## 3 Results and Discussion
### Antiferroelectricity of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) Ceramics
It can be seen that the absorption coefficient increased sharply within the wavelength range of 750 to 600 nm, reflecting the direct electron transition process from the valence band to conduction band. One stair step with a sharp change in the absorbance indicated that La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) is a direct bandgap semiconductor. The Tauc plot was used to fit the absorption spectrum so as to obtain the forbidden band width of the semiconducting La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) compound. The method is based on the formula proposed by [PERSON], [PERSON], and [PERSON] et al., which is as expressed as follows:  
\[\left(\alpha hv\right)^{\
icefrac{{1}}{{n}}}=A\left(hv-E_{g}\right)\]  
where \(\alpha\) is the absorption coefficient, \(h\) is [PERSON]'s constant, \(v\) is the frequency, A is a constant, and \(E_{g}\) is the semiconductor band gap. When the semiconductor is a direct bandgap semiconductor, \(n=1/2\). After processing the absorption spectrum data according to this formula, the data were plotted with \(\left(\alpha hv\right)^{\
icefrac{{1}}{{n}}}\) as the y-axis and \(hv\) as the x-axis. By linearly fitting the high-slope straight line segment in the figure and extending the fitted line to intersect the x-axis, the intersection is then determined to be the forbidden band width (band gap energy) of the La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) semiconductor. The final calculation result is shown in Figure 8B. In this way, the forbidden band width or the band gap energy of the La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) ceramic was experimentally determined to be 1.753 eV. For the magnetic measurement, the La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) powder was annealed by oxygen treatment for a total of 15 h in three steps. The magnetization strength M was then measured by a Vibrating Sample Magnetometer (VSM) in relation to the magnetic field strength H. The measurement results are shown in Figure 9, exhibiting a standard M-H hysteresis loop, indicating that La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) has a strong ferrimagnetic property. Upon the loop, the saturation magnetization \(M_{S}\) was estimated to be 56.402 emu/g, the remnant magnetization \(M_{r}\) was 23.593 emu/g, while the coercivity force \(H_{C}\) was 1063.9 Oe. Figure 8: (**A**) UV-vis-NIR optical spectrum of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) ceramics. (**B**) Tauc plot of \(\left(\alpha hv\right)^{2}\) versus _hv_ for La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) ceramics. This combination of ferrimagnetism and antiferroelectrics coexisting in one single phase suggests that La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) is a novel type of multiferroic candidate, exhibiting multiple functional responses to external excitation. Magnetoelectric Coupling and GMR Effect of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) Ceramics  
First of all, we measured the electrical polarization behavior of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) ceramics under different magnetic fields. The ceramic pellet was placed in a magnetic field whose direction was perpendicular to its surface, and then the magnetic field strength was continuously increased to 1140 mT in a step of 20 mT. After maintaining the magnetic field strength for about 30 s at each stair, the dielectric parameters were output through the LCR meter to the computer. The capacitance varying with the magnetic fields are shown in Figure 10. It can be seen from this figure that when the magnetic field (H) was increasing in the positive direction, the capacitance was increasing accordingly, and the capacitance reached 33.4 nF when the H field increased to 1140 mT. Afterwards, as the magnetic field reversely reduced down to about 360 mT, the capacitance further increased to 42.8 nF. Then, the capacitance began to decrease when the H field further reversely decreased.