3.1 Structural analysis
Figure 1 illustrates the XRD patterns and Rietveld refinement results of LFCO annealed at different temperatures of 750°C and 900°C. The best fit of the observed diffraction peaks is obtained with the orthorhombic structure with space group Pnma, which is the same as that of the members LaFeO3(PDF#24-1016) and LaCrO3(PDF#37-1493), which confirms that there is no impurity phase in the sample [14, 15]. Table 1 demonstrates the lattice parameters of LFCO-750 and LFCO-900. The average grain size of LFCO is calculated by the Bragg formula and Scheller formula, the grain size of the LFCO sample annealed at 750°C is 43.35nm, while the grain size of the LFCO annealed at 900°C is 40.38nm. As the annealing temperature increases, there is no significant difference in the cell volume.
Table 1
Lattice Parameters of LFCO-750 and LFCO-900.
Sample | Lattice parameters (Å) | Crystal cell volume(Å3) |
a | b | c |
LFCO-750 | 5.52 | 5.55 | 7.82 | 239.50 |
LFCO-900 | 5.52 | 5.54 | 7.81 | 239.08 |
3.2 Magnetic Properties
The hysteresis loops of LFCO-750 and LFCO-900 were recorded at 5K, within a magnetic field scope from − 7 to 7T, illustrated in Fig. 2. Both manifested distinct hysteresis loops, signifying that at 5K, both LFCO-750 and LFCO-900 were in a ferromagnetic condition. As the annealing temperature increased, the coercivity of the sample increased. Previous research indicates that coercivity is influenced by aspects like magnetic crystal anisotropy, microstrain, size distribution, anisotropy, and the size of the magnetic domain [16, 17]. This implies that the high-temperature annealing treatment altered the distribution and orientation of ions, thereby facilitating the ion spin excitation [18, 19]. At lower annealing temperatures, the hysteresis loop of the sample shows a more gentle shape, indicating weaker magnetic moment interactions within the sample. Whereas at elevated annealing temperatures, the hysteresis loop of the sample becomes sharper, indicating enhanced magnetic moment interactions within the sample. The hysteresis loop of LFCO-900 presents a wasp-waisted shape, leading to a decrease in rectangularity, which can be attributed to the formation of directionally ordered transitions within the domain wall consistent with the direction of electron spin, thereby fixing the entire domain wall [20].
Figure 3 depicts the variation curve of magnetization against temperature for LFCO-750 and LFCO-900 under an external magnetic field of 100 Oe. It is observable that LFCO-750 and LFCO-900 display a similar trend under a weak magnetic field, with magnetization rising as the temperature falls. At low temperatures, the magnetization of LFCO-900 increases due to the increased annealing treatment temperature, and the high-temperature annealing changes the crystallinity of the material, enhancing its stability [21]. Figure 4 represents the temperature-dependence curve of the demagnetization rate, with both LFCO-750 and LFCO-900 exhibiting a second-order paramagnetic to ferromagnetic transition at T ~ 290K [22]. At elevated temperatures, the superexchange effect of Fe3+ (3d5)-O (2p) Fe3+ (3d5) antiferromagnetic coupling takes the lead, with the material demonstrating paramagnetic properties. At temperatures below 290 K, because of the Dzialoshinski-Moriya interaction [23, 24], the Fe3+/Cr3+ spins slightly deviate from the antiparallel angle, resulting in weak ferromagnetism. As temperature further decreases, the Fe3+ (3d5)-O (2p) Fe3+ (3d5) ferromagnetic interaction becomes predominant, leading to an increase in magnetization.
Figure 5 shows the curve of magnetization versus temperature for LFCO-750 and LFCO-900 at 1000 Oe, indicating that the temperature-dependent magnetization curve of LFCO-750 shows a trend of first decreasing, then increasing, and then decreasing again. This is consistent with the research of Dahmani A et al [25] when the A site of the AB0.5B'0.5O3 perovskite system is a non-magnetic ion, the M-T curve shows two minimum points. This phenomenon is attributed to the change in the coercive field, that is, the first minimum temperature point in the magnetization versus temperature curve corresponds to the temperature at which the coercive field appears maximum. This primarily results from the thermal movement of domain walls that leads to a constant increase in the anisotropy between domains at lower temperatures. The increased anisotropy pins the orientation of the magnetic moments, thus rapidly reducing the magnetization. The M-T curve of LFCO-900 shows anomalous behavior in the 50K-150K range. This is attributable to the changes in magnetic entropy induced by annealing. The magnetic entropy hits a minimum of 50 K. At this temperature, the action of the external field on twisting the magnetic moment is strengthened, thereby enhancing the total magnetization. Once the temperature reaches 150 K, a minimum is observed once more, leading to a subsequent increase in magnetization [26].
3.3 Heisenberg Model Simulation for Disordered Systems
To support our viewpoint, we have employed a Monte Carlo simulation of the Heisenberg model and utilized the particle swarm optimization algorithm to fit disordered systems’ parameters (x,y) [27]. These two parameters have detailed explanations in our previous work [28, 29]. The Heisenberg Hamilton is :
$$\begin{array}{c}H=-\sum _{⟨i,j⟩}{J}_{ij}{S}_{i}{S}_{j}-h\sum _{i}{S}^{z}\#\end{array}\left(2\right)$$
where < i,j > represents the nearest neighbor of i site, Jij is the exchange constant of i and j site, h is the magnetic field actions on the z-axis, and Sz refers to the projection of spin on the z-axis. Based on the above formula, combined with the PSO algorithm, the values of (x,y) can be easily obtained and used to fit experimental data, which is ~(3.1,5.2). In the simulation, we chose a ratio of |JFeFe|:|JCrCr|:|JFeCr|=3:2:118, and all values were negatives. Analog details are presented in the literature [30]. Figure 6 shows the result of the calculation. Next, we will investigate the effect of external magnetic fields on magnetization. We compared the magnetization under zero-field conditions and h = 5, as shown in Fig. 7. The magnetization curve under applied field. The thermal dependence of the applied field is very similar to that of Fig. 5(a) because the magnetic entropy of the system simulated by the Heisenberg model is conventional. In contrast, the magnetic entropy of Fig. 5(b) exhibits significant anomalies. These anomalies are caused by annealing and the origin is currently unknown, requiring further research in the future.