## 3 Results and Discussion  
### Recombination Coefficients  
In order to compare the theory with the experiment, the data on wavelength-dependent RRCs and ARCs from [20] have been chosen as those providing the best correlation with independently estimated IQEs of state-of-the-art LEDs (see [10] for a summary of the reported efficiencies). These data were preliminarily refined to account for the wavelength dependence of light extraction efficiency [44] of the SQW LEDs studied in [20; 21]. Here, the QW width \(d\) = 2.5 nm was found to best reproduce all experimental current densities at which the LEDs approached maximum IQE values. The points corresponding to the refined data for LED structures grown on sapphire and silicon substrates are plotted in Figures 2 and 3 by circles and squares, respectively. Figure 2: Averaged square of electron envelope function computed for [0001]-polar and nonpolar SQW LED structures and localization radius (**a**) and theoretical and experimental \(B/C\) ratio of polar SQW LEDs (**b**) as a function of wavelength. Symbols are data from [20; 21]; dashed, dash-dotted, and solid lines in (**b**) correspond to the electron energy \(E_{e}\) of 26, 80, and 40 meV, respectively. In the plot (**a**), there is an arrow pointing out to the right vertical axis with the corresponding title. In the plot (**b**), red, orange, and purple lines are mentioned as “solid”, “dashed”, and “dash-dotted” lines, respectively. To calculate RRC and ARC by Equations (4) and (5), the electron envelope function was simulated using the SiLENSe 5.11 package [45]. Figure 2a compares \(\left\langle\psi^{2}\right\rangle/w_{e}\) values computed for [0001]-polar and nonpolar SQW LED structures. One can see that they differ by a factor of 6/23 at 400/600 nm, which originates from the electron and hole spatial separation in polar InGaN QW due to the built-in polarization field, becoming ever stronger with the emission wavelength. The present model predicts the \(B/C\) ratio to be practically independent of the hole localization radius and to depend on the material properties of InGaN alloys only. This prediction is validated in Figure 2b comparing the experimental and theoretical \(B/C\) ratios calculated for \(E_{e}\) varied from 26 meV (thermal energy at room temperature) to 80 meV. One can see that the experimental points corresponding to LED structures grown on sapphire substrates agree well with the theory, whereas the data for the structures grown on silicon substrates lie slightly beyond the theoretical predictions. In order to get a closer correlation between the absolute values of experimental and theoretical RCs, the parameter \(E_{t}\) was changed from 1.85 eV recommended in [31] to 2.0 eV, corresponding to the localization energy variation from 24 meV at 420 nm to 96 meV at 550 nm. These values are comparable with localization energies reported in [33] and associated with two- and three-atomic In clusters. The wavelength dependence of the respective localization radius is shown in Figure 2a. Here, the condition \(\left(a/d\right)^{2}\ll 1\), at which Equations (3) and (4) are valid, is met at the wavelengths longer than \(\sim\)435 nm. At shorter wavelengths, down to \(\sim\)410 nm, the approximations provide an inaccuracy of less than \(\sim\)20%, which is within the scatter of the measured recombination coefficients (see Figure 3). At \(E_{t}\) = 2.0 eV, the localization radius \(a\) varies from 0.92 nm at 420 nm to 0.47 nm at 550 nm, as is shown in Figure 2. The sub-nanometer scale of the localization radius variation agrees well with the mean dimensions (nearly equal to 2\(a\)) of the hole wave functions obtained in [35; 46; 47] by ab initio calculations. Figure 3 compares experimental and theoretical RRC and ARC calculated for [0001]-polar SQW LED structure vs. emission wavelength. All the calculations were carried out with wavelength-dependent material parameters, like electron and hole effective masses, energy gaps, [PERSON]'s energy \(E_{P}\), etc.