## 2 Model
### Radiative Recombination Coefficient  
In order to calculate the optical matrix element \(M_{eh}\), the envelope function \(\psi(z)\) is approximated by the first two terms of its Taylor series in the vicinity of \(z_{0}\). Due to the even symmetry of the hole wave function, the second term does not contribute to the matrix element, giving:  
\[M_{eh}=4P_{t}\bigg{(}\frac{2\pi}{w_{e}a^{3}S}\bigg{)}^{1/2}\frac{\psi(z_{0})}{ \left(a^{-2}+k^{2}\right)^{3/2}} \tag{3}\]  
where \(P_{t}\) is the Kane's matrix element coupled S-like electron Bloch function \(u_{e}\) with X(Y)-like hole Bloch function \(u_{hh}\) (see, e.g., [37]). Expression (3) is valid, if \(\left(a/d\right)^{2}\ll 1\). The accuracy of the above expression may be further improved by accounting for an additional (third) term in the Taylor series expansion of the electron envelope function. This results in substituting \(\psi(z_{0})\) with \(\psi(z_{0})+a^{2}\psi^{\prime\prime}(z_{0})\) in Equation (3), where \(\psi^{\prime\prime}\) is the second derivative of the electron envelope wave function. In this case, the model applicability becomes determined by a less strict inequality: \(\left(a/d\right)^{4}\ll 1\). A conventional procedure has been applied to calculate RRC, which is similar to that applied to free electron/acceptor recombination without the \(\mathbf{k}\)-selection rule [38; 39]. First, using the Fermi's Golden Rule and accounting for the equilibrium photon population, the recombination rate of a localized hole with electrons in all the allowed quantum states is found. Then, the recombination rate is averaged over all possible positions \(z_{0}\) of the localized hole inside the QW. Finally, dividing the above recombination rate by the electron concentration provides the RRC. Essential simplification can be achieved by substituting in the calculations the momentum-dependent electron kinetic energy \(\hbar^{2}k^{2}/2m_{e}\) (\(\hbar\) is the Planck constant and \(m_{e}\) is the electron effective mass) with a mean electron energy \(E_{e}\). This energy is equal to \(kT\) (\(k\) is the Boltzmann constant and \(T\) is temperature) in the case of non-degenerate electrons and nearly equal to \(\frac{1}{2}F_{n}\) (\(F_{n}\) is the quasi-Fermi level position counted from the conduction band edge), if electrons are degenerate. In this case, the optical matrix element \(M_{eh}\) defined by Equation (3) becomes momentum-independent and summation over all possible states of electrons in the QW can be reduced to their sheet concentration. As a result, the following analytical expression can be obtained for the sheet RRC:  
\[B_{sh}=32\pi\
u_{B}a^{3}\xi^{3}\Big{\langle}\psi^{2}\Big{\rangle}/w_{e} \tag{4}\]  
Here, \(\
u_{B}=\frac{2}{3}\alpha n_{r}E_{g}E_{P}/\hbar m_{0}c^{2}\) is the specific frequency of optical transitions, \(\xi=m_{h}E_{L}/\left(m_{h}E_{L}+m_{e}E_{e}\right)\) is the dimensionless energy factor, \(E_{L}=\hbar^{2}/2m_{h}a^{2}\) is the hole localization energy, \(\alpha\) is the fine-structure constant, \(n_{r}\) is the group refractive index, \(E_{g}\) is the InGaN energy gap, \(E_{P}=2m_{0}P_{t}^{2}\) is the energy associated with the [PERSON]'s matrix element \(P_{t}\), \(m_{0}\) is the electron mass, \(c\) is the light velocity, and \(\left\langle\psi^{2}\right\rangle\) is the square of the electron envelope wave function averaged over all possible positions \(z_{0}\) of the localized hole. Using the material-dependent frequency \(\
u_{B}\) provides compact expressions for the RRCs not only for the case considered above, but also for radiative processes with \(\mathbf{k}\)-selection rules involving free electrons and holes in bulk materials and QWs [31]. Averaging of the magnitude of the electron wave function over possible positions of localized holes should account for their distribution function.