\begin{appendices}
\chapter{Determination of on-site energies}
\label{chp:AppendixDeterminationOfOnsiteEnergies}

In this appendix we provide further information on the Coulomb matrix element calculations and the impact of the on-site energy, $W_0$, on the Auger rates. In the following we focus our attention on the general form of the Coulomb matrix elements,
\begin{eqnarray}
    M = \frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{r}} \int \int d^{3}\tilde{r}' d^{3}\tilde{r} \frac{\psi_{1}^{*}(\mathbf{\tilde{r}}')\psi_{2}(\mathbf{\tilde{r}}')\psi_{3}^{*}(\mathbf{\tilde{r}})\psi_{4}(\mathbf{\tilde{r}})}{|\mathbf{\tilde{r}} - \mathbf{\tilde{r}'}|} \,\, .
    \label{eq:CME_App}
\end{eqnarray}
Here, each $\psi(\mathbf{\tilde{r}})$ is a TB wave function, which are given by Eq.~(\ref{eq:TBWF}), i.e., by linear combinations of localised, atomic-like orbitals, $\phi_{\mathbf{R},\sigma}(\mathbf{\tilde{r}})$, $\sigma=\left\{s,p_x,p_y,p_z\right\}$, located at the lattice sites, $\mathbf{R}$. In the following we make the two-centre approximation discussed in detail in Ref.~\cite{ScSc2006}, so that the Coulomb matrix element, $M$, becomes:
\begin{eqnarray*}
    \nonumber
    M & \approx & \frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{r}}\sum_{\mathbf{R},\mathbf{R}'}\sum_{\alpha,\beta,\gamma,\delta} c^{*}_{\mathbf{R}',\alpha}c_{\mathbf{R}',\delta}c^{*}_{\mathbf{R},\beta}c_{\mathbf{R},\gamma}\\
    & & \times\int\int d^3r'd^3r\frac{\phi^*_{\mathbf{R}',\alpha}(\mathbf{r}')\phi_{\mathbf{R}',\delta}(\mathbf{r}')\phi^*_{\mathbf{R},\beta}(\mathbf{r})\phi_{\mathbf{R},\gamma}(\mathbf{r})}{|\mathbf{R}+\mathbf{r}-\mathbf{R}'-\mathbf{r}'|}\, .
\end{eqnarray*}
Here, we have decomposed the positions $\mathbf{\tilde{r}}$ and $\mathbf{\tilde{r}}'$ into the positions $\mathbf{R}$ and $\mathbf{R}'$ of the lattice sites and the positions $\mathbf{r}$ and $\mathbf{r}'$ inside the local tetrahedron of the lattice site. Due to the localised nature of the atomic-like orbitals, $\phi_{\mathbf{R},\sigma}(\mathbf{r})$, when the lattice sites at which the orbitals are localised are far apart, the exact structure of these $s$- and $p$-like orbitals is of secondary important. In doing so, one ends up with the approximation given in Eq.~(\ref{eq:CME}). 

However, when the situation $\mathbf{R}=\mathbf{R}'$ arises, this on-site energy needs to be treated carefully. In principle, the specific form of the atomic like basis states needs to be known. To this end, we now employ the often used approach~\cite{LeJo2001,LeKi2002} to approximate $\phi_{\mathbf{R},\sigma}(\mathbf{r})$ via Slater orbitals~\cite{Slat1930}, which can be written, in general, as:
\begin{equation}
    \phi_n(\mathbf{r})=R_n(r)Y_n(\theta,\phi)\,\, ,
\end{equation}
where the index $n$ denotes the orbital type, $Y_{n}(\theta,\phi)$ are spherical harmonics, and $R_{n}(r)=Dr^ae^{-br}$ are the radial parts; the constants $a$ and $b$ are given by Slater's rules~\cite{Slat1930} and $D$ is a normalisation constant. Given that we are evaluating on-site energies, and without loss of generality, we assume $\mathbf{R}=\mathbf{R}'=\mathbf{0}$. Equipped with these orbitals, in this second step, we expand the Coulomb potential in spherical harmonics~\cite{Jackson1999}:
\begin{eqnarray*}
    \frac{1}{|\mathbf{r} - \mathbf{r}'|} = \sum_{l=0}^{\infty} \frac{4\pi}{2l+1}\frac{r_{<}^{l}}{r_{>}^{l+1}} \sum_{m=-l}^{l} Y_{lm}^{*}(\mathbf{r}')Y_{lm}(\mathbf{r}) \,\, .
\end{eqnarray*}
Here, $r_{>}$ ($r_{<}$) are the length of the larger (smaller) of the vectors $\mathbf{r}$ and $\mathbf{r}'$. Using the above, and Eq.~(\ref{eq:TBWF}), the Coulomb matrix element, $M$, becomes, for the case of $\mathbf{R}=\mathbf{R}'=\mathbf{0}$:
\begin{eqnarray*}
    M & = & \frac{e^{2}}{4\pi\epsilon_{0}}\sum_{l,\{n\}}c^{*}_{n_1}c_{n_2}c^{*}_{n_3}c_{n_4}\int dr'\int dr r'^2R_{n_1}(r')R_{n_2}(r')r^2R_{n_3}(r)R_{n_4}(r)\\
    & & \times\frac{r_{<}^{l}}{r_{>}^{l+1}} \frac{4\pi}{2l+1}\sum_{m=-l}^{l}K_{n_1,n_2,l,m}K_{n_3,n_4,l,m}\,\, .
\end{eqnarray*}
The coefficients $K_{n,n',l,m}$ are the so-called Gaunt coefficients~\cite{Xu1996}. We define now:
\begin{eqnarray*}
    M & = & \sum_{\{n\}}c_{n_{1}}^{*}c_{n_{2}}c_{n_{3}}^{*}c_{n_{4}}\sum_{l}\mathcal{K}^l_{n_1,..,n_4}\times \mathcal{I}^l_{n_1,..,n_4}\\
    & = & \sum_{\{n\}}c_{n_{1}}^{*}c_{n_{2}}c_{n_{3}}^{*}c_{n_{4}}W_0\,\, .
\end{eqnarray*}
with
\begin{equation}
    \label{eq:Gaunt}
    \mathcal{K}^l_{n_1,..,n_4} = \frac{4\pi}{2l+1}\sum_{m=-l}^{l}K_{n_1,n_2,l,m}K_{n_3,n_4,l,m}\, ,
\end{equation}
and
\begin{equation}
\label{eq:radialpart}
    \mathcal{I}^l_{n_1,..,n_4} = C\int dr' r'^{2} R_{n_{1}}^{*}(r')R_{n_{2}}(r')\int dr r^{2} R_{n_{3}}^{*}(r)R_{n_{4}}(r)\frac{r_{<}^{l}}{r_{>}^{l+1}}\, ,
\end{equation}
with $C=e^{2}/(4\pi\epsilon_{0})$. The values for $\mathcal{K}^l_{n_1,..,n_4}$, Eq.~(\ref{eq:Gaunt}), can be obtained from the properties and values of the Gaunt coefficients. The radial part, Eq.~(\ref{eq:radialpart}), can be evaluated by integrating over a sphere of radius $s$, that has the same volume as a local tetrahedron in the wurtzite crystal~\cite{ScCz2002}:
\begin{equation*}
\begin{split}
    \mathcal{I}^l_{n_1,..,n_4} = & C\left(\int_{0}^{s}dr (r)^{l-1}R_{n_{3}}^{*}(r)R_{n_{4}}(r) \int_{0}^{r} dr'(r')^{2+l}R_{n_{1}}^{*}(r')R_{n_{2}}(r') \right.\\
    & \left. +\int_{0}^{s}dr (r)^{2+l}R_{n_{3}}^{*}(r)R_{n_{4}}(r) \int_{r}^{s}dr' (r')^{l-1}R_{n_{1}}^{*}(r')R_{n_{2}}(r')\right)\,\, .
\end{split}
\end{equation*}
We note that based on the Slater rules~\cite{Slat1930} and the fact that we are considering wave functions at the same lattice site, the radial part will be identical for the four wave functions involved in the above integral~\cite{ScSc2006}. 

This derivation for evaluating the on-site Coulomb energies allows us to now proceed as follows. Given that the underlying TB framework is an $sp^3$ TB model, and that when using Slater orbitals the radial function of the localised orbitals is the same, we restrict the spherical expansion to $l=0$ ($s$-like) contributions. Furthermore, we assume that the wave functions are constant over the sphere around the considered lattice site. In doing so, we find that 
\begin{eqnarray*}
    W_{0} = \frac{e^{2}}{4\pi\epsilon_{0}}\frac{9}{s} \frac{2}{15} \,\, .
\end{eqnarray*}
Evaluating $W_0$ for InN, we find that $W^\text{InN}_{0} \approx 16.5$ eV, while for GaN, given the smaller lattice constant compared to InN, we find $W^\text{GaN}_{0} \approx 18$ eV. While further refinements can be made in the evaluation of the on-site energies, these numbers are in good agreement with other calculations of unscreened on-site Coulomb matrix elements~\cite{ScCz2002}.

The above numbers for the on-site energy are determined for pure InN and GaN. In an (In,Ga)N/GaN QW, due to alloy fluctuations and strain effects, the volume of the local tetrahedron will vary. To test the impact of the on-site energy on the Auger rate, we have performed calculations using $W^\text{InN}_{0}$ and $W^\text{GaN}_{0}$ at low (T = 10 K) and high temperatures (T = 300 K). These test calculations revealed that the $eeh$ Auger coefficient increases by less than 3\% at temperatures of 10 K and 300 K when using $W^\text{GaN}_{0}$ instead of $W^\text{InN}_{0}$; the $hhe$ Auger coefficient increases by less than 13\% at the same temperatures again when using $W^\text{GaN}_{0}$ instead of $W^\text{InN}_{0}$. This analysis shows that precise knowledge of the on-site energy is of secondary importance and that the Auger rates are primarily affected by the long range character of the Coulomb interaction. For all calculations in the main part of the manuscript we have used $W^\text{InN}_{0}$, given that hole wave functions are strongly localised near In-N-In chains~\cite{KeZu2001}. Furthermore, the value $W^\text{InN}_{0}$ is expected to represent a lower bound on the on-site energies and thus also for the Auger rates. Larger $W_0$ values will, in general, result in a larger Auger rate compared to values obtained in the main text. 
\end{appendices}