## 4 Discussion  
When \(T_{\rm s}>155\) K, the shape of emission spectra changes significantly (see Figure 3B,C for example). It has a rising character when wavenumbers increase; however, the dips are observed at the energies of Reststrahlen regions in sapphire. Therefore, it seems that the emission signal reaching the detector is the product of a black-body spectral radiance \(B(\omega,T)\) and thermal emissivity \(\mathcal{E}(\omega,T)\) of the sample:  
\[S(\omega,T)=B(\omega,T)\cdot\mathcal{E}(\omega,T). \tag{1}\]  
According to Kirchhoff law, thermal emissivity is equal to sample absorptance \(\mathcal{E}(\omega,T)=\alpha(\omega,T)=1-t(\omega,T)-r(\omega,T)\), where \(t(\omega,T)\) and \(r(\omega,T)\) are the transmittance and reflectance of the sample. To determine \(t(\omega,T)\) and \(r(\omega,T)\) the numerical implementation of Rigorious Coupled-Wave Analysis (RCWA) method was used [15]. The sample itself was considered to stay in vacuum environment and its dielectric function \(\varepsilon_{\rm sl}(\omega)\) was modeled as a sum of dielectric  
Figure 2: (**A**) The temperature dependencies of pulsed emission signal \(S_{\rm E}\). (**B**) Pulsed current \(I_{\rm P}\) in the sample at 60 V (full symbols) and 90 V (open symbols) pump voltages. functions of each layer. To estimate the dielectric function of a single layer, the phononic term was involved, and taken in factorized form [16]:  
\[\varepsilon_{\text{ph}}(\omega)=\varepsilon_{\infty}\prod_{i=1}^{l}\frac{\omega_{ \text{TO},i}^{2}-\omega^{2}-j\omega\gamma_{\text{TO},i}}{\omega_{\text{LO},i}^ {2}-\omega^{2}-j\omega\gamma_{\text{LO},i}}. \tag{2}\]  
Here, \(\varepsilon_{\infty}\) denotes high frequency dielectric permittivity, \(\omega_{\text{TO(LO)}}\) and \(\gamma_{\text{TO(LO)}}\) - TO(LO) phonon energy and damping factor, respectively. The multiplication operation is performed over the number \(l\) of those IR-active optical phonon modes which create the dipole moment normal to \(c-\)axis (\(E_{l}\) symmetry). The temperature dependent values of \(\omega_{\text{TO(LO)}}\) and \(\gamma_{\text{TO(LO)}}\) for sapphire were taken from [16] and \(\varepsilon_{\infty}=3.08\). In case of GaN, however, damping weakly depends on temperature in the interval from 85 K to 200 K [17], and \(\gamma_{\text{TO}}=\gamma_{\text{LO}}=8\) cm\({}^{-1}\) was kept, whereas \(\varepsilon_{\infty}=5.35\). Two-dimensional electron thermal emission was not included in the calculations, as the intensity of such emission is at least a few orders of magnitude lower than the black-body radiation in a spectral range in which the experiment was conducted [18]. The black-body radiation of the sample was calculated using the parameters given above. The black-body spectral intensity is estimated as a difference of spectral intensity at a sample temperature at pumping pulse and the cryostat cold finger temperature: \(B(\widetilde{\
u},T)=B(\widetilde{\
u},T_{\text{p}})-B(\widetilde{\
u},T_{ \text{b}})\). An approximate value of \(T_{\text{p}}\) was found from sample resistance dependence on temperature (see Figure 1B). \(T_{\text{c}}=121\) K and 131 K corresponds to \(T_{\text{s}}=185\) K and 193 K when \(U_{\text{p}}=90\) V, and to \(T_{\text{s}}=162\) K and 167 K when \(U_{\text{p}}=60\) V. Theoretically calculated emission intensity is presented as solid curves in Figure 3. As it was assumed, its shape agrees well with the experimentally observed results, when the sample temperature is above 155 K. In addition, the dips in the Reststrahlen bands of sapphire are resolved in the spectra at 384-387 cm\({}^{-1}\) (l = 1 sapphire optical phonon mode) and 440-482 cm\({}^{-1}\) (l = 2). Also, the slope at 200-384 cm\({}^{-1}\) coincides with experimental results, therefore a good agreement between experimental and calculated data shows that at \(T_{\text{s}}>155\) K THz emission is caused by thermal radiation. Calculations were performed for the sapphire wafer only, and comparison showed, that the main contribution to emission spectra comes from sapphire substrate.