## 3 Results and Discussion
### Antiferroelectricity of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) Ceramics
\(R_{s}\) is the lead resistance, _ESL_ is the equivalent series inductance, \(R_{e}\) is the dielectric loss resistance, and \(C\) is the ideal capacitance. Figure 5:2 The Core–Core circle formed by the real and imaginary parts of the impedance as the \(M\)-field changed. To derive the maximum value of \(MR\), we treated \(MR\) as a function of frequency. Since the value of the complex impedance was very small as the frequency was larger than 300 kHz, and the inductive reactance in the capacitor gradually had a greater impact on the complex impedance, we chose to deal with the change in \(MR\) with a frequency below 300 kHz, and the result is shown in Figure 5:3 It can be seen that the \(MR\) increased with the enhancement of the magnetic field, and when the frequency was around 36.7 kHz, the \(MR\) reached a maximum of \(-110.6\%\) at a magnetic field of 1.1 T. However, when the frequency was increased to 40 kHz, the \(MR\) ratio gradually became smaller. This feature indicates that the response of the complex impedance to frequency was sensitive. In order to find the direct relationship between the complex impedance and magnetic field strength, we made a graph of the \(MR\) ratio versus magnetic field strength at different frequencies of 10 Hz, 100 Hz, 1 kHz, 10 kHz, 36.7 kHz, and 100 kHz. The results are shown in Figure 5:4 It can be seen that when the magnetic field was less than 600 mT, the slope of \(MR\) was basically 0. When the magnetic field was gradually larger than 600 mT, the value of the \(MR\) ratio started to gradually increase and the enhancement scale became bigger when the M field increased. Figure 5:5 Trend of \(MR\) (\(Z(B)-Z(0)/Z(B)\))% from 10 Hz to 300 kHz as the M field changes; (a) 0 mT, (b) 50 mT, (c) 100 mT, (d) 150 mT, (e) 200 mT, (f) 250 mT, (g) 300 mT, (h) 350 mT, (i) 400 mT, (j) 450 mT, (k) 500 mT, (l) 550 mT, (m) 600 mT, (n) 650 mT, (o) 700 mT, (p) 750 mT, (q) 800 mT, (r) 850 mT, (s) 900 mT, (t) 950 mT, (u) 1000 mT, (v) 1050 mT, (w) 1100 mT. Figure 5:6 Variation in dielectric loss resistance \(R_{e}\) with magnetic field strength. The inset in the figure corresponds to the curve of the lead resistance \(R_{s}\) as a function of the magnetic field strength. These behaviors can be explained as follows. In the GMR effect of La\({}_{0.75}\)Ba\({}_{0.25}\)Fe\({}_{12}\)O\({}_{19}\) ceramics, because the spins of different grains in the ceramics are randomly arranged, the spin-up and spin-down cancel each other in some way, and the electrons passing through each grain will have a certain amount of resistance, so when M = 0, the complex impedance is higher. When the magnetic field is applied, the spins in different directions begin to align in the direction parallel to the magnetic field. However, due to other resistances such as the interaction force between grains, when the magnetic field is small, there is not enough force to overcome the resistance of the material itself, so that the rate of change of the complex reluctance is small when the magnetic field is low. When the strength of the magnetic field increases to a certain level, the spins of the crystal grains begin to gradually accelerate to align in the direction of the magnetic field. Later, when the magnetic field strength becomes stronger and stronger, the spins of more crystal grains complete the alignment in the same direction as the magnetic field direction. In a parallel magnetic structure, electrons with the same spin direction as the magnetic grain can easily pass through the grain, so the impedance through the channel is reduced. The spin of the randomly arranged crystal grains is changed to the same spin with the magnetic field direction by the magnetic field, thereby reducing the resistance of the electrons in the magnetic field direction and increasing the mobility of the electrons.