## 3 Numerical Results and Discussion
### Dependence of the Band Gap Energy on Temperature, SPI and EPI  
The temperature dependence of the band gap is one of the most fundamental properties for semiconductors and has a strong influence on many applications. Let us emphasize that as the temperature increases, the band gap energy of the most semiconductors de creases. But, for example, in lead chalcogenides (PbS, PbSe, PbTe), the direct narrow band gap increases with increasing temperature [35]. Characteristic physical properties of the FMS, for example, the europium chalcogenides mainly come from the s-f interaction from the localized 4f electrons distributed between the conduction band (5d orbitals of Eu) and the valence band (2p (O\({}^{2-}\)) or 3p (S\({}^{2-}\)) orbitals). At first, we calculated the temperature dependence of the band gap energy \(E_{g}\) of FMS for the case \(I>0\), which is appropriate, for example, for EuO with a band gap of 1.12 eV. The results for \(E_{g}(T)\) are presented in Figure 1. It can be seen that \(E_{g}\) decreases with increasing temperature \(T\). The renormalization of the band gap at finite temperatures is due to the lattice expansion and the phonon-induced atomic vibrations. This is related to the renormalization of the magnetic interaction between the magnetic moments due to the SPI and the change of \(n_{\uparrow}-n_{\downarrow}\) when changing the EPI. The band gap is the energy difference between the valence band and the conduction band of a semiconductor. The band gap tends to decrease with increasing temperature because the atomic vibrations increase and the interatomic spacing becomes larger. The interaction between the phonons and the free electrons and holes also affects the band gap. It can be seen from Figure 1 that there is a small kink at the phase transition temperature \(T_{C}=69\) K due to the anharmonic spin-phonon interaction \(R\). For \(R=0\), the kink disappears. Both interactions--EPI and SPI--lead to reducing the band gap energy. The effect of the SPI is largest at low temperatures and vanishes at \(T_{C}\) because the magnetization is zero at \(T_{C}\), i.e., the magnetic ordering cannot influence the band gap (see Equation (5) for \(J^{eff}\)), whereas the EPI contributes also above \(T_{C}\). [PERSON] et al. [36] reported that the band structure of EuO and EuS is altered by magnetic ordering and is usually interpreted as conduction band splitting of spin-up and -down electrons. Moreover, we studied the temperature dependence of the band gap energy \(E_{g}\) for the case of \(I<0\), which can be appropriate for the antiferromagnetic semiconductor UTe. From Figure 2, it can be seen that the band gap \(E_{g}\) decreases again with increasing temperature. But there is a competition between the SPI and EPI. With increasing the SPI, the band gap energy \(E_{g}\) increases, whereas increasing the EPI leads to reducing \(E_{g}\). The influence of the SPI vanishes again at the phase transition temperature \(T_{N}=106\) K, whereas that of the EPI remains finite above \(T_{N}\). Unfortunately, there are no experimental data for UTe. Let us emphasize that in the antiferromagnetic semiconductor, MnTe is reported by [PERSON] et al. [37] to have an enhanced band gap \(E_{g}\) with increasing the SPI. In [15], we investigated the anharmonic spin-phonon and phonon-phonon interaction effects on optical phonon modes and spin wave excitations in FMS. It is shown that in order to explain the softening of the phonon mode, the SPI constant \(R\) must be negative, whereas for the hardening, \(R\) is positive. Therefore, we calculated the temperature dependence of the band gap energy \(E_{g}\) also for \(R<0\) for the two compounds EuO with \(I>0\) and UTe with \(I<0\). The results are presented in Figure 3 and Figure 4, respectively. It can be seen  
Figure 1: Temperature dependence of the band gap energy \(E_{g}\) of EuO with \(J=0.0001\) eV, \(I=0.2\) eV, \(S=7/2\), \(T_{C}=69\) K for different \(R>0\) and \(A\) values.