## 4 Modelling of Reflectance and Transmittance Spectra
Namely, Equations (5) and (6) were modified as:  
\[T=\frac{1}{2\pi}\int_{0}^{2\pi}\left|M_{11}^{\prime}(\delta)-\frac{M_{12}^{ \prime}(\delta)M_{21}^{\prime}(\delta)}{M_{22}^{\prime}(\delta)}\right|^{2}d \delta\, \tag{7}\]  
and  
\[R=\frac{1}{2\pi}\int_{0}^{2\pi}\left|\frac{M_{21}^{\prime}(\delta)}{M_{22}^{ \prime}(\delta)}\right|^{2}d\delta\, \tag{8}\]  
with the following changes of the Equations (1) and (4):  
\[\mathbf{M}^{\prime}(\delta)=\mathbf{D}_{6/5}\mathbf{P}_{5}^{\prime}(\delta) \mathbf{D}_{5/4}\mathbf{P}_{4}\mathbf{D}_{4/3}\mathbf{P}_{3}\mathbf{D}_{3/2} \mathbf{P}_{2}\mathbf{D}_{2/1}\mathbf{P}_{1}\mathbf{D}_{1/0}, \tag{9}\]  
and  
\[\mathbf{P}_{5}^{\prime}(\delta)=\left( \begin{array}{cc}e^{i(\varphi_{5}+ \delta/2)}&0\\ 0&e^{-i(\varphi_{5}+\delta/2)}\end{array} \right), \tag{10}\]  
-respectively. In such a way, we average the total transmittance and reflectance over all values of a phase \(\delta\) inside layer 5. Such calculation is appropriate when the double substrate thickness exceeds the coherent wavelength inside a material defined as \(2d\varsigma>(n_{5}\Delta k)^{-1}\), here \(\Delta k\) is the spectral resolution in wavenumbers. It should be noted that this condition was fulfilled over the entire spectrum range of interest. In our model, the barrier (layer 1) represented a stack of four upper layers in a real HEMT structure. The total thickness of the stack \(d_{1}=22.8\) nm was much less than the wavelength of THz radiation, enabling us to consider the stack as homogeneous media with an effective complex dielectric permittivity. Its magnitude \(\varepsilon_{1}\) is close to the dielectric constant of Al\({}_{0.25}\)Ga\({}_{0.75}\)N which is described as  
\[\varepsilon_{\text{AIA\'{e}AN}}=0.25\varepsilon_{\text{AIN}}+0.75\varepsilon_ {\text{GaN}}. \tag{11}\]  
Spectral dependences of AlN and GaN dielectric permittivities on the right side of Equation (11) are evaluated in one-phonon-resonance approximation:  
\[\varepsilon(\omega)=\varepsilon_{\infty}+\frac{(\varepsilon_{0}-\varepsilon_ {\infty})\omega_{\text{TO}}^{2}}{\omega_{\text{TO}}^{2}-\omega^{2}-i\gamma \omega}, \tag{12}\]  
where \(\varepsilon_{0}\) and \(\varepsilon_{\infty}\) are the low- and high-frequency dielectric constants, respectively, \(\omega_{\text{TO}}\) and \(\gamma\) are the oscillation frequency and the damping factor of TO phonons, respectively. The numerical values of these parameters were taken from literature [19; 20; 21] and found by fitting experimental data and are summarized in Table 2.  
The UID GaN (layer 2) and the high resistivity GaN:C buffer (layer 3) were described by Equations (1), (2) and (12). It is known that the lattice quality of the bottom part of a GaN buffer can be rather low due to lattice mismatch against the substrate material which results in a variation of the phonon damping factor \(\gamma\) along the growth direction of the structure [22]. For this reason, we introduced different damping factors \(\gamma_{2}\) and \(\gamma_{3}\) for the layers 2 and 3, respectively. The AlN nucleation layer was described by Equations (1), (2) and (12) using the values of \(\varepsilon_{0}\), \(\varepsilon_{\infty}\) and \(\omega_{\text{TO}}\) listed in Table 2. The damping factor of this layer \(\gamma_{4}\) was used as fitting parameter. The elements of the propagation matrix \(\mathbf{P}_{5}\) describing the substrate were calculated using the experimental data on a complex refractive index of 6H-SiC published recently in Refs. [23; 24]. The heterostructure under investigation includes one conducting interface only. This is the barrier/buffer interface. Its transfer matrix \(\mathbf{D}_{2/1}\) is determined by the conductivity of 2 DEG at the interface which was simulated using the Drude model:  
\[\sigma_{\text{s}}(\omega)=\frac{e^{2}N_{\text{s}}\tau}{m_{\text{e}}}\frac{1}{ 1-i\omega\tau}\;, \tag{13}\]  
where \(N_{\text{s}}\) is 2 DEG concentration, \(m_{e}\) is effective mass of 2D electrons and \(\tau\) is their relaxation time determined by the mobility: \(\tau=m_{\text{e}}\mu/e\).