## 2 Model
### Localization Energy  
A key point of any theory considering composition fluctuations in a semiconductor alloy is finding a correlation between its composition and the capability of the alloy to localize either electrons or holes. A straightforward approach for ternary alloys, like In\({}_{x}\)Ga\({}_{x-1}\)N, implies selecting an averaging volume \(V_{a}\) centered at the point with the radius-vector \(\mathbf{r}\), counting the number of atoms of a certain kind, e.g., those of In, contained in the volume, and then determining a local alloy composition \(x(\mathbf{r})\) as the ratio of the number of In atoms to the total number of all cations in the volume \(V_{a}\). At the next step, spatial variations of the conduction and valence band edges, \(E_{C}(\mathbf{r})\) and \(E_{V}(\mathbf{r})\), are determined using the obtained \(x(\mathbf{r})\) and known relationships between the alloy composition, its bandgap, and band offsets. Finally, Schrodinger equations are solved with \(E_{C}(\mathbf{r})\) and \(E_{V}(\mathbf{r})\) serving as the potential energy profiles for electrons and holes and confined states of the carriers are found, providing their localization energy. Such an approach has been used, in particular, in [36], where the composition fluctuations were evaluated experimentally using atomic probe tomography.  
A pitfall of the above approach is the choice of the averaging volume. Indeed, in view of general statistical trends, larger \(V_{a}\) results in lower relative composition fluctuations and, eventually, in a weaker carrier localization. In order to avoid the uncertainty in the choice of \(V_{a}\), the averaging volume should be found self-consistently with the solution of the Schrodinger equations in such a way as to nearly correspond to the volume occupied by a carrier wave function in the localized state. The above self-consistence exists ab initio in the method of optimal fluctuation (see [43] for a review of the method and relevant results), which predicts that the density of localized states \(g(E)\) vary with energy \(E\) counted from the edge of either the conduction or valence band towards the mid of the bandgap as \(g(E)\propto E^{3/2}exp\big{(}-\sqrt{E/E_{L}}\big{)}\) with the localization energy:  
\[E_{L}=E_{t}x^{2}(1-x)^{2},\ E_{t}=\frac{\beta^{4}m_{eff}^{3}}{18\pi^{2}\hbar^{6}n _{cs}}, \tag{6}\]  
where \(\beta\) is the rate of the corresponding band edge variation with the alloy composition \(x\), \(m_{eff}\) is the carrier effective mass, and \(n_{cs}\) is the concentration of cation sites in the crystal lattice. Expression (6) for the energy \(E_{t}\) is valid if: (i) composition fluctuations have a thermodynamic character; (ii) they occur in an ideal alloy, i.e., that having negligible enthalpy of mixing; and (iii) the number of In atoms in the optimal fluctuation \(N=xtn_{cs}V_{a}\gg N^{1/2}\). Estimations show that the latter two assumptions are not satisfied in real InGaN compounds. First, the InGaN alloys are far from ideal, tending to phase separation due to a large lattice constant mismatch between InN and GaN. Second, the typical number of In atoms in the optimal fluctuation is about six to eight because of the atomistic character of hole localization. This makes the condition \(N\gg N^{1/2}\) invalid. Therefore, Equation (6) is used in this study only as a general relationship between the localization energy \(E_{L}\) and alloy composition \(x\), whereas \(E_{t}\) is regarded as a fitting parameter. A similar approach has been previously applied to the bulk InGaN alloys [31].