## 3 Results and Discussion
The latter occurs despite the fact that both approaches have been developed for modeling trap levels that are strongly coupled with the lattice and, in principle, should give consistent results. Some possible explanations for this discrepancy are proposed below. In general, our observations on the \(\alpha_{\text{e}}^{0}\) data fitted with the Lucovsky model align with the model basis, where the potential of a deep trap center is of the ion core type or can be approximated as a delta function, neglecting any long-range Coulomb effects. This model is particularly relevant for hydrogenic-type defect levels, which are generally shallow and located near band edges. When a defect's wavefunction is delocalized, the electron density at the defect remains low, resulting in minimal influence on the local rearrangements of surrounding atoms, as observed for a trap with Ec-3.33 eV that is located just 70 meV above the valence band edge (V\({}_{\text{B}}\)). Therefore, it is entirely reasonable to classify this trap level as shallow. Consequently, the wavefunctions of midgap states are strongly localized, and the Lucovsky model is not applicable in this case. However, the E\({}_{\text{T}}\) determined from this model can still be used as a rough approximation. Furthermore, since the Lucovsky model assumes a purely electronic character for the ionization process, the dependence of the \(\alpha_{\text{e}}^{0}\) spectrum on temperature should remain unaffected. This is a distinctive feature of the Lucovsky model and can be easily verified to provide unambiguous proof of whether the trap level can be described with this model or not. Unfortunately, we could only carry out DLOS measurements at 293 K.  
In order to explain the discrepancies in the deep trap parameters obtained from the Chantre-Bois and Passler models, one should reconsider how electron-phonon coupling effects are implemented in both models. In the Passler model, to quantify the lattice-phonon interaction, the concept of an effective phonon energy, \(\varepsilon=\hbar\omega\), was proposed [7]. Therefore, \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model/Trap Concentration**} & \multicolumn{3}{c}{**Ec-2.05 eV**} & \multicolumn{3}{c}{**Ec-2.75 eV**} & \multicolumn{3}{c}{**Ec-3.33 eV**} \\ \cline{2-10}  & **E\({}^{0}\) (eV)** & **E\({}_{\text{T}}\) (eV)** & **d\({}_{\text{FC}}\) (eV)** & **E\({}^{0}\) (eV)** & **E\({}_{\text{T}}\) (eV)** & **d\({}_{\text{FC}}\) (eV)** & **E\({}^{0}\) (eV)** & **E\({}_{\text{T}}\) (eV)** & **d\({}_{\text{FC}}\) (eV)** \\ \hline Lucovsky & 2.05 & 2.05 & 0 & 2.75 & 2.75 & 0 & 3.33 & 3.33 & 0 \\ \hline Chantre–Bois & 2.28 & 1.88 & 0.4 & 3.07 & 2.42 & 0.65 & 3.35 & 3.35 & 0.002 \\ \hline [PERSON] & 2.14 & 1.99 & 0.15 & 2.91 & 2.61 & 0.3 & 3.33 & 3.33 & 0.001 \\ \hline Trap concentration (cm\({}^{-3}\)) & & \multicolumn{3}{c}{\(3\times 10^{14}\)} & \multicolumn{3}{c}{\(4\times 10^{15}\)} & \multicolumn{3}{c}{\(1\times 10^{16}\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of deep trap state parameters determined from Lucovsky, Chantre–Bois, and Passler models fitted to \(\alpha_{\text{e}}^{0}\) data obtained from DLOS spectra shown in Figure 5a–d. a magnitude of \(\varepsilon\) should be precisely determined and then used in the data fitting. In GaN, the phonon energy spectrum is generally known; however, it is rather broad and consists of low- and high-energy components, from 0 to 90 meV, as previously reported [26; 27]. If we assume that both components contribute equally, a good approximation could be \(\varepsilon\sim\) 45-50 meV, as used by [PERSON] in his original work [7]. However, it is highly recommended to check the calculations with other \(\varepsilon\)-values to ensure that this parameter has been correctly chosen. In our calculations, the [PERSON] model with \(\varepsilon=\) 70 meV provided the best agreement between the theory and the data and, thus, was used in the data fitting presented in Figure 5a-d.