## 3 Results and Discussion
### Optical Properties of MgBi\({}_{2}\)O\({}_{6}\)  
In order to understand the optical performance of semiconducting MgBi\({}_{2}\)O\({}_{6}\), we calculated its dielectric functions using the hybrid functional PBE0 method, and further computed its optical properties such as complex refractive index \(n\), extinction coefficient \(k\), reflectivity \(R\), absorption coefficients \(\alpha\), and electron energy-loss function \(L\). The calculated dielectric functions are depicted in Figure 5. It can be clearly seen that the static dielectric function is found to be 4.1 at 0 eV. With the increase of the polarization intensity and energy, the dielectric function, the real part \(\epsilon_{1}\) is also gradually increased and reaches a maximum value of 5.4 when the energy value is about 2.8 eV. When the photon energy reaches 1.6 eV, which is the width of electronic band gap, the electrons in the valence bands begin to excite and transit to conduction bands. As the carrier concentration increases, the degree of polarization decreases and the dielectric function decreases slightly. With the further increment of photon energy, the dielectric function values begins to fluctuate, which corresponds to the change of carrier concentration in the crystal. The imaginary part \(\epsilon_{2}\) reflects the transition between occupied and non-occupied electrons and can be used to characterize the light absorption behavior of the crystal. The first two peaks of the imaginary part of the dielectric function \(\epsilon_{2}\) are, respectively, at about 2.0 and  
Figure 4: Calculated electron localization function for (110) plane across Bi, Mg, and O atoms.  
3.0 eV, which probably results from the electron transition from the top of the valence bands to the bottom of the conduction bands.  
The calculated refractive index \(n\), extinction coefficient \(k\), absorption coefficient \(\alpha\), reflectivity \(R\), and electron energy loss function \(L\) of MgBi\({}_{2}\)O\({}_{6}\) are shown in Figure 6. As presented in Figure 6a, the higher value of n is in the energy range of 1.5-5.8 eV and the corresponding wavelength range is from 214 to 826 nm, indicating that MgBi\({}_{2}\)O\({}_{6}\) has a strong refractive effect in both ultraviolet (UV) and visible light region. The value of k rapidly starts to increase from 2.0 eV, also showing a response in the visible and UV light region. The calculated \(\alpha\) in Figure 6c shows that the light absorption edge is about 1.6 eV and is comparable with the band gap calculated by the PBE0 method. The value of optical absorption edge also agrees with literatural values [18; 20]. With the increase of energy, \(\alpha\) also gradually increases and a series of absorption peaks appear in the energy range from 1.6 to 27 eV. Combining with the calculated dielectric function and density of states, we can find that the first two absorption peaks at 4.5 and 6.0 eV are probably related to the electron migration of the O-2\(p\), Bi-6s, Bi-5\(d\), and Mg-2\(p\) states. The reflectivity \(R\) and the electron energy loss function \(L\) can be used to represent the resonant frequency of the incident light and the resonant frequency of the plasma. As shown in Figure 6d, the average value of \(R\) is about 13.8%, indicating that MgBi\({}_{2}\)O\({}_{6}\) can be used as a light absorbing material. The calculated \(L\) presented in Figure 6e completely locates in the continuous energy range of 0-45 eV, indicating that the characteristics of plasma oscillation in MgBi\({}_{2}\)O\({}_{6}\) are not obvious. This is strongly different from the behavior appearing in Bi\({}_{2}\)Sn\({}_{2}\)O\({}_{7}\)[31].  
Figure 5: Calculated complex dielectric function of MgBi\({}_{2}\)O\({}_{6}\).