in $B$ with increasing In content can be attributed to the increase in the strain-dependent piezoelectric polarisation field. Secondly, at low temperatures only states near the band edge are populated. These states are strongly localised, especially in the case of the holes~\cite{TaCa2016}. With increasing temperature, excited states may be populated, which results in an increase in the overlap of the electron and hole wave functions. This, in turn, can lead to the unusual behaviour that the $B$ coefficient increases with increasing temperature, T; as discussed above and observed in the experimental studies by Nippert \emph{et al.}~\cite{NiKa2016}. Figures~\ref{fig:BCBCRATVTHIGH} a), b) and c) display $B$, $C_{\text{tot}}$ and the $B/C_{\text{tot}}$ ratio as a function of the temperature (above 300 K) for the three different In contents considered, respectively; the $B/C_{\text{tot}}$ ratio has been \emph{normalised} to the respective values at 300 K. We stress that the carrier density is fixed and that we present the \emph{relative} change in $B/C_{\text{tot}}$ to gain insight into the \emph{temperature-dependent} competition between radiative and Auger recombination. As a result, Fig.~\ref{fig:BCBCRATVTHIGH} c) will be identical when plotting the \emph{relative} change of, for instance, $B/(C_{\text{tot}}n_{3D})$. The data show that $B$ increases slightly (Fig.~\ref{fig:BCBCRATVTHIGH} a)) while $C_{\text{tot}}$ varies weakly with T in the 25\% In case, or decreases with increasing temperature in the 10\% and 15\% In cases (Fig.~\ref{fig:BCBCRATVTHIGH} b)), when including outliers. 

Looking at the $B/C_{\text{tot}}$ ratio in Fig.~\ref{fig:BCBCRATVTHIGH} c), we always find values $>1$, noting again that this value is relative to the value at 300 K, which would indicate that the ``efficiency'' of the systems should increase with increasing temperature, even though, overall, the absolute magnitude of the $B/C_{\text{tot}}$ ratio is decreasing as In content increases (further discussion below). We note that the smaller rate of increase of the $B/C_{\text{tot}}$ ratio for the 25\% case is primarily explained by the weaker T dependence of the 25\% total Auger rate, in contrast to the decreasing rates for 10\% and 15\% In, at least when taking all alloy configurations for the 15\% In case into account. Excluding the two extreme configurations of the 15\% In case from the calculations, the situation slightly changes: while the $B/C_{\text{tot}}$ ratio is still larger than 1, the weaker T dependence of $C_{\text{tot}}$ in the 15\% In case leads to a $B/C_{\text{tot}}$ ratio similar to the 25\% In system.

Overall, given our finding of an increase of $B/C_{\text{tot}}$ with temperature above 300 K, one could expect that the likelihood of a photon being produced during a recombination event would increase relative to the likelihood of the carrier undergoing a nonradiative Auger process. Thus, our results indicate that the alloy-enhanced Auger processes studied here should not lead to a deterioration of device efficiency as temperature increases. Therefore, we conclude that the experimentally observed thermal droop~\cite{MeDe2020} is not caused by alloy-enhanced Auger recombination, and other, for instance, defect-assisted nonradiative recombination processes are more likely to be its origin.

Having analysed the impact of temperature on the radiative recombination rate, as well as its impact on the nonradiative Auger recombination rate, and the competition between the two rates, we now briefly turn to investigate the impact of In content on the recombination rates and their competition. We note again that with increasing In content the strain-dependent piezoelectric field increases, which in turn leads to a spatial separation of electron and hole wave functions and will thus modify the electronic structure of the wells. 

\begin{figure}[t!]
\centering
    \begin{tabular}{c}
        \includegraphics[width=0.63\columnwidth]{Figures/BCAndRatioVx_10_15_25_exclOutliers.eps} \\
        \includegraphics[width=0.63\columnwidth]{Figures/BCAndRatioVx_10_15_25_inclOutliers.eps} \\
    \end{tabular}
    \caption{Radiative coefficient, $B$, nonradiative Auger coefficient, $C_{\text{tot}}$, and ratio of these coefficients, $B/C_{\text{tot}}$, as a function of the In content, $x$, when a) \emph{excluding} and b) \emph{including }outlier configurations. The data are shown relative to their respective values at 10\% In. All data are given at a temperature of 300 K and a fixed carrier density of $n_{3D}=3.8\times10^{18}$ cm$^{-3}$.}
    \label{fig:RELBCRATVX}
\end{figure}

Figures~\ref{fig:RELBCRATVX} a) and b) show the change of $B$, $C_{\text{tot}}$ and $B/C_{\text{tot}}$ with increasing In content at a fixed temperature of 300 K and fixed carrier density; the data are shown \emph{relative} to the 10\% In system; for the 15\% In QW system we present the results when \emph{excluding}, Fig.~\ref{fig:RELBCRATVX} a), and \emph{including}, Fig.~\ref{fig:RELBCRATVX} b), the two ``outlier'' alloy configurations. Looking at the results when excluding the outliers first, Fig.~\ref{fig:RELBCRATVX} a), we find that when going from 10\% to 15\% In content, $C_{\text{tot}}$ drops faster than $B$, resulting in an increase of the $B/C_{\text{tot}}$ ratio. When increasing the In content further to 25\%, we find that $B$ continues to drop quickly while $C_{\text{tot}}$ shows only a weak In content dependence, slightly increasing by approximately 10\%. As a consequence, the $B/C_{\text{tot}}$ ratio drops, but this drop can be attributed to the reduction of $B$ when increasing the In content from 15\% to 25\%, and is not due to a significant increase in $C_{\text{tot}}$. 

Turning to our results when including the outliers in the 15\% In system, Fig.~\ref{fig:RELBCRATVX} b), we find that when increasing the In content from 10\% to 15\%, the $B$ coefficient decreases more quickly than the nonradiative Auger coefficient, $C_{\text{tot}}$. However, when increasing the In content from 15\% to 25\%, $B$ and $C_{\text{tot}}$ decrease at a similar rate. Between 15\% and 25\% In, the $B/C_{\text{tot}}$ ratio in Fig.~\ref{fig:RELBCRATVX} b) is approximately flat, or increases slightly, but is essentially independent of composition.

Overall, both investigations indicate that the so-called ``green gap''~\cite{TaDa2020}, i.e., the reduction of IQE in InGaN-based light emitters operating in the green to yellow spectral range, and thus at higher In contents (i.e. $>15$\%), may not be caused by \emph{alloy-enhanced} Auger recombination effects, since Figures~\ref{fig:RELBCRATVX} a) and b) both indicate that it is the reduction of $B$ and not an increase in $C$ that determines the composition dependence of $B/C_{\text{tot}}$. In other words, the alloy-enhanced Auger recombination may not present an intrinsic roadblock to achieving efficient recombination in the green to yellow range. Thus, in addition to the reduction of the radiative recombination rate when increasing the In content, other factors, such as an increasing defect density and the connected nonradiative recombination processes (e.g. \emph{trap-assisted} Auger recombination~\cite{DaYo2019TAAR} or Shockley-Read-Hall recombination~\cite{AlDr2016}) in higher In content samples are more likely to be responsible for the green gap problem. This suggests that by reducing defect densities and thus the connected nonradiative recombination in high In content systems, there may be a way forward to close the green gap. David \emph{et al.}~\cite{DaYo2019TAAR} comes to a similar conclusion and suggests that reducing the (point) defect density could close the green gap.

\section{Conclusion}
\label{sec:TempInGaNConc}

In this chapter we targeted an investigation of thermal droop in $c$-plane InGaN/GaN QWs. To do so, we disentangled the effects of changing temperature and carrier density by carrying out the analysis at a fixed, low carrier density of $n=3.8\times10^{18}$ cm$^{-3}$. We determined the temperature dependence of the radiative and Auger recombination rates for three different In contents in the well; 10\%, 15\% and 25\%. We found that the radiative rate increases with increasing temperature, in line with experimental studies. This unconventional behaviour stands in contrast to the expected decrease of radiative rate with increasing temperature predicted from a fundamental physics perspective. We found total radiative recombination coefficients of $B \approx 3\times10^{-11}$ cm$^{3}$/s (10\% In), $B \approx 5\times10^{-12}$ cm$^{3}$/s (15\% In) and $B \approx 4.5\times10^{-13}$ cm$^{3}$/s (25\% In) at a temperature of 300 K and the carrier density above. These values are averaged over 125 microscopic alloy configurations, used to investigate the impact of the alloy microstructure on the results. We found that at low temperatures, when mainly the strongly localised band edge states are populated, the alloy microstructure has a strong impact on the radiative rate, evidenced by the large variations in $B$ coefficient. As temperature increases, the spread in values decreases, due to the population of excited, less localised or even extended states that are less perturbed by the alloy disordered confining potential.

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

With the temperature dependence of the radiative and Auger rates determined, we were in a position to discuss the thermal droop. Our investigations showed that at temperatures greater than 300 K the radiative recombination rate slightly increases while the total Auger recombination rate (slightly) decreases. As mentioned above, this finding would indicate a positive impact of increasing temperature on device efficiency when only considering the Auger process as the main nonradiative recombination process. However, this stands in contrast to experimental studies which show that the device efficiency decreases with increasing temperature above 300 K. Therefore, we conclude that the alloy-enhanced Auger recombination process is not responsible for the thermal droop observed in $c$-plane InGaN/GaN quantum wells and other factors, such as carrier transport, defects, and defect-assisted Auger recombination, are more likely to explain the experimentally observed droop effect. Having determined the radiative and Auger rates for three different In contents, we also briefly commented on the decrease of efficiency observed in InGaN QW-based emitters when targeting longer wavelength emission (corresponding to higher In contents): the so-called ``green gap''.

We found that, when increasing the In content in the well, the radiative recombination rate drops. However, the analysis of the composition dependence of the Auger rate was complicated by the presence of ``outlier'' configurations with elevated Auger rates. Our subsequent analysis of these configurations, particularly our comparison to experimental data which is presented in detail in the following chapter, strongly suggested that the situation found in these outliers is not representative of InGaN/GaN QWs. Thus, when excluding them from the analysis we found that the Auger rate initially drops from 10\% to 15\% In in the well, but is composition independent when going to 25\% In. Thus, it is the reduction of the radiative recombination rate and not an increase in the Auger rate that determines the composition dependence of the competition between these processes. As a result, we expect that the reduction of IQE in InGaN-based light emitters operating in the green to yellow spectral range, and thus at higher In contents (i.e. $>15$\%), may not be caused by \emph{alloy-enhanced} Auger recombination effects. In other words, the alloy-enhanced Auger recombination may not present an intrinsic roadblock to achieving efficient recombination in the green to yellow range. Thus, in addition to the reduction of the radiative recombination rate when increasing the In content, other factors, such as an increasing defect density and the connected nonradiative recombination processes (e.g. \emph{trap-assisted} Auger recombination or SRH recombination) in higher In content samples are more likely to be responsible for the green gap problem. This suggests that by reducing defect densities and thus the connected nonradiative recombination in high In content systems, there may be a way forward to close the green gap. Having discussed the thermal droop and green gap in InGaN-based QWs, we now turn to an analysis of the carrier density dependent droop.

\chapter{Carrier density dependent recombination in InGaN QWs}
\label{chp:CarrDensInGaN}

The phenomenon of carrier density dependent droop is a significant roadblock to the optimisation of InGaN/GaN QW-based LEDs~\cite{Weisbuch2020Review}. To gain insight into the carrier density dependence of the IQE, as discussed above, we now investigate the radiative and Auger recombination rates as a function of carrier density, and the impact that the screening of the built-in polarisation field has on the electronic structure. The chapter is organised as follows. First we will detail the ingredients of the calculations, before turning to the impact of the polarisation field screening. Then, we will turn to the carrier density dependence of the Auger recombination coefficient, before finally comparing the Auger and radiative recombination rates with experimental IQE data.

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

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{Figures/ChheVT_15pc_3CfsHighlighted.eps}
    \caption{Hole-hole-electron Auger coefficient, $C_{hhe}$, as a function of temperature for 10 different microscopic alloy configurations of an In$_{0.15}$Ga$_{0.85}$N/GaN quantum well at a fixed, low carrier density of $n=3.8\times10^{18}$ cm$^{-3}$ ~\cite{McKi2022}.}
    \label{fig:CHHEVT15pcHighlights}
\end{figure}

In the previous chapter we studied InGaN/GaN QWs with In contents of 10\%, 15\% and 25\%. Here, we now focus on the 15\% In case (which has a well width of 3 nm) as this composition and well width are often found in experimental realisations of InGaN/GaN QWs~\cite{DaYo2019scaling,BaAh2021} and moreover, have been targeted for analysis by colleagues at the Universities of Manchester and Cambridge, which will allow us to compare theory and experiment~\cite{BaMc2023}. More details, e.g., about the simulation cell, can be found above in Chapter~\ref{chp:TempInGaN} and in Ref.~\cite{McKi2022}. 

The theoretical framework and QW model systems are essentially the same as above in Chapter~\ref{chp:TempInGaN}, however now we calculate 150 electron and hole state (in contrast to 100 above) to account for the occupation of states at very high carrier densities and a temperature of 300 K. \blue{We note that in LED devices, which are electrically driven, the operating temperatures can exceed 300 K. However, in the theory experiment comparison discussed below, we deal with an optically pumped QW system. In the experimental studies conducted by our colleagues at the University of Manchester, the temperature of the QW sample was held constant at 300 K. Therefore, we assume for our calculations a temperature of 300 K.} Furthermore, as discussed above in Chapter~\ref{chp:RecombinationRates}, with the introduction of charge carriers into the well, the built-in polarisation field will be screened. Thus, the calculation of 150 electron and hole states must be carried out for each carrier density, while taking the built-in field screening effect into account as outlined in Sec.~\ref{sec:RecombinationScreening}.

The computational expense of Auger rate calculations, given the large number of states and Coulomb matrix elements involved at high carrier densities ($\geq 1\times10^{19}$ cm$^{-3}$), is immense. Thus, we have selected three configurations from the previously studied 10, namely configurations (Cfs.) 2, 4 and 7 (also highlighted in Fig.~\ref{fig:CHHEVT15pcHighlights}). We have selected these configurations since Cfs. 2 and 7 give a very similar Auger coefficient at higher temperatures, similar to 8 of the 10 previously investigated systems. On the other hand, at lower temperatures, the behaviour of Cf. 2 ($C_{hhe}$ decreases with increasing T) and Cf. 7 ($C_{hhe}$ increases with increasing T) are opposites. Finally, we have selected Cf. 4 to study the carrier density dependence of one of the configurations with a significantly larger Auger coefficient, which may be connected to defect-related/trap-assisted Auger processes discussed previously and in the literature~\cite{ZhTu2022,DaYo2019TAAR}. \blue{As mentioned previously, a number of possible defect/impurity states exist in InGaN systems such as nitrogen interstitials and unintentionally incorporated Ca or Mg dopants~\cite{HaCa2018,ShWi2017,VaNe2004}.} A configuration with such a high Auger coefficient may dominate the recombination process in InGaN QWs. However, as we will discuss in more detail below, comparing the results on the carrier density dependence of $C$ with experimental data gives very little indication that this is the case, and suggests that Cfs. 2 and 7 describe the most commonly found situation in InGaN QWs~\cite{BaMc2023}. However, it is still important to investigate how the Auger rate of such an ``extreme'' configuration changes with carrier density. Overall, the three selected configurations furnish a good overview of how the Auger coefficient can change with carrier density in In$_{0.15}$Ga$_{0.85}$N/GaN QW systems.

\section{Results}
\label{sec:CarrDensInGaNResults}

Before turning to the carrier density dependence of the Auger coefficients, and subsequently the competition between the radiative and Auger recombination rates as carrier density increases, we first analyse how the screening of the built-in field affects the electronic structure of the QW. 

\subsection{Impact of polarisation field screening on electronic structure}

\begin{figure*}[t!]
    \centering
    \includegraphics[width=\columnwidth]{P3F2c.png}
    \caption{Isosurface plots of the ground state electron (red) and hole (blue) charge densities for Configuration 2 with a built-in field determined for varying carrier densities in an In$_{0.15}$Ga$_{0.85}$N/GaN quantum well. The light (dark) isosurfaces correspond to 10\% (40\%) of the maximum value of the respective charge density. In the ``Top View'', the black dashed square indicate the limits of the simulation cell and the dimensions of a well width fluctuation (black dashed circle) considered. In the ``Side View'', the black dashed lines give an indication of the quantum well interfaces.}
    \label{fig:CF2ChargeDensities}
\end{figure*}

Figure~\ref{fig:CF2ChargeDensities} depicts the ground-state electron (red) and hole (blue) charge densities for Cf. 2 for varying carrier densities (increasing from left to right). We have selected Cf. 2 here since (i) at low temperatures its Auger coefficient $C_{hhe}$ is very similar to Cf. 4, which gives a significantly larger Auger coefficient over the full temperature range, and (ii) $C_{hhe}$ decreases with increasing temperature for Cf. 2, thus also providing insight into the importance of excited states (further discussion below). Looking first at the ``Top View'' (first row, view along $c$-axis), for lower carrier densities the electron charge density (indicated in red in Fig.~\ref{fig:CF2ChargeDensities}) is mainly localised by the well width fluctuation (WWF) and with increasing carrier density starts to ``delocalise'' in the $c$-plane. However, the alloy microstructure is of secondary importance when compared to holes (charge densities indicated in blue in Fig.~\ref{fig:CF2ChargeDensities}). Here, the hole charge density is strongly localised, and remains so even at higher carrier densities. 

However, looking at the ``Side View'' (second row), while still being strongly localised by the alloy fluctuations, the hole ground state localisation region changes (when comparing charge densities for built-in fields determined at carrier densities of $1\times10^{18}$ cm$^{-3}$ and $5\times10^{19}$ cm$^{-3}$). We attribute this to the following. At low carrier densities, the electrostatic built-in field forces both electron and hole wave functions to be localised at the lower (holes) and upper (electrons) QW barrier interfaces. Thus, only the alloy microstructure at and near these interfaces plays a role for alloy-induced carrier localisation effects. However, when the built-in field is reduced by screening effects, local potential ``pockets'' away from the interfaces now also play a role for carrier localisation effects. This is an important observation as it means that screening of the built-in field not only changes the electron and hole wave function overlap  {(both in-plane and out-of plane)}, but it can be accompanied by significant changes in the electronic structure of the well in general.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.6\columnwidth]{P3F3c.png}
    \caption{Isosurface plots of the five energetically lowest hole charge densities, $|\psi^{1,2,3,4,5}_h|^2$, for Configuration 2 of an In$_{0.15}$Ga$_{0.85}$N/GaN quantum well; the built-in potential has been determined for a carrier density of $n = 5\times10^{18}$ cm$^{-3}$. The light (dark) isosurfaces correspond to 10\% (40\%) of the maximum value of the respective charge density.} 
    \label{fig:CF2HoleStates}
\end{figure}

Moreover, Fig.~\ref{fig:CF2ChargeDensities} also reveals the expected effect that the built-in polarisation field spatially separates the electron and hole wave functions at lower carrier densities along the $c$-axis. With increasing carrier density the built-in field is screened so that the spatial separation between electron and hole wave functions, along the growth direction, is reduced. As a result, the electron and hole wave function overlap is increased. Overall, and based on the method described above, we find that the screening of the built-in field becomes noticeable (i.e. reduction of the electrostatic built-in field by, for example, more than 10\%) for carrier densities exceeding $n=5\times10^{18}$ cm$^{-3}$; this finding is in line with recent literature data~\cite{DaYo2020review,HaCh2022}.

However, with increasing carrier density and/or temperature, not only ground states, but also excited states are important for Auger recombination, especially for holes where one is left with strong alloy-induced carrier localisation effects. Figure~\ref{fig:CF2HoleStates} displays the first five hole states for Cf. 2, with the built-in field determined for a carrier density of $n = 5\times10^{18}$ cm$^{-3}$. Several features of these states are important. Firstly, Fig.~\ref{fig:CF2HoleStates} clearly shows that not only the ground state exhibits strong localisation effects, but also the first few excited states. Secondly, one can see that while the ground state, $\psi^{1}_{h}$, and the first excited state, $\psi^{2}_{h}$, are localised by alloy fluctuations, they localise in different spatial in-plane regions. This result is not captured by a calculation where these localisation features are neglected, e.g., a standard one-dimensional continuum-based description (e.g. a virtual crystal approximation (VCA)) of InGaN/GaN QWs, which is often used in literature to evaluate Auger recombination~\cite{BeGo2010,HaMo2006,TaSu2017}. Thus, Coulomb matrix elements involving wave functions localised in different in-plane regions are strongly reduced in magnitude compared to a VCA description of the same system. However, Fig.~\ref{fig:CF2HoleStates} also reveals that the second excited state, $\psi^{3}_{h}$, is strongly localised and localises in the same spatial position as the ground state, $\psi^{1}_{h}$; this situation could for instance be approximated or described by a ``local'' quantum dot with a ground and an excited state. This also indicates that Coulomb matrix elements involving these states may be strongly enhanced compared to a VCA description. The situation is further modified and complicated by the spatial separation of the third and fourth excited hole states, which have multiple localisation centers, sometimes spatially overlapping with other hole states, and sometimes not.

Overall, the results presented above indicate that while the wave function overlap \emph{along} the $c$-axis increases with increasing carrier density, the wave function overlap between hole states \emph{within} the $c$-plane may strongly depend on the alloy microstructure. This should at least be the case for lower temperatures and lower carrier densities since here mainly the localised hole states near the ``band'' edge are important. As a consequence, and even though with increasing carrier density more states are involved in the Auger rate, Eq.~\eqref{eq:RateAuger}, their contribution may be small due to the spatial in-plane separation of the (hole) states. Furthermore, it is important to note that to evaluate the Auger \emph{coefficient}, the rate is divided by the carrier density cubed. Thus, at lower temperatures, the Auger coefficient may increase or decrease with increasing carrier density, depending on whether $n^{3}$ or the number and magnitude of the involved states/Coulomb matrix elements dominate the Auger rate, Eq.~\eqref{eq:RateAuger}. Finally, the evolution of the Auger coefficient may be further impacted by the fact that the electronic structure changes with increasing carrier density as seen above in Fig.~\ref{fig:CF2ChargeDensities}, where, for instance, the hole ground state at lower carrier densities is different from the hole ground state in the high carrier density regime and the same may be the case for some excited states.

In general, at higher temperatures and/or higher carrier densities, where the localised states are saturated, and one is approaching the situation that the wave functions for electrons and holes are more delocalised, the alloy microstructure is expected to be less important. Thus, more macroscopic effects, e.g., the spatial separation of electrons and holes along the $c$-axis, become important.  

Equipped with this insight and taking these considerations as a whole into account, we present now in a first step the calculated total Auger coefficient, $C_{\text{tot}}$, as a function of the carrier density, $n$. We start our analysis with low temperature \mbox{(T = 10 K)} results, before turning to room temperature data \mbox{(T = 300 K)}. Finally, we investigate in more detail the contribution from electron-electron-hole ($C_{eeh}$) and hole-hole-electron ($C_{hhe}$) processes to the total Auger coefficient, \mbox{$C_{\text{tot}}=C_{eeh}+C_{hhe}$}.

\subsection{Auger recombination}

\begin{figure}[t!]
\centering
    \includegraphics[width=0.7\columnwidth]{P3F4c.png}
    \caption{Total Auger coefficient, $C_{\text{tot}}$, as a function of carrier density, $n$, at a temperature of T = 10 K for three different microscopic alloy configurations (Cfs.) for a $c$-plane In$_{0.15}$Ga$_{0.85}$N/GaN quantum well with a well width of 3 nm.}
    \label{fig:CTOTCarrierDensity10K}
\end{figure}

Figure~\ref{fig:CTOTCarrierDensity10K} displays the total Auger coefficient, \mbox{$C_{\text{tot}}$}, for a temperature of T = 10 K as a function of carrier density, $n$; the data are shown for the three microscopic alloy configurations considered in this study (see discussion above). This figure reveals that for carrier densities below \mbox{$n=1\times10^{19}$ cm$^{-3}$}, the alloy microstructure significantly impacts not only the magnitude of $C_{\text{tot}}$, but also how $C_{\text{tot}}$ evolves with increasing carrier density $n$. At low carrier densities, the behaviour seen already in Fig.~\ref{fig:CHHEVT15pcHighlights}, is here also visible, namely a large spread in the values of $C_{\text{tot}}$. As carrier density in the well increases up to \mbox{$n=1\times10^{19}$ cm$^{-3}$}, the configurations show different behaviours. In the case of Cf. 7, $C_{\text{tot}}$ decreases up to \mbox{$n=1\times10^{18}$ cm$^{-3}$} before starting to increase. Although at low carrier densities Cfs. 2 and 4 exhibit Auger coefficients of similar magnitudes, their carrier density dependence is quite different: $C_{\text{tot}}$ for Cf. 2 decreases when the carrier density increases up to \mbox{$n=1\times10^{19}$ cm$^{-3}$}; $C_{\text{tot}}$ for Cf. 4 has an almost constant (slight decrease) behaviour up to a carrier density of $n=1\times10^{18}$ cm$^{-3}$ and a ``kink'' in the evolution of $C_{\text{tot}}$ at \mbox{$n=5\times10^{18}$ cm$^{-3}$}. Overall, and as discussed above, several factors play an important role (e.g. number of states involved, wave function localisation effects and associated magnitudes of the Coulomb matrix elements, potential changes in the electronic structure, scaling by 1/$n^{3}$) in determining how the Auger coefficient evolves with increasing carrier density, which makes it difficult to deconvolute the impact of these different factors.

However, Fig.~\ref{fig:CTOTCarrierDensity10K} also shows that for carrier densities \mbox{$\geq 1\times10^{19}$ cm$^{-3}$}, and independent of the configuration, the Auger coefficients, $C_{\text{tot}}$, start to increase with increasing carrier density. Moreover, at the highest carrier density considered here, we find that all three alloy configurations give, to a first approximation, very similar Auger coefficients. Overall, we attribute this to the following two effects. Firstly, with increasing carrier density the electrostatic built-in field is significantly screened (at a carrier density of \mbox{$n=5\times10^{19}$ cm$^{-3}$} the built-in field is reduced by a factor of around 2 when compared to a situation that neglects any screening), which increases the electron and hole wave function overlap, as indicated in Fig.~\ref{fig:CF2ChargeDensities}. As a consequence, one may expect the increase observed in $C_{\text{tot}}$. Furthermore, with increasing carrier density the localised states become saturated and for all configurations more delocalised electron and hole states become populated and thus start to contribute. These more delocalised states are less impacted by the alloy microstructure and thus less prone to in-plane carrier separation. All in all, with increasing carrier density the Auger coefficient increases, and differences between the three different alloy configurations are ``smeared'' out, resulting in the observed very similar results for the Auger coefficient at higher carrier densities.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.7\columnwidth]{P3F5c.png}
    \caption{Total Auger recombination coefficient, $C_{\text{tot}}$, as a function of carrier density, $n$, at a temperature of T = 300 K for three different microscopic alloy configurations (Cfs.) for a $c$-plane In$_{0.15}$Ga$_{0.85}$N/GaN quantum well with a well width of 3 nm.}
    \label{fig:CTOTCarrierDensity300K}
\end{figure}

Based on these considerations, one may also expect that at elevated temperatures, for which the carriers are distributed over a larger range of (localised hole) states than at lower temperatures, variations in Auger coefficient for each configuration become less pronounced. This is reflected in Fig.~\ref{fig:CTOTCarrierDensity300K}, which depicts $C_{\text{tot}}$ as a function of the carrier density at a temperature of T = 300 K. For carrier densities \mbox{$\leq1\times10^{18}$ cm$^{-3}$} all configurations show, in general, a (slight) increase with increasing carrier density. Turning to the magnitude of the Auger coefficient in the lower carrier density regime of \mbox{$n\leq1\times10^{18}$ cm$^{-3}$}, we find that Cfs. 4 and 7 are only slightly affected by increasing the temperature from T = 10 K (see Fig.~\ref{fig:CTOTCarrierDensity10K}) to T = 300 K when compared to Cf. 2. For Cf. 2 we observe a noticeable decrease in the magnitude of $C_{\text{tot}}$ when increasing the temperature from T = 10 K (see Fig.~\ref{fig:CTOTCarrierDensity10K}) to T = 300 K. This reflects the results already seen above in Chapter~\ref{chp:TempInGaN} and summarised in Figs.~\ref{fig:CHHEVT15pcBreakdown} and~\ref{fig:CHHEVT15pcHighlights}. 

In the carrier density range of \mbox{$n=1\times10^{18}$ cm$^{-3}$} to \mbox{$n=5\times10^{18}$ cm$^{-3}$}, the Auger coefficient $C_{\text{tot}}$ exhibits only a slight carrier density dependence. Turning to the results when increasing the carrier density beyond \mbox{$n=5\times10^{18}$ cm$^{-3}$}, we observe that for all configurations studied, $C_{\text{tot}}$ starts to increase. Moreover, and similar to the low temperature results, Cfs. 4 and 7 give very similar Auger coefficients at the high carrier density of \mbox{$n=1\times10^{20}$ cm$^{-3}$}, and Cf. 2 is of the same order of magnitude. Overall, we attribute this to similar effects discussed above. Firstly, at these higher carrier densities the localised states are, to a large extent, saturated. Thus, with increasing carrier density more delocalised states, which are less sensitive to the alloy microstructure, contribute to the Auger rate. Secondly, at high carrier densities the intrinsic built-in field is reduced, and in conjunction with the population of the more delocalised states, the electron and hole wave function overlap along the growth direction is increased, which results in larger Coulomb matrix elements and consequently an increase in the Auger rate and thus the coefficient $C_{\text{tot}}$.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{P3F6c.png}
    \caption{Percentage contribution to the total Auger coefficient, $\Delta C_{\alpha} = \left(C_{\alpha}/C_{\text{tot}}\right)$, from the electron-electron-hole (\mbox{$\alpha = eeh$}, represented by triangles) and hole-hole-electron (\mbox{$\alpha = hhe$}, represented by circles) coefficients as a function of carrier density, $n$, at T = 10 K (filled symbols, solid line) and T = 300 K (open symbols, dashed line) for a $c$-plane In$_{0.15}$Ga$_{0.85}$N/GaN quantum well with a well width of 3 nm. The data are averaged over three microscopic configurations.}
    \label{fig:PercentCTOT}
\end{figure}

So far we have only discussed the \emph{total} Auger coefficient, $C_{\text{tot}}=C_{eeh}+C_{hhe}$, but not the individual contributions from the electron-electron-hole, $C_{eeh}$, and hole-hole-electron, $C_{hhe}$, processes. As discussed above in Chapter~\ref{chp:TempInGaN} (see also Ref.~\cite{McKi2022}), at lower carrier densities, the $C_{hhe}$ part dominates the magnitude of the total Auger coefficient, $C_{\text{tot}}$, and $C_{eeh}$ is of secondary importance. This is consistent with the DFT studies in Ref.~\cite{KiSt2015} on InGaN \emph{bulk} systems and also theoretical work on $c$-plane InGaN QWs using a modified continuum-based model and analytic considerations~\cite{JoTe2017,DaYo2019scaling}. However, experimental investigations by Nirschl~\emph{et al.}~\cite{NiBi2016} on InGaN QWs, concluded that the $eeh$ Auger rate is equal to, or larger than, the $hhe$ rate. The work by David \emph{et al.}~\cite{DaYo2019scaling} discusses in general that screening the built-in field increases the relative importance of the $C_{eeh}$ contribution. Jones \emph{et al.}~\cite{JoTe2017}, using a modified continuum-based model to describe the Auger recombination, conclude when studying \emph{non-polar} InGaN QWs, which do not exhibit any macroscopic polarisation field~\cite{ScTa2015}, that indeed screening of the polarisation field may lead to the situation found in the experiments.

Thus, to gain insight into this question, Fig.~\ref{fig:PercentCTOT} presents the (percentage) contribution from the electron-electron-hole, $\Delta C_{eeh}(n)= \left(C_{eeh}(n)/C_{\text{tot}}(n)\right)$, and hole-hole-electron, $\Delta C_{hhe}(n)=\left(C_{hhe}(n)/C_{\text{tot}}(n)\right)$, processes to the total Auger coefficient, $C_{\text{tot}}$, as a function of the carrier density, $n$. The data are shown for low (T = 10 K) and room temperatures (T = 300 K). At low carrier densities (\mbox{$n\leq5\times10^{18}$ cm$^{-3}$}) and independent of the temperature, Fig.~\ref{fig:PercentCTOT} reveals that $C_{hhe}$ dominates the total rate, being over 95\% of $C_{\text{tot}}$. This finding is consistent with our previous results~\cite{McKi2022} and literature theoretical data~\cite{JoTe2017,DaYo2019scaling} on $c$-plane InGaN/GaN QWs. In the higher carrier density regime, i.e., when increasing the carrier density beyond \mbox{$n=5\times10^{18}$ cm$^{-3}$}, $C_{eeh}$ increases slightly, relative to the contribution from $C_{hhe}$; the general trend is thus consistent with expectations in, for example, Ref.~\cite{DaYo2019scaling}. 

At high carrier densities, $C_{hhe}$ still contributes $\geq 90\%$ of $C_{\text{tot}}$ which stands in contrast to the experimental results of Nirschl \emph{et al.}~\cite{NiBi2016}, and the calculations by Jones \emph{et al.}~\cite{JoTe2017} for non-polar InGaN QWs. We note that the electronic structure of a non-polar QW system is different to that of a $c$-plane well~\cite{ScTa2015}. Thus, Auger rates may behave differently in a non-polar well when compared to $c$-plane systems with partially screened built-in fields. Further studies comparing atomistic and modified continuum-based models are required to shed light on the importance of electron-electron-hole Auger processes in polar $c$-plane and non-polar InGaN QWs.

Having calculated the carrier density dependence of the Auger coefficient, the competition with the radiative recombination process as a function of carrier density is essential for understanding the ``droop'' phenomenon. Thus, in our next step, we focus on the carrier density dependence of the competition between the radiative and Auger recombination rates and how our theoretical data compares to experimental data obtained by collaborators at the University of Manchester on samples grown at the University of Cambridge.

\subsection{Competition between radiative and Auger recombination - theory experiment comparison}

With the theoretical data on the carrier density dependence of the radiative and Auger rates determined, we now seek further insight into the role of carrier localisation in causing droop. Furthermore, given the recent literature discussions of defect-assisted Auger recombination, we aim to disentangle the importance of the extrinsic, defect-related process from the intrinsic, alloy-enhanced Auger process. To do so, we have carried out a comparison between theoretical and experimentally determined efficiency droop data in collaboration with colleagues at the University of Manchester, using InGaN/GaN MQW samples grown at the University of Cambridge. 

To investigate the impact of defect-assisted Auger processes the samples were designed to have the same density of extended defects but different densities of point defects. Full details of the comparison and sample growth can be found in Ref.~\cite{BaMc2023}. Here, the main features are summarised. The samples contain five approximately 3 nm wide InGaN QWs embedded in approximately 7 nm wide GaN barriers. The In content in the wells was determined to be 17\% $\pm$ 2\%. Thus, the well width and In content are in line with those targeted in our theoretical framework. The samples emit in the green spectral range. The density of point defects is controlled by varying the growth temperature, which has been shown to inversely correlate with the density of point defects, i.e., higher growth temperatures lead to lower point defect densities~\cite{HaKa2016}. The samples are labelled by the growth temperature T$_{G}$ in degrees Celsius. In order of increasing expected point defect density the samples studied are labelled 716, 706 and 698. 

The samples are simplified structures compared to those employed in actual LEDs, for instance lacking an AlGaN electron blocking layer. However, they consist of 5-period multi-QW structures, in contrast to the single QW structure investigated theoretically. The purpose of this design is so that PL measurements can be taken of carriers excited in the QW regions alone, so-called resonant PL since the excitation energy is below the band gap energy of the GaN barrier. The use of resonant PL is employed to isolate the recombination processes occurring in the QW regions, thus disentangling the IQE from external effects such as transport losses or inhomogeneous carrier distribution across multi-QW structures, and is a technique that has been employed previously~\cite{ShMu2007,BiNi2013}. 

\blue{As discussed above in Chapter~\ref{chp:TempInGaN}, it was found that at 300 K and a carrier density of $n=3.8\times10^{18}$ cm$^{-3}$, for a 15\% In QW, eight of the 10 investigated configurations showed similar values for $C_{\text{tot}}$, while two were classified as outliers. So far in this chapter we have analysed three configurations, two ``standard'' configurations, Cfs. 2 and 7, and one ``outlier'', Cf. 4. The analysis below was carried out including and excluding this outlier, allowing us to identify the potential impact of a much larger Auger rate stemming from the ``outlier'' configuration. Several things are important to note here: given the numerical expense of the calculations, the statistical importance of the outlier configuration cannot be quantified based on the theoretical calculations alone. However, such an extreme/outlier configuration should dominate the nonradiative recombination process in the QW system studied experimentally, given that the outlier configuration exhibits a much larger Auger coefficient. Bearing in mind that there is no fitting involved in our theory experiment comparison, one should see clear indications of such an increased nonradiative process, meaning that the Auger coefficients of the ``standard''/non-outlier configurations would not explain the experimental data. Furthermore, the spot size of the optical beam is much larger than the simulation cell. Therefore, it will sample a large number of ``local'' alloy configurations simultaneously. Again, if such an outlier configuration would be realised, it should dominate the recombination process, as it could act as a ``sink'' for carriers due to the much larger associated Auger rate in a specific spatial region. Finally, to further shed light onto the question of the presence of configurations classified as ``outliers'' in the theoretical model, the measurements have been repeated for different spots on the sample; this accounts for possible experimental bias in which a region on the sample exhibiting a non-extreme Auger rate could be selected for analysis by luck.}

\blue{Building on this, our theory experiment comparison gives very little evidence for the presence of the configurations regarded as outliers in the theoretical model, and the experimental data is best described by the ``standard'' configurations. We note that all this does not rule out entirely the presence of these extreme configurations as it may depend on growth conditions and In content. Indeed, other recent experimental studies suggest that defect-related/trap-assisted Auger recombination only accounts for a small fraction of the total nonradiative recombination rates~\cite{LiDe2023}. Finally, as already discussed above, we cannot classify the statistical significance of the outlier configurations. Based on the 10 configurations studied initially we find 2 outliers out of 10, i.e., 20\%. However, if we would repeat the calculations for a much larger number of configurations, we may still find these configurations but at a much lower percentage and may indeed also theoretically confirm that these extreme configurations are unlikely to appear in a random alloy. In summary, our combined theory experiment comparison indicates that the Auger rate is best described by the ``standard'' configurations and that the outlier configurations are statistically less likely, even though their existence in an alloy cannot be fully ruled out.}

\begin{figure}[t!]
\centering
    \begin{tabular}{c}
        \includegraphics[width=0.75\columnwidth]{Figures/chp6f1a.eps} \\
        \includegraphics[width=0.75\columnwidth]{Figures/chp6f1b.eps}
    \end{tabular}
    \caption{a) \emph{Theoretical} radiative and Auger recombination coefficients as a function of carrier density and b) \emph{theoretical} radiative, Auger and SRH recombination rates (per volume) as a function of carrier density. For both a) and b) the radiative and Auger recombination data are averaged over Cfs. 2 and 7 for an InGaN/GaN QW with 15\% In at T = 300 K. The value of SRH coefficient, $A=2\times10^{6}$ s$^{-1}$, has been estimated experimentally (see text for details) and lies within the range of values reported in the literature.}
    \label{fig:coeffsAndRates}
\end{figure}

Before turning to discuss our theory experiment comparison in more detail, in our theoretical framework we have determined the $B$ and $C$ coefficients, so we begin by presenting the theoretical $B$ and $C$ coefficients (the calculation of which is described above in Chapter~\ref{chp:RecombinationRates}) as functions of carrier density at T = 300 K in Fig.~\ref{fig:coeffsAndRates} a). Both $B$ and $C$ have been averaged over Cfs. 2 and 7. As mentioned above in Sec.~\ref{sec:LEDsModellingEfficiency}, in the simplest formulation of the ABC model, the $A$, $B$ and $C$ coefficients are carrier density independent. Here, we see a significant deviation from this assumption for the $B$ and $C$ coefficients, particularly as carrier density increases above approximately $n=1\times10^{18}$ cm$^{-3}$. This strong dependence of the coefficients on carrier density is largely driven by the built-in polarisation field screening we previously discussed in Sec.~\ref{sec:RecombinationScreening}. Above approximately $n=5\times10^{18}$ cm$^{-3}$ the reduction of the $c$-direction separation of the electron and hole wave functions due to screening effects enhances the wave function overlap, leading to an increase in not only the Auger coefficient, but also the the radiative coefficient. Thus, we expect that both the radiative and the Auger recombination rate will increase as carrier density increases. While we may expect an increase in efficiency when the radiative rate increases due to the screening of the polarisation field, this will be counteracted by an increase in the Auger rate as well.

In order to compare our theoretical results with experimentally determined IQE data, within the ABC model, we require some knowledge of the Shockley-Read-Hall (SRH) $A$ coefficient. We have not calculated this parameter but typical values are reported in the range of $A = 1\times10^{5}$ -- $1\times10^{7}$ s$^{-1}$~\cite{ScBi2013,PiRo2015,NiKa2016TRPL}. Thus, in Fig.~\ref{fig:coeffsAndRates} b), using a typical value of SRH coefficient, $A=2\times10^{6}$ s$^{-1}$, the radiative and Auger recombination rates are presented alongside an illustrative SRH recombination rate. Figure~\ref{fig:coeffsAndRates} b) shows that as the carrier density increases above $n=1\times10^{18}$ cm$^{-3}$ the radiative recombination rate overtakes the SRH recombination rate, while as the carrier density increases past $n=5\times10^{19}$ cm$^{-3}$ the Auger recombination rate exceeds the radiative recombination rate. As a consequence we would expect a reduction in efficiency in this high carrier density regime.

Figure.~\ref{fig:coeffsAndRates} b) also shows that in the high carrier density regime (above approximately $n=1\times10^{19}$ cm$^{-3}$) the SRH recombination rate is already approximately an order of magnitude smaller than both the Auger and radiative recombination rate, and becomes negligible at higher carrier densities. Thus, we expect that the behaviour of the IQE will be dominated by the radiative and Auger recombination rates at high carrier density. The $A$ value used here is often reported in literature and used in theoretical analyses, as mentioned above, thus, it is not, for instance, an extreme value. 

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{Figures/chp6f2.eps}
    \caption{Theoretically determined IQE as a function of carrier density, $n$, for varying values of SRH coefficient, $A$.}
    \label{fig:IQEVaryingSRHCoeff}
\end{figure}

\blue{Although we now expect that at high carrier densities the IQE is determined by $B$ and $C$, the value of $A$ may impact the IQE behaviour in the ``intermediate'' carrier density range (i.e. $n=1\times10^{18}$ -- $1\times10^{19}$ cm$^{-3}$), thus, for instance, changing the onset of the droop regime. To investigate this, Fig.~\ref{fig:IQEVaryingSRHCoeff} shows the IQE as a function of carrier density using our calculated $B$ and $C$ coefficients and a range of $A$ coefficients. Figure~\ref{fig:IQEVaryingSRHCoeff} demonstrates that the impact of $A$ becomes significant below $n=1\times10^{19}$ cm$^{-3}$, and further shows the robustness of the high carrier density behaviour of the IQE against varying values of $A$. Further details on the determination of the $A$ coefficient from experimental data is given in Ref.~\cite{BaMc2023}. Lastly, Fig.~\ref{fig:IQEVaryingSRHCoeff} also demonstrates that already the $B$ and $C$ coefficients calculated in our TB framework predict droop, without considering any defect related Auger processes; a threshold for droop onset between approximately $n=5\times10^{18}$ cm$^{-3}$ and $n=1\times10^{19}$ cm$^{-3}$ is found in our calculations.}

\blue{Of course, it is possible to fit the IQE curve with $A$, $B$ and $C$ as free, adjustable parameters, thus extracting a single, carrier density \emph{independent} value for each coefficient that can describe the IQE curve. However, it has been found that when such a fitting procedure is carried out on a specific experimental sample, the same parameter set may not describe different samples~\cite{WePi2015}. This approach also neglects the fundamental underlying physics of III-N heterostructures, particularly the screening of the built-in field, which also depends on the carrier density. We have also investigated the impact of using constant (i.e. carrier density independent) values for $B$ and $C$ that are evaluated at a fixed carrier density from our model. As an example, using the values of $B$ and $C$ determined at $n=5\times10^{18}$ cm$^{-3}$ shifts the droop onset threshold carrier density from approximately $n=7.5\times10^{18}$ cm$^{3}$ to approximately $n=2\times10^{18}$ cm$^{-3}$, and moreover, reduces the slope of the theoretical IQE in the high carrier density regime by approximately 13\%. Thus, the carrier density dependence of the $B$ and $C$ coefficients is an important component in understanding droop and the role that Auger recombination plays for this effect.}

To be able to compare theory with experiment, a key ingredient is knowledge of the carrier density in the QW. In our theoretical framework the free carrier density is input to the model, which is fixed to the required value. Experimentally, determining the carrier density in the QW is more challenging~\cite{WePi2015,DaYo2020review,BaAl2016}. The PL measurements are taken in a pulsed excitation regime. In this framework a 150 fs pulse of light illuminates the sample and the PL measurement is taken over the course of many of these pulse cycles. The reported measure of PL intensity is then the intensity of emitted light integrated over a single pulse. A pulse cycle begins when the sample is illuminated by light, causing excitation. The initial carrier density, $n_{0}$, can be determined from the absorbed energy density of the pulse as the excitation occurs on a much faster timescale than recombination, i.e., the initial carrier density is not affected by recombination dynamics. The carriers then recombine, which leads to a reduction of the carrier density accompanied by a reduction of the PL intensity.

It is common for the integrated PL intensities to be divided by the initial carrier density, essentially giving a carrier density-averaged PL intensity. As a result, reported IQE($n$) values are averaged over $n$. However, to reliably compare the \emph{fixed} carrier density in the theoretical approach to the variable carrier density in the experimental situation, the IQE at a fixed $n$ must be determined. R. Barrett and D. Binks at the University of Manchester developed an approach to determine IQE($n_{0}$)~\cite{BaMc2023,BarrettThesis}. A brief description of the idea underlying the experimental approach is given below. Starting from the ABC model, the carrier density decay is governed by
\begin{equation}
    \frac{dn}{dt} = -\left(An(t) + B(n(t))[n(t)]^{2} + C(n(t))[n(t)]^{3}\right) \,\, .
    \label{eq:dndt}
\end{equation}
One can assume that the photon emission rate per volume, $\phi(t)$, is essentially the rate of spontaneous radiative recombination,
\begin{equation}
    \label{eq:PhotonEmissionRate}
    \phi(t) = Bn^{2} = B(n(t))[n(t)]^{2} \,\, .
\end{equation}
Building on this, we can find an expression for the emitted photon density, $\Phi$, as it is proportional to the integral of $\phi(t)$ over a pulse. Thus, 
\begin{align}
    \label{eq:PhotonDensity}
    \Phi &= \int_{0}^{\infty} \phi(t) dt \,\, , \nonumber\\
    &= \int_{0}^{\infty} B(n(t))[n(t)]^{2} dt \,\, , \nonumber\\
    &= \int_{n_{0}}^{0} B(n)n^{2} \left(-\frac{dn}{An + B(n)n^{2} + C(n)n^{3}}\right) \,\, , \nonumber\\
    &= \int^{n_{0}}_{0} \left(\frac{B(n)n^{2}}{An + B(n)n^{2} + C(n)n^{3}}\right) dn \,\, , \nonumber\\
    &= \int_{0}^{n_{0}} \text{IQE}(n) dn \,\, ,
\end{align}
where we have used Eq.~\eqref{eq:dndt} to carry out the change of variable. Finally, by taking
\begin{equation}
    \label{eq:IQEn0}
    \frac{d\Phi}{dn_{0}} = \text{IQE}(n_{0}) \,\, ,
\end{equation}
the IQE at $n_{0}$ can be determined given experimental data describing $\Phi$.

\begin{figure}[t!]
\centering
    \begin{tabular}{c}
        \includegraphics[width=0.75\columnwidth]{Figures/chp6f3a.eps} \\
        \includegraphics[width=0.75\columnwidth]{Figures/chp6f3b.eps}
    \end{tabular}
    \caption{a) Experimental IQE($n_{0}$) for all three samples (filled symbols) with the theoretically determined IQE($n_{0}$) (black line) using a value of $A=2\times10^{6}$ s$^{-1}$ for the SRH coefficient. b) Experimental IQE averaged over a pulse for all three samples (filled symbols) along with the theoretically determined IQE averaged over a pulse (black line). For more details see main text. The calculations use a value of $A=2\times10^{6}$ s$^{-1}$ for the SRH coefficient.}
    \label{fig:IQEN0AndAverageIQE}
\end{figure}

Having done so, Fig.~\ref{fig:IQEN0AndAverageIQE} a) shows the experimental IQE($n_{0}$) in filled symbols calculated using Eq.~\eqref{eq:IQEn0} and the theoretically determined IQE($n_{0}$) calculated using the definition of IQE (see Eq.~\eqref{eq:PhotonDensity}) and our calculated $B(n)$ and $C(n)$ values determined above, along with a value of $A=2\times10^{6}$ s$^{-1}$ for the SRH coefficient, as discussed above. Due to the process of numerical differentiation when applying Eq.~\eqref{eq:IQEn0} to the experimental data, there is additional scatter introduced to the data. To combat this, a Savitsky-Golay filter has been used to smooth the data~\cite{PrTe1990}. Overall, Fig.~\ref{fig:IQEN0AndAverageIQE} a) shows a close agreement between theory and experiment in the high carrier density regime, along with the very similar behaviours of the IQE at high carrier densities between different samples, i.e., different defect densities. We will return to discuss the similarity of the behaviour of the IQE in the high carrier density regime below.

Figure~\ref{fig:IQEN0AndAverageIQE} a) shows the IQE at the beginning of a pulse. However, it is also possible to evaluate the average IQE over the course of a pulse. To do so experimentally, the integrated intensity is divided by the initial carrier density, as mentioned above. Theoretically, the average IQE can be determined from Eq.~\eqref{eq:PhotonDensity} and dividing by $n_{0}$. This can then be compared to the reported average experimental PL integrated intensity. Doing so yields the data presented in Fig.~\ref{fig:IQEN0AndAverageIQE} b) and shows again that the theoretically determined $B$ and $C$ coefficients, in conjunction with the experimentally verified $A$ values, reproduce the experimental data closely in the high carrier density regime. It is important that the $B$ and $C$ parameters used here stem from the independent calculations carried out above, and are not free fitting parameters.

Overall, our analysis highlights that the carrier density dependence of the recombination coefficients is key, and experimental studies should not assume carrier density independent coefficients. This comparison also suggests that carrier localisation and polarisation field screening effects play an important role in causing droop in InGaN-based LEDs, primarily due to the impact of these factors on Auger recombination. This is further suggested when we recall that the experimental samples were grown so as to have differing point defect densities. In the high carrier density regime in Figs.~\ref{fig:IQEN0AndAverageIQE} a) and b), there is no significant variation in the experimental data between the three samples. This data is also consistent with theoretical data that does not include defect-assisted processes. Thus, all this highlights that, at least for the samples studied here, defect-assisted Auger processes are likely to be of secondary importance in the droop carrier density regime. 

\section{Conclusion}
\label{sec:CarrDensInGaNConc}

In this chapter we presented an atomistic theoretical analysis of nonradiative Auger recombination processes in $c$-plane InGaN/GaN QWs as a function of carrier density. The model included the screening of the electrostatic built-in field with increasing carrier density. The investigation in Chapter~\ref{chp:TempInGaN} showed that at low temperatures and carrier densities, the Auger coefficients are strongly impacted by the alloy microstructure. However, as temperature and carrier density are increased and the localised states are saturated, the different microscopic alloy configurations exhibit very similar Auger coefficients. Moreover, we found that the contribution from the electron-electron-hole Auger process is of secondary importance when compared to the hole-hole-electron process, and its relative importance increases only slightly with increasing carrier density. We found Auger coefficients that vary from $C_{\text{tot}}\approx1\times10^{-31}$ cm$^{6}$/s at a carrier density of $n=1\times10^{17}$ cm$^{-3}$ to $C_{\text{tot}}\approx2\times10^{-30}$ cm$^{6}$/s at $n=1\times10^{20}$ cm$^{-3}$. Thus, based on the work of Meneghini \emph{et al.} this is already indicative that our calculated Auger coefficient values should lead to a significant droop. 

Equipped with the theoretical data on the radiative and Auger recombination coefficients as functions of carrier density, we targeted a comparison to experimental data. Using samples provided by colleagues in the University of Cambridge, optical studies were carried out in the University of Manchester. Manchester established a novel method to carefully determine the carrier density in InGaN QWs when using pulsed excitation schemes. As literature data recently indicated the presence of defect-assisted Auger recombination processes in InGaN-based devices, special attention was paid to the impact of point defect density and carrier localisation effects on the droop carrier density regime behaviour of the IQE. 

To do so, three different samples where grown at different growth temperatures to modify the density of point defects in the well. Using these samples, the experimental data revealed that independent of sample (i) droop onset occurs at similar carrier densities in the well and (ii) the high carrier density regime evolution of the integrated PL intensity was basically identical. All this indicated that trap-assisted (defect related) Auger recombination seems to be of secondary importance for the studied samples. 

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

\chapter{Temperature dependent radiative and Auger recombination in AlGaN/AlN QWs}
\label{chp:TempAlGaN}

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

As discussed above in the Prologue, III-N semiconductors can, in principle, be used to create efficient light emitters operating from the deep UV spectral range all the way to the red emission wavelength window. The material of choice is InGaN for visible emitters and AlGaN alloys for UV emitters. However, in comparison to InGaN systems, AlGaN QWs have received far less attention, particularly in terms of understanding the fundamental properties of AlGaN-based emitters. AlGaN-based QWs have recently attracted significant research attention for use in UV LEDs, including potentially into the deep UV range ($\lambda$ < 280 nm)~\cite{AmCo2020}. Over the last few years, spurred by a renewed interest in developing fast, efficient and safe methods of anti-microbial sterilisation due to the COVID-19 pandemic~\cite{PiPi2023} interest in these systems has grown rapidly.  

The UV spectral range is generally divided into three sub-ranges, UVA (400 nm to 315 nm), UVB (315 nm to 280 nm) and UVC (also called the deep UV range, < 280 nm). All three UV regimes have numerous applications. For instance, UVA has applications in the curing of polymers, resins and inks, while UVB is used in phototherapies for the treatment of various skin diseases, as well as to generate extra nutrients in vegetables when used in plant growth lighting. Relevant to COVID-19, it has been found that UV light with wavelengths near 265 nm is the most effective for anti-microbial sterilisation, and, in general, UVC emitters in the wavelength range of 250 nm to 280 nm have been targeted for use in large-scale water, sewage and other sterilisation treatment applications. An overview of these applications can be found in Ref.~\cite{AmCo2020}.

Overall, UV LEDs could form the basis for robust, long-lived and wavelength tunable light-emitting devices for use in a wide range of applications. They may provide a route to UV light emission that is more efficient and much more environmentally friendly than conventional UV emission technologies (e.g. those that incorporate mercury~\cite{AmCo2020}). Although SQW and MQW AlGaN-based UV LED systems have been realised in research settings, they generally suffer from very low wall plug efficiencies (WPEs), especially when compared to high quality commercial InGaN-based devices. This disparity in efficiency may stem from a number of sources, for instance due to large defect densities when growing on sapphire substrates or issues with \emph{p}-type doping~\cite{AmCo2020}. A comprehensive discussion of the current state-of-the-art, along with details of the technical challenges facing UV LEDs, can be found in Ref.~\cite{Kneissl2016}, and we will briefly mention some examples of these challenges now.

For example, the WPEs of AlGaN-based UVB and UVC LEDs currently lag behind UVA devices, and so industrial scale production of AlGaN QW-based UVB and UVC LEDs has not become widespread, although several companies have started to commercialise UV LED devices (particularly in the UVA range)~\cite{AmCo2020}. UVB and UVC LEDs may suffer from poorer light extraction efficiencies in comparison to UVA LEDs due to the higher Al compositions required to access smaller wavelengths, along with the differences in electronic structure between AlN and GaN mentioned above and discussed in, for instance, Ref.~\cite{FiSc2022}. These shorter wavelength emitters also currently suffer from high threading dislocation densities, which contribute to stronger nonradiative recombination~\cite{AmCo2020}. Understanding the fundamental properties of these emitters is crucial to overcoming these efficiency bottlenecks.

Although there have been studies investigating the fundamental material properties and electronic structures of bulk AlN and AlGaN~\cite{ChHa1986,RuAl1994,LiBo2002,CoSc2015}, relatively few studies target an understanding of the electronic and optical properties of AlGaN QWs, despite their potential for use in energy efficient and environmentally friendly UV LEDs~\cite{Kneissl2016}. For instance, even though AlGaN QW systems also exhibit both thermal~\cite{DeMe2017,IsYo2020,MuTa2020} and carrier density dependent droop~\cite{NiTo2018,IwBa2015,HaTa2017,YuSh2015,ShYa2009}, the fundamental origin of either form of droop in AlGaN QWs is still unclear. Furthermore, knowledge of the processes contributing to InGaN QW efficiency droops cannot be directly applied due to the fundamental differences of AlN compared to InN and GaN alloys, for example differences in structural or polarisation properties, or the more complicated orbital character of the valence band edge in AlN. In this chapter, we seek further insights into the electronic structure of AlGaN QWs, the effect of carrier localisation in these systems and the resulting impact on the radiative and Auger recombination rates. Before proceeding, we briefly discuss some recent relevant literature results on AlGaN QWs.

Generally, most experimental studies reporting data on optical properties in AlGaN-based QWs do not go beyond the ABC model of recombination described above~\cite{NiTo2018,IsYo2020,HaTa2017,YuSh2015,DeCh2022}. Other approaches use continuum-based models, such as Ref.~\cite{HaRu2019}, that do not take alloy fluctuations into account. In InGaN QW systems the effect of carrier localisation is reflected in, for instance, the ``S''-shaped dependence of the PL peak on temperature, and also in the broadening of low temperature PL peaks (see above in Sec.~\ref{sec:LEDsIIINMaterials}). Both of these features have also been observed in AlGaN-based QWs, indicating that AlGaN systems also experience carrier localisation effects~\cite{FrNi2020PLSShape,FrNi2020PLBroadening}. As we have seen above in Chapters~\ref{chp:TempInGaN} and~\ref{chp:CarrDensInGaN}, in the case of InGaN-based systems, carrier localisation effects can have a major impact on not only the electronic structure, but also the recombination processes and efficiency of these systems. Thus, an understanding of the impact of carrier localisation on the underlying electronic structure of AlGaN QWs is required before we can target optical properties. As we have discussed throughout this thesis, the accurate description of carrier localisation effects in III-N QWs requires, ideally, an atomistic resolution.

Recently, the work of Ref.~\cite{FiSc2022} has targeted this gap in understanding by employing a similar theoretical framework to that described above for InGaN QWs; i.e., an $sp^{3}$ empirical tight-binding model in conjunction with a valence force field model, as well as a local strain and polarisation theory. In that work, AlGaN QWs of varying Al contents embedded in pure AlN barriers were investigated. With the assumption of random alloy fluctuations in the well region, a number of microscopic configurations were investigated and the carrier ground and excited states determined for each. 

They found a broadening of the fundamental transition energy of the wells due to the impact of the alloy microstructure, and that although there is variation in both the electron and hole ground state energies, the impact is larger for the holes. Analysis of the charge densities of the ground states showed the impact of random alloy fluctuations on the carrier wave functions, including the very strong hole localisation. Notably, at higher Al contents, $\geq$ 50\%, the ground state electron wave function was also affected by the alloy microstructure in some configurations, however this effect was less pronounced for excited electron states. This stands in contrast to typical InGaN QWs (i.e. with In contents from 10\% to 20\%), in which the impact of the alloy fluctuations is generally small for all electron states, although, as the authors point out, a direct comparison between the systems is not possible due to the differences in low band gap material content and the experimentally challenging nature of growing a high quality InGaN QW with an In content $\geq$ 50\%. 

Thus, the work in Ref.~\cite{FiSc2022} shows that alloy fluctuations lead to carrier localisation effects in AlGaN QWs, and can affect the electronic structure. Based on our experience with InGaN QWs in Chapters~\ref{chp:TempInGaN} and~\ref{chp:CarrDensInGaN} we expect that the recombination processes in AlGaN QWs will also be impacted by carrier localisation. With regard to carrier density dependent droop in InGaN QWs, our results in Chapter~\ref{chp:CarrDensInGaN}, consistent with the literature, show that the Auger recombination process (or a related process) is one of the most important factors. Despite the studies mentioned earlier in this section reporting carrier density dependent (and thermal) droop in AlGaN QWs, the importance of Auger recombination in causing these efficiency droops has not been studied in detail. 

In order to shed light onto these questions about the links between both types of droop and the recombination processes, the corresponding recombination coefficients must be determined. As we have previously discussed, the \emph{experimental} determination of recombination coefficients is, in general, non-trivial and requires fitting to experimental data that can give rise to much uncertainty. The few studies targeting an analysis of Auger recombination report significant Auger coefficients in AlGaN QWs of $C_{\text{tot}}\sim 1$ -- $4\times10^{-30}$ cm$^{6}$/s~\cite{NiTo2018,HaTa2017,YuSh2015}. However, issues with the use of the temperature dependent PL approach used to determine the efficiency droop have also been raised by Nippert \emph{et al.}~\cite{NiTo2018}. This approach may introduce errors as low temperature efficiency is assumed to be perfect. 

In their study of 1.5 nm wide, $c$-plane Al$_{0.45}$Ga$_{0.55}$N QWs in pure AlN barriers, Nippert \emph{et al.} reported that the process governing efficiency droop is indeed an internal loss mechanism (i.e. is not related to carrier injection or light extraction)~\cite{NiTo2018}. Furthermore, their results indicated that this internal loss mechanism is, in fact, Auger recombination. With careful experimental analysis they extracted a value for the Auger coefficient of $C = 3.4\times10^{-30}$ cm$^{6}$/s. They also determined a value for the radiative coefficient of $B = 2.4\times10^{-11}$ cm$^{3}$/s. Other experimental reports investigating recombination rates targeted wider wells ($\sim$ 2.4 nm) compared to Nippert \emph{et al.} above, such as in Ref.~\cite{YuSh2015}. However, in that study, the Al contents of both the well and barriers differ to the studies above, having 57\% Al in the well region and 81\% Al in the barrier. Furthermore, although Ref.~\cite{YuSh2015} reports radiative ($B=2.13\times10^{-11}$ cm$^{3}$/s) and Auger ($C=2.88\times10^{-30}$ cm$^{6}$/s) coefficients, the assumption of carrier density independent coefficients is made in order to fit the coefficients to experimental IQE data.

Another experimental work highlighting the impact of carrier localisation effects investigated a range of QW widths between 1.5 nm and 3 nm, with similar Al content in the well ($\sim$ 50\%) and pure AlN barriers~\cite{FrNi2020LocalisationWellWidth}. They found that well width strongly impacts the strength of carrier localisation due to the strength of the built-in polarisation field. However, that study did not report radiative or Auger recombination rates. We will return to the discussion of experimental results for $B$ and $C$, as well as the impact of carrier localisation, below. However, we note that generally experimental studies reporting recombination rates continue to work in the ABC model~\cite{NiTo2018,IsYo2020,HaTa2017,YuSh2015,DeCh2022} and thus we will also continue in that framework.

Before discussing the impact of carrier localisation on the electronic structure of the AlGaN QWs studied here, and subsequently the temperature dependence of the radiative and Auger recombination rates, we briefly discuss the theoretical framework for the study and the QW model systems targeted.

\subsection{Theoretical framework and quantum well model systems}

As mentioned above, when investigating carrier localisation effects in III-N QWs, ideally an atomistic resolution is required. We note that an approach using a modified continuum-based model can be found in Ref.~\cite{RuKa2020}, in which the authors take localisation of the hole wave function into account as an input parameter to the model, while assuming that the electron wave functions are completely delocalised. As we have discussed above, the work of Finn \emph{et al.} in Ref.~\cite{FiSc2022} suggests that, to an extent, electron wave functions may also be impacted by alloy fluctuations. We have seen in previous chapters, and it is also discussed in the literature~\cite{ChOD2021}, that continuum-based approaches may underestimate the impact of carrier localisation. Thus, we approach the problem with the atomistic $sp^{3}$ empirical tight-binding (TB) model described above in Chapter~\ref{chp:ElecStruc}. The model was modified for the analysis of AlGaN/AlN QWs and wave functions and energies were determined by R. Finn in the Photonics Theory Group at the Tyndall National Institute. We now aim to use these existing wave functions and energies as input to the framework for determining the recombination rates described above in Chapter~\ref{chp:RecombinationRates}. 

Before proceeding, we briefly recap the ingredients to the model used and established by R. Finn to determine the wave functions and energies. In line with experimental studies on the structural properties of AlGaN QWs, a random alloy distribution of Al and Ga atoms in the well was assumed~\cite{RiMa2016}. The wave functions and energies for 150 electron and 150 hole states for 20 alloy configurations (i.e. microscopic distributions) were calculated. The simulation supercells had dimensions of ($10\times9\times10$) nm$^{3}$ (containing approximately 82,000 atoms). Due to the computational load of Auger recombination rate calculations, in this study we focus on the QW structures investigated by Finn \emph{et al.} in Ref.~\cite{FiSc2022} that are similar in composition to those in the experimental work of Frankerl \emph{et al.}~\cite{FrNi2020LocalisationWellWidth} and Nippert \emph{et al.}~\cite{NiTo2018}. Here, the active region consists of a single approximately 2.85 nm wide Al$_{0.5}$Ga$_{0.5}$N QW embedded in pure AlN. This Al content is similar to the 45\% Al QW investigated by Frankerl \emph{et al.} and Nippert \emph{et al.}. However, the wells investigated here are (slightly) wider, compared to the 2.5 nm wells in Ref.~\cite{FrNi2020LocalisationWellWidth} and the 1.5 nm wells investigated in Ref.~\cite{NiTo2018}. We will discuss below how the obtained results may be affected by the well width. 

Using the TB wave functions and energies as input, we employ the same framework detailed in Chapter~\ref{chp:RecombinationRates} for the determination of the $B$ and $C$ coefficients in InGaN systems, with modifications for AlGaN. We continue to use a Sellmeier-type description of the wavelength dependent refractive index, $n(\lambda)$, but using the coefficients for AlN of $A = 1.00$, $B = 3.12$ and $C = 138.0$ nm given in Ref.~\cite{AnNa2003}. Previously, for InGaN/GaN QW systems we used the high frequency dielectric constant of the barrier (GaN) material ($\epsilon_{\infty}(\text{GaN}) = 5.5$), as this approach was employed in other studies of InGaN/GaN systems~\cite{KiSt2015}. In this study, we assume a 50/50 ratio of AlN and GaN such that $\epsilon_{\infty}(\text{Al}_{0.5}\text{Ga}_{0.5}\text{N}) = 5.0$, based on a dielectric constant for AlN of $\epsilon_{\infty}(\text{AlN}) = 4.64$. This value for AlN is determined from the average of the values reported in Ref.~\cite{BeFi1998DielectricConstants} for two theoretical investigations ($\epsilon_{\infty}^{\text{theo, a}}(\text{AlN}) = 4.61$; $\epsilon_{\infty}^{\text{theo, b}}(\text{AlN}) = 4.62$) and an experimental investigation ($\epsilon_{\infty}^{\text{exp}}(\text{AlN}) = 4.68$). 

We have also adapted the 1D self-consistent \sch-Poisson framework for determining the screening of the built-in polarisation field when charge carriers are introduced into the well. In InGaN QWs we saw above that the screening of the built-in field can have a significant impact on the electronic structure above a carrier density of approximately $n=1\times10^{19}$ cm$^{-3}$~\cite{McKi2023}. To adapt the model for AlGaN QWs we have used parameters for spontaneous and piezoelectric polarisation for AlGaN linearly interpolated from AlN and GaN parameters available in Ref.~\cite{CaSc2013local} for the determination of the polarisation field. We have used band offsets available in Ref.~\cite{CoSc2015} and average strain fields extracted from our TB model. Having adapted the framework, we determined that at the low carrier density investigated here, $n = 2.5\times10^{18}$ cm$^{-3}$, there is no significant screening of the built-in field. This carrier density has been chosen as it is in the range widely investigated and reported as typical for AlGaN QWs~\cite{NiTo2018,MuTa2020,HaTa2017,DeCh2022,HaRu2019}.

We note that in InGaN QWs it has been suggested that at carrier densities below approximately $n=1\times10^{18}$ cm$^{-3}$ Coulomb enhancement of the radiative recombination coefficient may be observed~\cite{DaYo2019manybody}. Carrying this assumption over to the AlGaN systems here, we expect that Coulomb effects are of secondary importance at the carrier density we investigate. However, future studies may target this question explicitly for AlGaN QW systems. Having now discussed the QW structures and theoretical framework used to determine the electronic structure, along with how AlGaN specific aspects are included in the calculation of the radiative and Auger coefficients in AlGaN/AlN QWs, we are in a position to analyse and investigate the impact of carrier localisation on the results. 

\section{Results}
\label{sec:TempAlGaNResults}

We begin by first looking at the electronic structure of $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QWs, before turning to the radiative and Auger coefficients, and the competition between the radiative and Auger coefficients as a function of temperature. 

\subsection{Electronic structure}

The electronic structure can strongly affect the recombination rates, where the recombination rates are determined as above in Chapter~\ref{chp:RecombinationRates}, particularly via the wave function overlap. By looking at the charge densities of the electronic states we can gain some first insights into the impact of carrier localisation on the electronic structure and thus on the recombination rates. In the following we will discuss the electronic structure and point out some expected consequences for the radiative and Auger recombination rates along the way.

When considering the Auger recombination rates, the estimation is complicated by the need to factor in the electron-electron wave function overlap and the hole-hole wave function overlap for the $eeh$ and $hhe$ Auger recombination processes, respectively. As discussed above, for the configurations investigated here, \emph{excited electron} states tend to not be greatly perturbed by the alloy disorder and are generally delocalised over the growth plane. Thus (and as we will see below), variations in both radiative and Auger recombination rates between microscopic configurations are primarily driven by variations in hole localisation characteristics (i.e. how strongly localised the hole states are, where they are localised in relation to the electron or other hole states, hole state energies, etc.). On top of this, given our observations of hole localisation being linked to enhancement of the $hhe$ Auger process, as well as the observation of a much larger $hhe$ Auger rate in comparison to the $eeh$ rate in InGaN/GaN QWs above in Chapter~\ref{chp:CarrDensInGaN}, in the analysis that follows, we focus mainly on the nature of hole states, and only briefly discuss the electron ground state.

\begin{figure}[t!]
\centering
    \begin{tabular}{|c|c|c|}
    \hline Cf. 3 & Cf. 5 & Cf. 6 \\
    \hline \multicolumn{3}{|c|}{Side View} \\
    \hline \includegraphics[width=0.3\columnwidth]{50pc_cf3_1e_3h_side_cropped.eps} & \includegraphics[width=0.3\columnwidth]{50pc_cf5_1e_3h_side_cropped.eps} & \includegraphics[width=0.3\columnwidth]{50pc_cf6_1e_3h_side_cropped.eps}\\
    \hline \multicolumn{3}{|c|}{Top View} \\
    \hline \includegraphics[width=0.3\columnwidth]{50pc_cf3_eGS_h123_topRotated.eps} & \includegraphics[width=0.3\columnwidth]{50pc_cf5_eGS_h123_topRotated.eps} & \includegraphics[width=0.3\columnwidth]{50pc_cf6_eGS_h123_topRotated.eps} \\
    \hline
    \end{tabular}
    \caption{Isosurface plots of the charge densities of the electron ground state (red), hole ground state (blue), first excited hole state (green) and second excited hole state (purple) for three configurations (Cfs. 3, 5 and 6) of a 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QW. The light (dark) isosurfaces correspond to 10\% (40\%) of the respective maximum charge density.}
    \label{fig:alElecStruc}
\end{figure}

Figure~\ref{fig:alElecStruc} shows isosurface plots of the ground state electron (red) and hole (blue) charge densities, along with the first (green) and second (purple) excited hole states for a sample of three microscopic alloy configurations from the 20 investigated. Figure~\ref{fig:alElecStruc} shows both a ``Side View'', in which we can see the $c$-direction separation of the electron and hole states due to the built-in field, and also a ``Top View,'' in which the varying in-plane separation of both carriers can be observed. The dark isosurfaces represent 40\% of the maximum value of the charge density, and the light isosurfaces represent 10\%. 

Starting with some common features of the configurations (Cfs.), in the ``Top View'' we can see that all three Cfs. exhibit strong hole localisation, not only of the ground state (blue) but also the first two excited states (green and purple). Also, in all three Cfs., the electron ground state (red) is less perturbed by the alloy fluctuations compared to the hole states. However, this effect is less pronounced for Cf. 6, for example, compared to Cf. 3, i.e., the electron ground state in Cf. 6 appears to be more strongly localised than in Cf. 3.

Continuing to look at the Top View, although there is strong hole localisation exhibited by all three Cfs., in Cfs. 5 and 6 the hole ground state and first excited state are spatially co-localised, in contrast to Cf. 3. In Cf. 3 however, the second excited hole state is co-localised with the hole ground state, while these hole states are spatially separated in Cfs. 5 and 6. The Top View of the charge density visualisations is also indicative of the spatial separation of the electron and hole ground states, although the spatial relationship between these states varies significantly between configurations. Before moving on, we note that in Cfs. 3 and 5 the hole ground state has one localisation centre. In Cf. 6 the hole ground state appears to have two localisation centres in the bottom right of the Top View figure, both spatially separated from the electron ground state. However, in the bottom left of the Cf. 6 Side View figure, we can see that part of the hole ground state is localised under the electron ground state, and so is not visible in the Top View. This spatial co-localisation of the electron and hole ground states may have a significant impact on the wave function overlaps.

Overall, it is clear that the alloy microstructure will strongly impact the hole states. Furthermore, we have seen that the alloy microstructure can also impact the electron ground state, and thus the spatial relationship between the electron ground state and valence ``band'' edge hole states. Although it is possible to pick out some general features of the wave function localisation from the plots in Fig.~\ref{fig:alElecStruc}, we now seek to quantify and gain further insight into the wave function overlap via calculating and analysing the dipole matrix elements between states.

The dipole matrix element (DME) between an electron and a hole state is proportional to the wave function overlap between those states and is an essential ingredient in the calculation of the radiative recombination rate (see Sec.~\ref{sec:RecombinationRadiative}). The DMEs between the electron ground state and the 10 energetically lowest hole states for Cfs. 3, 5 and 6 are presented as a function of the hole state number, $j$, in Fig.~\ref{fig:alDMEsvHoleState}. The first three DMEs shown in Fig.~\ref{fig:alDMEsvHoleState}, for instance, are related to the overlaps between the electron ground state ($i=0$) and the first three hole states (ground, $j=0$; first excited $j=1$; second excited, $j=2$) displayed above in Fig.~\ref{fig:alElecStruc}. 

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{DMEvsStateHoleVar_logy.eps}
    \caption{Dipole matrix elements, $|d^{e,h}_{ij}|^{2}$, (see Sec.~\ref{sec:RecombinationRadiative}) between electron ground state and varying hole states ($j$) for three configurations (Cfs. 3, 5 and 6) of a 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QW.}
    \label{fig:alDMEsvHoleState}
\end{figure}

Beginning with the DMEs between the electron and hole ground states, the effect of spatial co-localisation/separation becomes clear. For example, above we noted that for Cf. 6 although the two main localisation centres of the hole ground state were spatially separated from the electron ground state, the third localisation centre was actually co-localised with the electron ground state. In Fig.~\ref{fig:alDMEsvHoleState} we see a large DME for the hole ground state ($j=0$) in Cf. 6 in comparison to the hole ground state DMEs for Cfs. 3 and 5. In stark contrast, for Cf. 3 we noted that the electron and hole ground states were spatially separated, which is reflected in the much smaller ground state DME for Cf. 3 compared to Cfs. 5 and 6. Cf. 5 then lies somewhere in the middle. From this analysis we may expect that, at least at low temperatures (T $\approx$ 10 K), when mainly the ground states are populated, the radiative recombination rate for Cf. 6 will be larger than Cf. 5, and the rate for Cf. 5 will be larger than for Cf. 3.

As temperature is increased above 10 K, excited states will be populated and thus will contribute to the radiative recombination rate. Looking at the DMEs between the electron ground state and excited hole states ($j>0$), Fig.~\ref{fig:alDMEsvHoleState} shows that the alloy microstructure impacts the DMEs involving excited hole states, as well as the hole ground state. Looking at Cf. 6, there is an overall decreasing trend of DME with increasing hole state number, at least until the 7$^{\text{th}}$ excited hole state. For hole states energetically higher than the 7$^{\text{th}}$ excited state the DMEs begin to increase again but excited electron states will also begin to contribute and the situation becomes more complicated due to the Fermi-Dirac statistical weightings. Thus, the related $B$ coefficient may exhibit very difference trends for different alloy configurations. A similar argument may hold for Cf. 3, leading to a contrasting trend where the radiative recombination rate will increase in the low temperature range due to the increasing trend of DME with increasing hole state number seen in Fig.~\ref{fig:alDMEsvHoleState}. For Cf. 3, for hole states with $j>2$, the trend of DME with hole state number is less clear than in the Cf. 6 case. However, again, at temperatures high enough for these states to be significantly populated the Fermi-Dirac weighting and contributions from excited electron states complicate the analysis. For Cf. 5, in the low temperature range we may expect a decrease of radiative recombination rate with temperature, due to the decreasing trend of DME with increasing hole state number up to $j=2$. 

So far we have only discussed our expectations for the impact of the alloy microstructure on the radiative recombination rate, based on the impact of alloy fluctuations on the DMEs. As mentioned above, extending this analysis to Auger recombination is complicated by the contributions from electron-electron (for $eeh$ Auger) and hole-hole (for $hhe$ Auger) wave function overlaps. On top of the added complexity due to the introduction of electron-electron and hole-hole overlaps, the temperature dependence of not only the $eeh$ and $hhe$ Auger rates, but also the radiative rate, will be affected by the energetic distribution of states. The population of higher lying states is described by Fermi-Dirac statistics and will depend on this energetic spacing (see Sec.~\ref{sec:RecombinationRadiative}). Before discussing our results for the radiative and Auger recombination rates directly, we briefly look at the energy level spacing of the hole states of the three configurations above, to gain some insight into how the carrier localisation effects impact the energetic separation of states. We note that the conduction states are generally distributed in a similar manner when comparing different configurations and behave as might be expected from a VCA description. This is shown in more detail by Finn \emph{et al.}~\cite{FiSc2022}.

\begin{figure}[t!]
\centering
    \includegraphics[width=0.75\columnwidth]{vbEdges.eps}
    \caption{Energy difference, $\Delta E_{\text{hole}}$, between the hole ground state and excited hole states for Cfs. 3, 5 and 6 for a 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QW. The data are normalised to the energy of the hole ground state of Cf. 3. States within 100 meV of the Cf. 3 hole ground state energy are shown. Horizontal blue lines represent the energy differences between hole states and the Cf. 3 hole ground state.}
    \label{fig:alVBedges}
\end{figure}

Figure~\ref{fig:alVBedges} shows a representation of the difference in energy between the hole ground state and excited hole states for Cfs. 3, 5 and 6. The data are normalised to the Cf. 3 hole ground state, and states within 100 meV of this reference level are shown. We see that there is a significantly different valence ``band'' structure between configurations. This is particularly clear for Cf. 6, in which the first excited hole state is very closely spaced in energy to the hole ground state ($\Delta E = 4$ meV), in stark contrast to Cf. 3 ($\Delta E = 19$ meV) and Cf. 5 ($\Delta E = 13$ meV). Such a close energy spacing results in the first excited state being significantly populated even at T = 10 K. Thus, we might expect the radiative and Auger recombination rates to be enhanced in Cf. 6. For the radiative rate, even at T = 10 K there will be a contribution from not only the hole ground state, but also the first excited hole state, in contrast to Cfs. 3 and 5. For the $hhe$ Auger rate, the contribution from the first excited hole state may bring a further enhancement effect due to the strong co-localisation of the hole ground state and first excited hole state seen in Fig.~\ref{fig:alElecStruc}, via the hole-hole overlap entering the $hhe$ Auger rate calculation. 

The energy level spacings of Cfs. 3 and 5 differ from Cf. 6 but also from each other. In Cf. 3, as mentioned, the first excited hole state is approximately 20 meV below the hole ground state, but subsequent pairs of hole states are closely spaced. For example, the second and third excited holes states ($\Delta E = 2$ meV), the fourth and fifth excited hole states ($\Delta E = 1$ meV) and the sixth and seventh excited hole states ($\Delta E = 2$ meV) are closely spaced. Although, the spacing between the last two pairs (fourth/fifth and sixth/seventh) is only 4 meV so this may be closer to a group of four states. Regardless, in contrast, the energy spacing in Cf. 5 is more uniform down to states approximately 90 meV below the reference level. 

With this contrast in mind, we might expect that as temperature increases, in Cf. 3 the states in pairs will be populated in quick succession, and so we might expect a stronger increase in the radiative and Auger rates compared to Cf. 5. This may be compounded by the generally increasing trend of DME with hole state number seen above in Fig.~\ref{fig:alDMEsvHoleState}. In Cf. 5 the situation is less clear although our expectation above, i.e., that there is an initial decrease in the radiative recombination rate (based on the DMEs in Fig.~\ref{fig:alDMEsvHoleState}) as temperature increases, may still be reasonable. Subsequently, as temperature continues to increase, higher lying, delocalised hole states (with larger overlaps with the electron states compared to the strongly localised hole states of Cf. 5) will start to contribute, potentially leading to an increase in the radiative recombination rate with increasing temperature for Cf. 5. Overall though, it is difficult to predict how the system will behave with increasing temperature. The impact of the spatial co-localisation or separation of states introduces significant complexity when considering the population of excited states.

Overall, we have seen that alloy fluctuations can strongly impact the energetic distribution, as well as the spatial distribution of states (e.g. hole states can co-localise, electron and hole states are spatially separated along the growth direction but can co-localise in the growth plane, etc.), and thus, the wave function overlap. We also discussed the impact that we expect from these features (i.e. the spatial and energetic distributions of states and the wave function overlaps of states) on the radiative and Auger recombination processes. With these effects in mind, we now turn to look at the coefficient of radiative recombination, $B$, as a function of temperature. Subsequently we will discuss the results for the Auger recombination coefficients, $C_{eeh}$ and $C_{hhe}$, as a function of temperature, as well as the total Auger coefficient, $C_{\text{tot}}$, and lastly the competition between $B$ and $C_{\text{tot}}$.

\subsection{Radiative recombination}

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{BperConfigT_SellCorr.eps}
    \caption{Radiative recombination coefficient, $B$, as a function of temperature for 20 microscopic configurations of a 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QW. The average value of the coefficients (Average) taken over the 20 configurations is also presented. The carrier density is $n=2.5\times10^{18}$ cm$^{-3}$. Configurations used in Auger recombination study (Cfs. 3, 5, and 6) are highlighted, see main text for details.}
    \label{fig:alBvT}
\end{figure}

We begin our discussion of the radiative recombination rate by presenting the $B$ coefficient as a function of temperature for a 2.85 nm wide Al$_{0.5}$Ga$_{0.5}$N/AlN QW in Fig.~\ref{fig:alBvT}. We note a large variation in magnitude in the value of $B$ at low temperatures (T = 10 K). We attribute these variations to differences in the alloy microstructure between configurations. At low temperatures mainly the ``band'' edge states are populated, which we saw above, for three example configurations in Fig.~\ref{fig:alElecStruc}, are significantly impacted by the alloy fluctuations. Returning to Fig.~\ref{fig:alBvT}, we see that the variation in $B$ between configurations at a given temperature decreases as temperature increases. This is due to the population of higher lying, less localised states. These excited states, due to their extended nature, are not as strongly affected by the underlying variations in alloy disorder potential. This effect is also seen for $B$ as a function of temperature in InGaN/GaN QWs above (see Sec.~\ref{sec:TempInGaNResults}).

In terms of the evolution of $B$ with temperature, we see that on average, $B$ increases slightly with increasing temperature. The magnitude of the increase is similar to that observed in InGaN QWs (see Fig.~\ref{BversusT_lowest} in Sec.~\ref{sec:TempInGaNResults}), and stands in contrast to the usually expected decrease in $B$ with increasing temperature observed for other, more conventional III-V semiconductor QW structures (see discussion in Sec.~\ref{sec:TempInGaNResults}). We note that only one configuration shows the opposite trend, Cf. 6, with $B$ decreasing as temperature increases across the whole temperature range studied. 

For 50\% of the configurations, $B$ increases with increasing temperature across the whole studied temperature range. For the other 50\%, there is an initial decrease of $B$ with increasing temperature between 10 K and approximately 60/120 K, before $B$ begins to increase with increasing temperature for higher temperatures. Generally, we also attribute this behaviour to the population of higher lying, less localised states as temperature increases. We will discuss the differing behaviour of Cf. 6 below. We will also analyse the behaviours of Cfs. 3 and 5, as these three configurations (Cfs. 3, 5 and 6) represent the wide range of possible situations seen for $B$ as a function of temperature in Fig.~\ref{fig:alBvT}. Configuration 6 is strictly decreasing with increasing temperature, Cf. 3 is strictly increasing and Cf. 5 initially decreases before starting to increase subsequently. 

For the majority of configurations, as temperature increases excited hole states that are less localised will be populated with two resulting effects. Firstly, there will be more states contributing to the radiative recombination rate. Secondly, these less localised states (including less localised hole states) may have larger wave function overlaps with the extended electron states (see the DMEs in Fig.~\ref{fig:alDMEsvHoleState}). Thus, the evolution of the $B$ coefficient indicates that these effects start to dominate for temperatures greater than approximately 120 K.

However, as mentioned, Cf. 6 does not follow this generally increasing trend and the behaviours of $B$ for Cfs. 3 and 5 are also different from Cf. 6, and each other. So, we now examine the dependence of the $B$ coefficient on temperature for each of these configurations. For Cf. 6, we noted above that we expected an overall larger $B$ coefficient at low temperature (T = 10 K) compared to Cfs. 3 and 5 due to the spatial co-localisation of the electron ground state and part of the hole ground state (see Fig.~\ref{fig:alElecStruc}). On top of this, we noted that the first excited hole state is also strongly localised and energetically close to the hole ground state, and so will contribute to $B$ even at T = 10 K, which further contributes to enhancing it compared to the other configurations. This effect explains why the 10 K $B$ coefficient for Cf. 6 is much larger than Cfs. 3 and 5 in Fig.~\ref{fig:alBvT}. However, as temperature increases we would (as is the case for the other configurations) expect that the higher lying states, being less localised, contribute larger DMEs, on average, and thus $B$ should increase. 

To understand the behaviour of Cf. 6 we consider again the idea that the excited, extended states are less perturbed by the alloy microstructure. So, we expect that the DMEs of the excited states of Cf. 6 may be, on average, similar to the DMEs of the excited states of Cfs. 3 and 5. Indeed, for Cf. 6 the large electron hole ground state DME is larger than the average value of the DMEs between the electron ground state and all of the hole states of Cf. 6. Thus, as temperature increases, the higher lying states are given larger weightings by the Fermi-Dirac statistics at the cost of reducing the contributions from the strongly localised states. As a consequence, the importance of the strongly localised states for the overall $B$ coefficient is diminished and Cfs. 3, 5 and 6 all give similar values in the high temperature range. For Cf. 6 this effect manifests as a decrease in $B$ as temperature increases. For Cfs. 3 and 5 the opposite is true and the electron hole ground state DME is smaller than the respective averages of the DMEs between the electron ground state and the hole states. Thus, overall, as temperature increases, DMEs for excited states contribute more. These contributions are larger compared to the band edge state DMEs, and they now contribute more, leading to an increase in $B$.

Turning to look at the low temperature ($\lessapprox$ 180 K) behaviours of $B$ for Cfs. 3 and 5 we see different evolutions with increasing temperature. For Cf. 3, from T = 10 K to T = 60 K, there is a slight increase in $B$, before a much stronger increase up to T = 180 K. For Cf. 5, in contrast, $B$ decreases from T = 10 K to T = 60 K, before a much slower increase up to T = 180 K. We attribute this difference to effects described above, i.e., that the first few DMEs in Fig.~\ref{fig:alDMEsvHoleState} are increasing for Cf. 3 and decreasing for Cf. 5, and due to the structure of the valence ``band'' in Fig.~\ref{fig:alVBedges}. For Cf. 3, pairs of states will begin to contribute as temperature increases, whereas the more uniform energetic distribution of states in Cf. 5 means only single states may contribute. Thus, we expect that the less strong increase of $B$ with temperature above T = 60 K (for Cf. 5 compared to Cf. 3) is due to the valence ``band' structure. 

We turn now to the overall magnitude of the $B$ coefficient and how it compares to experimental results. Overall, factors such as Al content, carrier density and well width will significantly affect the built-in potential and thus the electronic and optical properties of AlGaN QWs. For instance, experimentally, Frankerl \emph{et al.} investigated a range of well widths (from 1.5 nm to 3.0 nm) with similar structural characteristics to the QWs studied here ($\sim$ 45\% Al content, AlN barriers) and found that efficiency decreases, which could be linked to a decrease in the $B$ coefficient~\cite{FrNi2020LocalisationWellWidth}. From our discussion of the built-in field in InGaN/GaN QWs above, we expect that a stronger built-in field will lead to a smaller $B$ coefficient, and that a wider well will also lead to a smaller $B$ coefficient due to the larger potential drop between the QW interfaces and resulting increase in carrier separation along the growth direction. For AlGaN/AlN QWs, theoretically, Rudinsky \emph{et al.} confirmed this in their investigation of 1.5 nm and 2.5 nm wells, both with 45\% Al in the well and pure AlN barriers~\cite{RuKa2020}. Thus, a one-to-one comparison between our theoretical results and literature data for QWs with different structural parameters (such as Al content and well width) will be complicated by the impact of the built-in field. Bearing this in mind, we discuss now general trends. 

Nippert \emph{et al.} report values of $B = 2\times10^{-11}$ cm$^{3}$/s for a 1.5 nm wide Al$_{0.45}$Ga$_{0.55}$N/AlN QW at room temperature (T $\approx$ 300 K) and a carrier density of $n=1.8\times10^{18}$ cm$^{-3}$~\cite{NiTo2018}. However, uncertainties in determining the $B$ coefficient experimentally are mentioned in that work, such that the values may be as low as $B = 6\times10^{-12}$ cm$^{3}$/s. Given the wider, 2.85 nm, well investigated here compared to the experimental work of Nippert \emph{et al.}, we expect that our values for $B$ should be lower than those reported in Ref.~\cite{NiTo2018}. From our calculations we determined an average value for $B$ of $B \approx 4\times10^{-13}$ cm$^{3}$/s at a temperature of T = 300 K and a carrier density of $n=2.5\times10^{18}$ cm$^{-3}$ for the Al$_{0.5}$Ga$_{0.5}$N/AlN QWs studied here. Again, although differences in carrier density and Al content will also impact any comparison, this value for $B$ is consistent with the expected trend due to the significant difference in well width between theory and experiment.

Comparing our results now to the theoretical work of Rudinsky \emph{et al.}~\cite{RuKa2020}, they report values of $B$ for a 1.5 nm well, structurally similar to that investigated by Nippert \emph{et al.}, on the order of $B\sim 1\times10^{-11}$ cm$^{3}$/s. Thus, our results are still consistent with the expected larger $B$ coefficients for narrower wells. However, when investigating a 2.5 nm well, they find $B\sim 5\times10^{-12}$ cm$^{3}$/s, approximately an order of magnitude larger than the $B$ value determined here. It is important to note that the model employed in Ref.~\cite{RuKa2020} takes the strength of hole localisation as input to an essentially continuum-based model, and electrons are assumed to be entirely delocalised. On top of this, the model requires assumptions about the dependence of the h-DOS on energy, as detailed in Ref.~\cite{Karpov2018AlGaN}, and essentially averages the localisation radius of the ensemble of localised hole states. Thus, the strength of hole localisation may be underestimated.

Lastly, it is now interesting to compare, but also highlight differences, to InGaN QW systems in terms of the obtained $B$ coefficients. Just as in the case of InGaN/GaN QWs, the interplay of carrier localisation and its impact on the electronic structure can be complex. We find here $B$ coefficients smaller than those calculated for InGaN QWs above, although a direct comparison with the systems studied above in Chapters~\ref{chp:TempInGaN} and~\ref{chp:CarrDensInGaN} is difficult to make. This is in part due to the fundamental differences in the AlGaN and InGaN systems but also due to differences in well width and composition. Even with the same well width and equivalent ``low band gap'' material composition, there will still be significant differences in the magnitude of the built-in polarisation field between an AlGaN QW and an InGaN QW due to differences in spontaneous and piezoelectric polarisation (see discussion in Sec.~\ref{sec:LEDsIIINMaterials}). 

Notwithstanding these differences, we expect that the smaller $B$ coefficient in the AlGaN QWs studied here may be due to an overall stronger built-in polarisation field. To gain some insight into this, we have estimated the magnitude of the built-in polarisation field in a 1D VCA-like description by following the process in Ref.~\cite{CaSc2011}. We find that, to a first approximation, the built-in polarisation field in the Al$_{0.5}$Ga$_{0.5}$N/AlN QWs studied here is approximately twice as strong as in the In$_{0.15}$Ga$_{0.85}$N/GaN QWs studied in Chapters~\ref{chp:TempInGaN} and~\ref{chp:CarrDensInGaN}. Using the 1D VCA-like model in Ref.~\cite{CaSc2011}, and described above, we find that for a 2.85 nm wide Al$_{0.5}$Ga$_{0.5}$N/AlN QW the potential drop across the well is $\varphi = 1.46$ V, compared to a 3 nm In$_{0.15}$Ga$_{0.85}$N/GaN QW which exhibits a potential drop of $\varphi = 0.55$ V. In this 1D, VCA approach we take a linear interpolation of the material parameters in the well, and so the large differences in potential drop stem primarily from the much larger difference in low band gap material composition between the well and barriers in the AlGaN system compared to the InGaN system, as opposed to arising from differences in, for example, the magnitude of the spontaneous polarisation in GaN versus InN or AlN. In the presence of a stronger built-in polarisation field, carriers will be more strongly localised at opposite interfaces of the well, as well as more strongly localised within local potential ``pockets'', leading to lower wave electron hole wave function overlaps and thus a lower radiative recombination rate. 

We now have a general understanding of the temperature dependence of the radiative recombination rate. To study the thermal droop in AlGaN QWs we need also to determine the temperature dependence of the Auger recombination rates. This is the topic of the next section.

\subsection{Auger recombination}

Given the large computational expense of Auger rate calculations, we seek a way to restrict the analysis to a subset of configurations. As discussed above, Cfs. 3, 5 and 6 are representative of the different impacts that the alloy microstructure can have on the evolution of $B$ with temperature. Thus, we focus now on these configurations. As well as representing the different trends of $B$ with temperature, these three configurations also represent the highest, lowest and average value of $B$ at each temperature data point. We begin our discussion of the Auger recombination rate in 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QWs by discussing the breakdown of the total Auger rate into contributions from the two Auger processes considered here, electron-electron-hole ($eeh$) and hole-hole-electron ($hhe$) Auger recombination. Figures~\ref{fig:alChhevT} a) and b) show the $hhe$, $C_{hhe}$, and $eeh$, $C_{eeh}$, Auger coefficients, respectively, for the three configurations along with the average value of the coefficients (Average) taken over the configurations considered, for the QW systems described above as functions of temperature at a carrier density of $n=2.5\times10^{18}$ cm$^{-3}$. Before discussing the breakdown we note some general features.

\begin{figure}[t!]
    \centering
    \begin{tabular}{c}
        \includegraphics[width=0.75\columnwidth]{alChhevT.eps}\\
        \includegraphics[width=0.75\columnwidth]{alCeehvT.eps}
    \end{tabular}
    \caption{a) Hole-hole-electron ($hhe$) Auger recombination coefficient, $C_{hhe}$, and b) electron-electron-hole ($eeh$) Auger recombination coefficient, $C_{eeh}$, as a function of temperature for Cfs. 3, 5 and 6 of a 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QW, along with the average value for the coefficients (Average) taken over the three configurations considered. The carrier density is $n=2.5\times10^{18}$ cm$^{-3}$.}
    \label{fig:alChhevT}
\end{figure}

Recalling that the total Auger recombination coefficient, $C_{\text{tot}}$, is related to the $eeh$ and $hhe$ coefficients via $C_{\text{tot}}=C_{eeh}+C_{hhe}$, we first note that Figs.~\ref{fig:alChhevT} a) and b) show that $C_{hhe} \gg C_{eeh}$. Looking at the average of the configurations, at T = 300 K we find values of $C_{hhe} = 5\times10^{-32}$ cm$^{6}$/s. But for $C_{eeh}$, at T = 300 K a value of $C_{eeh} = 6\times10^{-36}$ cm$^{6}$/s is found on average. Being 4 orders of magnitude larger than $C_{eeh}$, $C_{hhe}$ makes up well over 99\% of $C_{\text{tot}}$. This is a similar result to the InGaN QWs studied above, where we found that $C_{eeh}$ is smaller than $C_{hhe}$ by approximately the same factor ($\approx$ 4 orders of magnitude) across the same temperature range and at a similar carrier density. There has not yet been a study that estimates the relative magnitudes of $C_{eeh}$ and $C_{hhe}$ in AlGaN QWs. Thus, in the following, we will primarily focus on $C_{hhe}$, as this will determine the behaviour of $C_{\text{tot}}$.

We note that a common approach to estimating the Auger coefficients is via the triple wave function overlap~\cite{JoTe2017}. For instance the $hhe$ Auger coefficient may be estimated via
\begin{equation}
    \label{eq:AugerTripleIntegral}
    C_{hhe} \sim \int \psi^{e}\psi^{h}\psi^{h} \,\, .
\end{equation}
Thus, as mentioned previously, we must now also account for the wave function overlap of hole states. With this in mind, we turn to Fig.~\ref{fig:alChhevT} a) and $C_{hhe}$ as a function of temperature for Cfs. 3, 5 and 6. At low temperature (T = 10 K), the relative differences in magnitude of $C_{hhe}$ between the three configurations follows from our analysis above of the impact of alloy fluctuations on the electronic structure. At T = 10 K, $C_{hhe}$ for Cf. 6 may be larger than for Cfs. 3 and 5 due primarily to the very close energetic separation between the hole ground state and first excited state. Not only will this extra excited hole state now contribute even at T = 10 K, but due to the spatial co-localisation of the hole ground state and the first excited hole state, the $\psi^{h}\psi^{h}$ overlap in Eq.~\eqref{eq:AugerTripleIntegral} will be further enhanced, in contrast to Cfs. 3 and 5, wherein there will not even be a contribution from the first excited hole state at T = 10 K. For Cfs. 3 and 5, the relative magnitudes of $C_{hhe}$ may be driven by the electron hole wave function overlaps for the band edge states.

In terms of the temperature dependence of $C_{hhe}$ for the three configurations, we see different behaviours for each. Overall, as temperature increases, $C_{hhe}$ for Cf. 3 increases, in contrast to Cfs. 5 and 6 which see a decrease of $C_{hhe}$ with increasing temperature. The low temperature (T < 180 K) behaviour of $C_{hhe}$ is also different between Cfs. 5 and 6, even though overall $C_{hhe}$ decreases in the temperature range studied for those two configurations. In Cf. 5, $C_{hhe}$ decreases over the whole low temperature range, whereas $C_{hhe}$ for Cf. 6 initially increases with increasing temperature from T = 10 K to T = 60 K, before decreasing for all subsequent higher temperatures. 

Turning first to Cf. 3, as temperature increases, the first excited hole state will be populated. However, looking at the Top View of the charge densities in Fig.~\ref{fig:alElecStruc} suggests that the first excited hole state of Cf. 3 is spatially separated from the hole ground state, leading to a small hole-hole wave function overlap between these states. Thus, in contrast to the radiative recombination rate for Cf. 3 above, in which there was a strong increase of $B$ as temperature increased from T = 60 K to T = 180 K, for $C_{hhe}$ there is not an increase until much higher temperatures. As temperature initially increases above T = 10 K, for $C_{hhe}$ there is a competition between the larger electron hole wave function overlaps (see Fig.~\ref{fig:alDMEsvHoleState}) that begin to contribute, and the smaller wave function overlap between the hole ground state and first excited state due to the spatial separation of these states, which is not a factor in the calculation of the radiative coefficient. Subsequently, as the second excited hole state begins to contribute at higher temperatures, the value of $C_{hhe}$ begins to increase due to the spatial co-localisation of this state with the hole ground state.

For Cf. 5, $C_{hhe}$ decreases as temperature increases across the whole temperature range. At least in the low temperature range (T < 180 K), when mainly ``band'' edge states are populated, we might expect this given the spatial separation of the second excited hole state charge density from the ground and first excited hole states in Fig.~\ref{fig:alElecStruc}. Furthermore, the DMEs between some of the higher lying excited hole states with the electron ground state are smaller than the DME between the electron ground state and the hole ground state, as per Fig.~\ref{fig:alDMEsvHoleState}. Thus, the decrease of $C_{hhe}$ between T = 10 K and T = 180 K may stem primarily from the population of higher lying, spatially separated hole states, via the smaller hole-hole overlap entering Eq.~\eqref{eq:AugerTripleIntegral} and also the smaller electron hole overlaps. As temperature increases beyond T = 180 K we might expect an increase of $C_{hhe}$ due to the population of higher lying, delocalised states which have, on average, larger overlaps compared to the ``band'' edge states. However, the large number of states populated above T = 240 K makes an explicit analysis difficult, so instead we note that the $C_{hhe}$ coefficient for Cf. 5 appears to start to plateau above T = 240 K, potentially due to the increasing contributions from these higher lying states and the lower ``weight'' of the band edge states.

Turning lastly to Cf. 6, as temperature is initially increased from 10 K to 60 K we see an increase in $C_{hhe}$, likely related to the strong hole-hole wave function overlap between the hole ground state and the first excited hole state. Subsequently, as temperature increases further and the second excited hole state is populated, we might expect this decrease due to the spatial separation of this state from both the hole ground state and first excited state. Interestingly, even at elevated temperatures (T $\gtrapprox$ 240 K), when we would expect the $C_{hhe}$ coefficient to reflect the more homogenised nature of the excited, extended states by converging to a value similar to the Cf. 3 and 5 values, as the radiative recombination rate does above, we see instead that $C_{hhe}$ for Cf. 6 remains elevated across the whole investigated temperature range. 

A similar behaviour was observed in our investigation of the temperature dependence of $C_{hhe}$ in In$_{0.15}$Ga$_{0.85}$N/GaN QWs, discussed above in Chapter~\ref{chp:TempInGaN}. In that context we found two configurations exhibited significantly larger $C_{hhe}$ values compared to the majority of the investigated configurations. This phenomenon was linked to very strongly localised hole states, both ground and first excited. Both of these states sat energetically far above the valence ``band'' edge, compared to the valence ``band'' edge states of all the other configurations studied. Effectively, these states were energetically in the band gap and we suggested that they may be related to so-called trap-like states. Returning to the AlGaN case here, in contrast, these hole states lie (energetically) \emph{within} the broadened valence ``band'' edge (see Fig.~\ref{fig:alVBedges}), and so we do not associate them with trap-like states. Rather, it is the co-localisation of the hole states, coupled with their small energetic separation, that drives the elevated Auger rate. Furthermore, $C_{hhe}$ for Cf. 6 is elevated over the whole temperature range, showing that even at higher temperatures when excited, extended states are populated, the underlying contributions from strongly localised states are still significant, and can vary between configurations.

As mentioned, $C_{eeh}$, presented in Fig.~\ref{fig:alChhevT} b), is approximately four orders of magnitude smaller than $C_{hhe}$ and thus is of secondary importance. We note that the alloy microstructure can be seen to play a role in $C_{eeh}$ at low temperatures, but this effect diminishes at higher temperatures likely due to the contribution of higher lying, extended states that are less perturbed by the alloy microstructure. Overall though, the behaviour of $C_{eeh}$ is of secondary importance for the discussion of the total Auger rate. Thus, we only discuss $C_{hhe}$ in detail and now move on to the temperature dependence of the total Auger coefficient, $C_{\text{tot}}$.

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{alCtotvT.eps}
    \caption{Total Auger recombination coefficient, $C_{\text{tot}}$, as a function of temperature for Cfs. 3, 5 and 6 of a 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QW, along with the average of the configurations (Average). The carrier density is $n=2.5\times10^{18}$ cm$^{-3}$.}
    \label{fig:alCtotvT}
\end{figure}

Figure~\ref{fig:alCtotvT} shows the total Auger recombination coefficient for the three configurations investigated, along with the average of the configurations. At 300 K, the average value is $C_{\text{tot}} = 5\times10^{-32}$ cm$^{6}$/s. As the total rate is dominated by $hhe$ Auger process, the evolution of $C_{\text{tot}}$ with increasing temperature is essentially the same as for $C_{hhe}$. Nippert \emph{et al.}~\cite{NiTo2018} report a value of $C_{\text{tot}} = 3.4\times10^{-30}$ cm$^{6}$/s, approximately two orders of magnitude greater than the values determined here, but for narrower wells of well width 1.5 nm. Rudinsky \emph{et al.} \cite{RuKa2020}, using their modified continuum-based model that includes hole localisation, also significantly underestimate the total Auger coefficient (finding values of $C_{\text{tot}}$ in range of $C_{\text{tot}} = 7\times10^{-32}$ -- $4\times10^{-31}$ cm$^{6}$/s) in comparison to the experimental values of Nippert \emph{et al.} (see Fig. 3 (b) in Ref.~\cite{RuKa2020}). However, when targeting wider QWs similar to our study (with a well width of 2.5 nm, 45\% Al in the well and AlN barriers) they find total Auger coefficients in the range of $C_{\text{tot}} = 5\times10^{-32}$ -- $2\times10^{-31}$ cm$^{6}$/s at carrier densities between $n=1\times10^{18}$ -- $1\times10^{19}$ cm$^{-3}$. Thus, the trend of wider wells leading to smaller recombination coefficients is seen theoretically, and when accounting for difference in well width it seems reasonable to suggest that the theoretical models are in relative agreement. When comparing to the experimental results, however, a number of other factors may impact the values, which we will now discuss.

Rudinsky \emph{et al.} suggest that the inclusion of an AlN cap in the experimental setup of Nippert \emph{et al.} may alter the band structure of all of the MQWs in the structures, leading to non-uniform carrier distributions across the wells. Furthermore, as we have discussed above, determining the carrier density in III-N QW systems can be challenging. The carrier density in Ref.~\cite{NiTo2018} was determined by fitting the SRH $A$ and Auger $C$ coefficients to carrier density dependent IQE data, while assuming that $A$ and $C$ are carrier density \emph{independent}. As we have seen above for InGaN QWs, this may not be the case for $C$ (or $B$), which will impact the determined carrier density and also the value of $B$ extracted (which depends on $C$ in the fitting). We note also that in the experimental data, defect-assisted Auger processes may be an important factor which we do not account for here. Lastly, due to the computational expense of Auger rate calculations we focused here on only three configurations. Future studies may target a larger number of configurations to gain a better understanding of the range of expected Auger coefficients, as well as investigating the impact of defect-assisted processes. With our knowledge of the temperature dependence of the radiative and Auger recombination rates, we are now in a position to discuss the impact of Auger recombination on the thermal droop in AlGaN QWs.

\subsection{Competition between radiative and Auger recombination}

\begin{figure}[t!]
    \centering
    \includegraphics[width=0.75\columnwidth]{relBCrat_highT.eps}
    \caption{Ratio of $B$ and $C_{\text{tot}}$, $B/C_{\text{tot}}$, as a function of temperature for a 2.85 nm wide $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QW. The data are normalised to the ratio at 300 K. The carrier density is fixed at $n=2.5\times10^{18}$ cm$^{-3}$.}
    \label{fig:alBCratvT}
\end{figure}

Experimentally, it has been found that in UV LEDs, as temperature increases, especially above 300 K, IQE decreases. This can be seen, for instance, in Refs.~\cite{DeMe2017} and~\cite{IsYo2020}. Figure~\ref{fig:alBCratvT} shows the ratio of the $B$ and $C_{\text{tot}}$ coefficients as a function of temperature in the range in which thermal droop has been observed experimentally~\cite{DeMe2017}. The data are normalised to the value of the ratio at 300 K. We note as well that as we are varying temperature at a fixed carrier density a plot of $B/C_{\text{tot}}$ will be equivalent to a plot of $B/(C_{\text{tot}}n)$, i.e., any factor of $n$ will only scale the results. 

We see that, on average, $B/C_{\text{tot}}$ is increasing relative to its value at 300 K. Thus, we could expect the likelihood of a recombination event to result in the emission of a photon to increase relative to the likelihood of a recombination event resulting in a ``hot'' Auger carrier as temperature increases. This would indicate that device performance improves, in a first approximation. However, this stands in contrast to the decrease in device performance observed experimentally, thus, our results suggest that the here considered Auger recombination process is probably not directly responsible for thermal droop in AlGaN/AlN QWs, as is suggested in the experimental studies above~\cite{IsYo2020}. It is suggested in Refs.~\cite{DeMe2017} and~\cite{IsYo2020} that a different nonradiative process external to the active region is the underlying cause of thermal droop in AlGaN QWs.

However, we note that the large band gap of AlGaN QWs may result in the generation of very high energy Auger carriers, so-called ``hot'' Auger carriers. Interactions between these high energy carriers and the crystal structure may produce defects~\cite{GlRu2022,RoPi2023}, having a knock-on effect on the SRH-related $A$ coefficient and potentially other processes such as defect-assisted Auger recombination or carrier injection. Furthermore, both Refs.~\cite{DeMe2017} and~\cite{IsYo2020} suggest that thermal droop is linked to recombination processes occurring outside of the active region. Thus, the generation of hot Auger carriers, which may undergo thermionic escape and reach regions of the LED structure outside of the active region, may still have an effect on the thermal droop. Further studies of the impact of such hot Auger carriers on device performance are thus required.

In this chapter we have discussed \emph{thermal} droop in AlGaN-based QWs. However, \emph{carrier density dependent} droop (as was discussed above for InGaN QWs) has also been observed in AlGaN QW-based systems~\cite{NiTo2018}. Although we have not yet carried out a study on the evolution of the radiative or Auger recombination rates with carrier density, we briefly note that for InGaN QWs, values of $C_{\text{tot}}$ above approximately $1\times10^{-31}$ cm$^{6}$/s are considered generally to be large enough to impact the carrier density dependent efficiency droop, due to the cubic dependence of the Auger rate on carrier density. However, the droop phenomena also depends on the carrier density dependence of the $B$ coefficient. So, given that we find smaller values for $B$ here than in the InGaN systems studied above, the threshold of $C_{\text{tot}} = 1\times10^{-31}$ cm$^{6}$/s may be an overestimation for AlGaN QWs. Thus, the values of $C_{\text{tot}}$ determined here could be large enough to contribute to droop. 

Furthermore, as we have seen in the case of InGaN QWs, screening of the built-in polarisation field as carrier density increases can lead to a significant enhancement of both the radiative and Auger recombination rates. Thus, the Auger coefficients determined here may represent a lower bound for $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QWs, and will increase with increasing carrier density. We note that it is commonly assumed that similar values of $A$ can be found in AlGaN/AlN QWs as can be found in InGaN QWs (see, for instance, Refs.~\cite{NiTo2018},~\cite{HaTa2017} and \cite{YuSh2015}). This assumption for $A$, coupled with the smaller $B$ and $C_{\text{tot}}$ values determined in this work for AlGaN systems compared to InGaN systems, would suggest that at least in the ABC model, AlGaN QWs should enter the droop regime at higher carrier densities. Lastly, variations in carrier density through, for instance, an AlGaN MQW structure may lead to regions in which the carrier density is larger and so Auger recombination may be enhanced, negatively impacting device performance. Overall though, further studies targeting an analysis of carrier density dependent droop in AlGaN QWs are required.

\section{Conclusion}
\label{sec:TempAlGaNConc}

In this chapter we applied our atomistic TB model to the investigation of the impact of random alloy fluctuations on the electronic structure, radiative and Auger recombination coefficients and the competition between these recombination processes as a function of temperature in 2.85 nm $c$-plane Al$_{0.5}$Ga$_{0.5}$N/AlN QWs. We found that alloy disorder leads to strong carrier localisation of not only the hole ground state, but also excited hole states. Electron states may also be impacted by carrier localisation, but to a lesser extent than holes.

Turning to the radiative recombination coefficient, $B$, we found that overall, at a carrier density of $n=2.5\times10^{18}$ cm$^{-3}$, the $B$ coefficient slightly increases with increasing temperature. The calculations showed that the alloy microstructure plays a crucial role at low temperatures as mainly the strongly localised ``band'' edge states are occupied. As temperature increases and higher lying, extended states are populated, the spread in $B$ values is strongly reduced, as the details of the alloy microstructure impact the extended states less than the strongly localised states. 

We found an average value of $B \approx 4\times10^{-13}$ cm$^{3}$/s at T = 300 K and $n = 2.5\times10^{18}$ cm$^{-3}$. Experimental studies reporting values of $B$ are sparse. Data on $B$ for structurally similar QWs (i.e. AlN barriers, similar Al contents in the well) is only available for much smaller well widths. That study yields a higher value for $B$ in comparison to our data. However, this is expected as well width is a major factor that can impact the magnitude of $B$ due to the separation of electron and hole states along the growth direction associated with the built-in polarisation field. Thus, we expect that narrower wells will have larger $B$ coefficients. Future studies can now target an investigation of the dependence of $B$ on well width. 

Having discussed the radiative recombination rate via the $B$ coefficient, we then turned to the temperature dependence of the Auger recombination processes. We calculated both the $eeh$ and $hhe$ Auger rates as a function of temperature. We found that both processes are strongly impacted by the alloy microstructure and the localisation of holes. Overall, the $hhe$ Auger coefficient, $C_{hhe}$, was found to be greater than the $eeh$ Auger coefficient, $C_{eeh}$, by four orders of magnitude. Thus, the total Auger rate is dominated by the $hhe$ Auger rate. The total Auger coefficient at a carrier density of $n=2.5\times10^{18}$ cm$^{-3}$ and a temperature of T = 300 K was found to be, averaged over three microscopic configurations, $C_{\text{tot}} = 5\times10^{-32}$ cm$^{6}$/s. As in the case of the $B$ coefficient above, one of the few experimental investigations of $C_{\text{tot}}$ was carried out for AlGaN QW systems with smaller well widths. In a narrower well, we also expect that the reduction in the spatial separation of the electron and hole states along the growth direction will lead to an overall increase in the Auger rate. We found here Auger coefficients smaller than the experimental values, in line with this expected trend. However, we noted the significant experimental uncertainties in determining the radiative and Auger recombination coefficients more generally. We also found that strongly localised hole states can lead to significantly enhanced Auger rates. Further studies of the impact of carrier localisation on the recombination processes in AlGaN-based QWs are required to shed light on the role of alloy-enhanced Auger in contrast to, for instance, defect-assisted Auger.

Having determined the temperature evolution of the radiative and nonradiative Auger recombination rates, we investigated the competition between these two processes via the $B/C$ ratio. In principle, we expect that an increase of the $B/C_{\text{tot}}$ ratio could correspond to an improvement in device performance. Experimentally it is found that device performance deteriorates as temperature is increased, including into the device operating temperature range of T > 300 K. If Auger recombination was a driving cause of thermal droop we would thus expect a decrease of $B/C_{\text{tot}}$. We found that the $B/C_{\text{tot}}$ ratio increases with increasing temperature at high temperatures (T > 300 K). Thus, our findings stand in contrast to experimental reports, indicating to us that the here considered Auger recombination process is not directly causing thermal droop. However, we note that Auger recombination could have an indirect impact on device performance due to the generation of high energy Auger carriers which may cause defects in the material, thus increasing nonradiative recombination pathways and hampering efficiency.