## 4 Modelling of Reflectance and Transmittance Spectra  
The reflectance and transmittance spectra of the samples were modelled using the transfer matrix method [17]. We considered the normal incidence of electromagnetic radiation considering the following layers: (0) vacuum, (1) barrier, (2) top part of the buffer (UID GaN), (3) bottom part of the buffer (GaN:C), (4) nucleation layer, (5) substrate and (6) vacuum. Layer sketches are presented in the insets of respective Figure 3 and Figure 6.  
Radiation propagation through the \(i\)-th layer (\(i\) = 1-5) is described by the propagation matrix \(\mathbf{P}_{i}\):  
\[\mathbf{P}_{i}=\left( \begin{array}{cc}e^{i\varphi_{i}}&0\\ 0&e^{-i\varphi_{i}}\\ \end{array} \right), \tag{1}\]  
where the phase shift \(\varphi_{i}\) of the electric field vector is given by  
\[\varphi_{i}=\frac{\omega}{c}\widetilde{n}_{i}d_{i}\;, \tag{2}\]  
\(\widetilde{n}_{i}\equiv n_{i}+ik_{i}\) and \(d_{i}\) are the complex refractive index and thickness of the layer, respectively. Complex dielectric permittivity \(\varepsilon_{i}\) of the \(i\)-th layer is related to \(\widetilde{n}_{i}\) by the Fresnel equation: \(\widetilde{n}_{i}^{2}=\varepsilon_{i}\). For a conducting interface (\(i\)+1)/\(i\) the transfer matrix \(\mathbf{D}_{i+1/i}\) was written as [18]  
\[\mathbf{D}_{i+1/i}=\frac{1}{2\widetilde{n}_{i+1}}\left( \begin{array}{cc} \widetilde{n}_{i+1}+\widetilde{n}_{i}-\frac{4\pi\varepsilon_{\mathrm{s}}}{c} &\widetilde{n}_{i+1}-\widetilde{n}_{i}-\frac{4\pi\varepsilon_{\mathrm{s}}}{c} \\ \widetilde{n}_{i+1}-\widetilde{n}_{i}+\frac{4\pi\varepsilon_{\mathrm{s}}}{c} &\widetilde{n}_{i+1}+\widetilde{n}_{i}+\frac{4\pi\varepsilon_{\mathrm{s}}}{c} \\ \end{array} \right), \tag{3}\]  
where \(\sigma_{\mathrm{s}}\) is the surface conductivity of the interface. For a non-conducting interface, Equation (3) can be used with \(\sigma_{\mathrm{s}}\) = 0. The transfer matrix \(\mathbf{M}\) for the whole structure is related to above matrices as  
\[\mathbf{M}=\mathbf{D}_{6/5}\mathbf{P}_{5}\mathbf{D}_{5/4}\mathbf{P}_{4}\mathbf{ D}_{4/3}\mathbf{P}_{3}\mathbf{D}_{3/2}\mathbf{P}_{2}\mathbf{D}_{2/1}\mathbf{P}_{1} \mathbf{D}_{1/0}, \tag{4}\]  
Figure 3: Reflectance of the samples, HBR (**a**) and QWR (**b**), at room temperature. Wavenumbers and corresponding photon energies are denoted as \(k\) and \(\hbar\omega\), respectively. Solid lines represent experimental data. Dashed lines demonstrate results of the spectra fitting at the best-fit parameters. Inset in panel (**a**) demonstrates the multi-layer model used for the spectrum simulation for the QWR sample. Inset in panel (**b**) shows the model used for the HBR sample. The numbers in brackets indicate the layers of the model described in Section 4.  
and the transmittance \(T\) and reflectance \(R\) can be found by  
\[T=\left|M_{11}-\frac{M_{12}M_{21}}{M_{22}}\right|^{2}, \tag{5}\]  
and  
\[R=\left|\frac{M_{21}}{M_{22}}\right|^{2}, \tag{6}\]  
respectively. Generally, the transmittance and reflectance spectra simulated using Equations (5) and (6) demonstrate the short-period oscillations due to multiple passes of radiation inside a thick substrate. Such F-P oscillations are clearly seen on experimental spectra in the case of sufficiently high resolution [8; 9; 19]. In our experiments, we used a reduced spectral resolution that eliminated F-P oscillations. To perform corresponding transformation of the simulated spectra we considered the substrate (layer 5) as an \"incoherent layer\" [17].