The XRD patterns of the samples La_1-xCa_xMnO₃ (x = 0.1, 0.2, 0.3) are shown in Figure 1. After fitting to the space group Pbnm and JCPDS No 89-0662, all the peaks matched the standard orthorhombic structure of perovskite manganite nanoparticles [6]. At diffraction angles 2θ = 22.50°, 32.85°, 40.45°, 48.30°, and 58.75°, the peaks were indexed to (101), (121), (220), (202), and (042) planes. No additional and unidentified peaks were found, revealing that impurities were not present in the samples. It shows that the system is structurally single phase. The range of crystallite sizes was 25–32 nm using Debye–Scherrer’s formula (D = 0.9λ/βcosθ) (Safeen et al., 2022a; Safeen et al., 2022b), where “D” represents the crystallite size, “λ” is the wavelength of the x-ray, “θ” is the Bragg diffraction angle, and “β” is the full width at the half maximum (FWHM) of the diffraction peak (Asghar et al., 2020).

SEM images for La_1-x Ca_xMnO₃ (x = 0.1, 0.2, or 0.3) are featured in Figures 2A–C, which shows agglomerative particles that were, on average, 33–46 nm in size with a cage-like surface morphology. The samples were bulky and porous as gas was released in large amounts during heat treatment and sonication. The samples overall have a sphere-shaped cage-surface morphology.

EDX spectra for La_1-x Ca_xMnO₃ (x = 0.1, 0.2, or 0.3) is shown in Figures 3A–C, which shows the presence of La, Mn, O, and Ca peaks, confirming elemental compositional and phase purity. No extra peaks related to impurities can be observed. The atomic percentage of La, Ca, Mn, and O obtained from EDX spectra are in close agreement with the initial values used during synthesis. Elemental percentages of samples are shown in Table 1.

The field-dependent magnetization M(H) behavior of La_1-x Ca_xMnO₃ (x = 0.1, 0.2, and 0.3) at 77 K is shown in Figure 4, which shows that coercivity increases from x = 0.1 to x = 0.3. All the M(H) loops showed ferromagnetic behavior, as confirmed by the M(T) study. The enhanced coercivity could be attributed to single domain to multidomain magnetic behavior and randomly oriented equiaxial cubic magneto-crystalline anisotropy due to larger grain size (Chinnasamy et al., 2003). Additionally, the saturation magnetization increased with the concentration of x due to random canting of the grain surface spin occurring from the competition between ferromagnetic (FM) and anti-ferromagnetic (AFM) exchange interactions close to the surface. The M_r/M_s ratio was initially constant but decreased with doping concentration. Atoms have several domains of huge size, so magnetization reversal is gained by domain wall motion, thus coercivity is small. Magnetization reversal remains uniform for the single domain. For instance, it is necessary to respond with this field when the size of the spherical particle is larger for the identical spin rotation. Hence, initially, the energy wall for magnetization rotation increases with increasing particle size within a range of particle sizes. Therefore, coercivity increases because of the increase in particle size (Chinnasamy et al., 2003; Shah et al., 2012). The values of different parameters, such as M_s, M_r/M_s, M_r and H_c, obtained from the magnetic measurement of the synthesized nanoparticles are shown in Table 2.

The temperature-dependent magnetization M(T) for three samples at a magnetic field of 70 Oe is shown in Figure 5. Paramagnetic (PM) to ferromagnetic (FM) transition was observed due to cooling at a particular temperature of the material. The Curie temperatures T_c (the temperature representing the peak of -dM/dT in the M vs T curve) were 258 K, 255 K, and 250 K for x = 0.1, 0.2, and 0.3, respectively. The fall in T_c was caused by a decline in double exchange due to a reduction in mobility electrons and bandwidth. This reduction is up on rise in bond-length and decline in bond-angle of Mn-O-Mn (Shah, 2011).

Examination of M(T) behavior of La_1-xCa_xMnO₃ (x = 0.2 and 0.3) at a field of 70 Oe for FC and ZFC conditions to determine the behavior of magnetic spin freezing is shown in Figure 6. A split in the magnetization field cooled versus temperature and magnetization zero field cooled versus temperature curves over a wide temperature range. Upon cooling, the magnetic moment increased for all samples up to the lowest temperature for FC curves. A phase transition from paramagnetic to ferromagnetic was observed, but ZFC was different from FC, in which magnetic moments decreased as the temperature fell. We observed a huge difference in magnetization at lower temperatures between FC and ZFC curves. The dissimilarity between ZFC and FC magnetization was due to the significant spin disorder and competing interaction between different phases (Hasanain et al., 2004). For FC curves, the maximum values of magnetization were 8.47 emu/gm and 12.16 emu/gm for x = 0.2 and 0.3, respectively, while for ZFC curves, these values were 3.49 emu/gm and 5.49 emu/gm, respectively. As the temperature increased this difference minimized, and finally both curves merged. Such a behavior can be explained by the fact that when temperature is low the magnetic moment becomes frozen so that there is no order state in the long range. As a result of this spin disorder, strong competition occurred among ferromagnetic double-exchange (DE) and anti-ferromagnetic super-exchange (SE) so that a specific spin could obtain contradictory information, such as how to align with its neighbors due to exchange energy (Xia et al., 2017a). It may not be feasible to choose an assured spin configuration for the system to be able to minimize its energy. This decrease of the magnetic moment is due to frustration. Between 240 K and 258 K, there is a delay in the increase in the magnetic moment during transition. The super exchange and strong double exchange magnetic interaction between Mn⁴⁺ and Mn³⁺ ions reduces the increase due to low temperature. As a result, there is competition between two phases and a spin-glass-like phase occurs due to the generation of some degree of frustration resulting from such a FM and AFM interaction. When the metallic-like conduct disappears, distinctive glass-like behavior is observed with small ions (Shah, 2011).

There are two phases in which ac susceptibility measurements of the samples can be taken, namely, the in-phase and out-of-phase behavior as a function of temperature. Figure 7 shows the in-phase part of ac susceptibility. A sharp increase was observed at approximately 275 K, peaking at approximately 250 K. After the peak, there was a steady fall of ferromagnetic ac susceptibilities, with a sharp decline starting at approximately 100 K. A phase transition from the paramagnetic to the ferromagnetic phase occurred for all the samples in the transition region. Figure 7 shows that as Ca content increases, susceptibilities and sharpness in the transition region systematically decrease. Additionally, a decrease from the peak to the smallest value was observed. An increase of Ca concentration results in dissipation and spin disorder at low temperatures (Shah and Safeen, 2012). The presence of Ca results in a distortion not found in the XRD pattern. There is a possibility that an AFM arrangement is favored by local distortion catalyzed by Ca ions. Ferromagnetic clusters are found in the paramagnetic phase region. These ferromagnetic clusters increase in volume so that there is a complete ferromagnetic transition. However, in the clustering region, a continuous fluctuation occurs at high temperatures (Shah and Hasanain, 2010).

The behavior of the imaginary part of ac susceptibility is quite dissimilar from the real part as it sharply decreases at high T_c. The values of T_c for three samples (x = 0.1, 0.2, and 0.3) were 272.01 K, 271.20 K, and 265.87 K, respectively, as shown in Figure 8. The out-of-phase part of susceptibility occurs because of spin direction and/or fluctuation losses. At phase transition, these losses become large. Within the low temperature region, the generation of losses is dependent on the merger of small clusters to form the major clusters. The fluctuation in the magnetization is proportional to the loss components for a system that is going through a magnetic transition, as described by the fluctuation dissipation theorem (Karoblis et al., 2020). The dynamical changes occurring in the critical region are related to the lost components. The development of dynamical correlation of the spin occurs due to energy losses that appear in the behavior. The variation of the out-of-phase part of ac susceptibility is shown in Figure 9. The peaks in the out-of-phase part of susceptibility occur near the T_c, where M(T) shows no specific structure except for the decrease of the variations in magnetization. The losses continue growing to a point at which the magnetization starts to rise sharply. This occurs during the persistent fast growth of small clusters in this range. These small clusters decrease in number at lower temperatures (Supplementary Material S1).

To study the thermal fluctuation mechanism, the four-probe method was employed to determine the temperature-dependent electrical resistivity of La_0.8Ca_0.2MnO₃ nanoparticles. Figure 10 shows the effect of the applied magnetic field on resistivity in different temperature ranges. The values of resistivity were 1.4 Ω-cm with the cooling protocol and 1.5 Ω-cm with heating. This result shows that the transition temperature for the cooling curve rose as the magnetic field increased from 0 to 3 KOe. Additionally, resistivity decreased upon application of the magnetic field, as the microstructure of the large-range grain domain was parallel to the applied field. When the magnetic field was increased, the large-range magnetic array was increased, resulting in a decrease in resistivity. A rapid change in metal-to-insulator transition peak temperature (T_p) was observed. At room temperature, the sample was a semiconductor and had a temperature above the peak, i.e., at T_p = 170 K it exhibited metallic behavior. This alteration is related to magnetic ordering. The double-exchange mechanism defines the co-occurrence of FM and metallic behavior. At the highest value of Hund’s exchange energy, the conductivity maximizes. Thus, ferromagnetism is associated with metal-like behavior. T_p is rather different for heating and cooling curves (Hao et al., 2013). T_p is low compared to normal. This drop of T_p is caused by the decrease of oxygen (Van Cuong and Kim, 2009). Identical and smaller particles have unlike stoichiometric oxygen. The deduction of oxygen reduces Mn⁴⁺ to Mn³⁺. It preserves the electrical neutrality of the assemblage. The elimination of the oxygen bond stops the conduction pathways for e_g electrons. For the large range, the insulating part is further enhanced in the high-temperature range. With the rise in temperature, these defects decrease resistivity, resulting in an insulating appearance.