## 3 Results and Discussion
### Antiferroelectricity of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) Ceramics
After the magnetic field weakened to 0 and then increased in the reverse direction, the capacitance started to increase again, until the magnetic field increased to about \(-\)360 mT, and the capacitance increased to 43.4 nF again. After that point, the capacitance began to decrease with the increase in the H field. When the magnetic field increased to \(-\)1140 mT, the capacitance decreased to 34 nF, and then as the magnetic field went back to 0 mT, the capacitance further decreased to 30.2 nF. The varying track of the capacitance with the recycling H field looked like a hysteresis loop, which could be designated to be a magnetically induced electric polarization (_P_-_H_) hysteresis loop, which is similar to a _P_-_E_ or _M_-_H_ loop. The change in capacitance with the H field was caused by the magnetoelectric coupling effect, thus forming a pear-loop-shaped diagram as shown in Figure 10.  
Figure 9: Magnetic hysteresis loops of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) specimen. Since the giant magnetoresistance effect was discovered by [PERSON] et al. in 1988, the GMR effect has been widely used in hard disk drive (HDD) heads [50]. We measured the complex impedance of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) under different magnetic fields by using a Microtest 6630 LCR meter. We placed the pellet sample coated with silver electrodes on both surface sides in a space between two magnets, whose magnetic field was normal to the surface of the specimen. Then we measured its complex resistance, which changed with the H field under different frequencies. However, if the capacitor was not ideal, it will have parasitic parameters, such as equivalent series inductance ESL and equivalent series resistance \(ESR\), so the capacitor can be simplified as shown in Figure 4:0 So, the impedance of the actual capacitor can be expressed as:  
Figure 4:1  
The modulus of impedance is:  
Figure 4:2  
So, it can be seen from this formula that when the frequency is small, the complex impedance decreases as the frequency increases. This is because the current leads the voltage, which is a typical capacitor charging characteristic, and the capacitive reactance plays a major role, and it is also the part we need to test and deal with. When the frequency increases to about 1 MHz, the complex impedance has an increasing process. This process is due to the excessively high frequency, which causes the voltage to lead the current, which is a typical behavior of inductors applying voltage. When the complex impedance reaches the minimum, the capacitive reactance and the inductive reactance are offset to 0, and the phase of the complex impedance is 0, showing a pure resistance characteristic. We mainly discussed the behavior of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) in the frequency range where capacitive reactance dominated. Figure 4:3 Variation diagram of the capacitance of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) ceramics under different magnetic fields. The change in complex impedance with different \(B\) fields is shown in Figure 4:4 It can be seen that the complex impedance largely decreased with the increasing the magnetic field at a low frequency, while in the same magnetic field, the complex impedance sharply decreased with the frequency. At 10 Hz, the complex impedance under the 0 T magnetic field was 1682 \(\Omega\). As the magnetic field increased to 1.1 T, the complex impedance was reduced to 1065 \(\Omega\). The change rate of \(Z\) under M (\(\Delta Z=[Z(B)-Z(0)/Z(B)]\%\)) was \(-58\%\). This \(MR\) ratio is similar to the GMR magnitude of Pr\({}_{0.775}\)Sr\({}_{0.225}\)MnO\({}_{3}\). However, Pr\({}_{0.775}\)Sr\({}_{0.225}\)MnO\({}_{3}\) shows GMR with this ratio only below \(-73\)\({}^{\circ}\)C, while La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) demonstrated this GMR ratio at room temperature.