## 3 Results and Discussion
### Electronic Properties of MgBi\({}_{2}\)O\({}_{6}\)  
To study the electronic property of tetragonal MgBi\({}_{2}\)O\({}_{6}\), the electronic band structure along high symmetry directions in the Brillouin zone (BZ) and density of states are calculated. The band structure calculated by conventional DFT (dotted line in Figure 2) shows that there is a bit of overlap between the conduction band and valence band, indicating the metallic feature of MgBi\({}_{2}\)O\({}_{6}\). This contradicts the experimental findings. It is well known that DFT calculations do not take into account the effects of electron excitation and thus tend to underestimate the electronic band gap. To acquire the more accurate results, the calculations based on the HF PBE0 method are further performed. The HF corrected band structure (solid line in Figure 2) shows that MgBi\({}_{2}\)O\({}_{6}\) is a direct semiconductor with a band gap of about 1.62 eV, which is well consistent with the experimental measured data (1.6\(\sim\)1.8 eV) [18; 19; 20] and larger than the previously calculated value (1.10 eV) based on the HF method [21]. The valence band near the Fermi level (\(E_{\rm F}\)) is flat, while the dispersion of the conduction band close to the \(E_{\rm F}\) is relative strong. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Lattice Constants**} & \multicolumn{6}{c}{**Atomic Coordinates (Fractional)**} \\ \cline{2-6}  & **Atom** & **Site** & \(x\) & \(y\) & \(z\) \\ \hline \(a=4.920\) (4.826) & Bi & \(4e\) & 0 & 0 & 0.333 (0.332) \\ \(c=9.924\) (9.719) & Mg & \(2a\) & 0 & 0 & 0 \\  & O1 & \(4f\) & 0.302 (0.305) & 0.302 (0.305) & 0 \\  & O2 & \(8j\) & 0.309 (0.307) & 0.309 (0.307) & 0.336 (0.335) \\ \hline BiO\({}_{6}\) octahedron & & & MgO\({}_{6}\) octahedron & & \\ Bi-O(1) & 2.156 (2.114) \(\times\) 2 & Mg-O(1) & 2.107 (2.084) \(\times\) 2 & \\ Bi-O(2) & 2.153 (2.102) \(\times\) 2 & Mg-O(2) & 2.103 (2.073) \(\times\) 4 & \\ Bi-O(3) & 2.141 (2.084) \(\times\) 2 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimized structural parameters of MgBi\({}_{2}\)O\({}_{6}\), Bi-O and Mg-O interatomic distances. The experimental values [17] are listed in brackets for comparison. Figure 1: Crystal structure of the trirutile-type MgBi\({}_{2}\)O\({}_{6}\) represented as two different views. The effective mass (\(m^{*}\)) of carriers is an important parameter, because it determines the transfer and separation efficiency of electrons and holes, and directly influences the photophysical properties of semiconductors. \(m^{*}\) can be calculated as follows:  
\[\left(\frac{1}{m^{*}}\right)_{ij}=\frac{1}{\hbar^{2}}\frac{\partial^{2}E(k)}{ \partial k_{i}k_{j}},\ i,j=x,\ y,\ z, \tag{1}\]  
where \(E_{n}(k)\) represents the band energy, \(k\) is the wave vector, and \(\hbar\) is the reduced Planck constant. For MgBi\({}_{2}\)O\({}_{6}\), the effective electron mass (\(m_{e}\)\({}^{*}\)) at the conduction band and hole mass (\(m_{h}\)\({}^{*}\)) at the valence band at the \(\Gamma\) point are calculated and listed in Table 2. The calculated tensors of \(m^{*}\) clearly show that \(m_{e}\)\({}^{*}\) is fairly isotropic and \(m_{h}\)\({}^{*}\) shows a relatively strong anisotropy. \(m_{h}\)\({}^{*}\) along the [001] direction is larger than those along the [100] and [010] directions. The values of \(m_{e}\)\({}^{*}\) in Table 2 are also comparable to the previously calculated value (0.277) for MgBi\({}_{2}\)O\({}_{6}\)[21]. Moreover, the values of \(m_{h}\)\({}^{*}\) are distinctly larger than those of \(m_{e}\)\({}^{*}\), which indicates that the mobility of holes at the valence band is obviously slower than that of electrons at the conduction band. The big difference in mobility between electrons and holes is undoubtedly beneficial to the separation of charge carriers and the reduction of the recombination rate of electron-hole pairs.