## 2 Structures, Synthesis, and Magnetic Properties of Fe\({}_{3}\)O\({}_{4}\) Nanoparticles
When the size of particles is adequately small (r \(<\) r\({}_{\rm SP}\)), thermal energy overcomes anisotropy energy at the blocking temperature (T\({}_{\rm B}\)), and nanoparticles become superparamagnetic. In our context, superparamagnetic NPs (at RT) should exhibit a blocking temperature much lower than room temperature. The single-domain size (r\({}_{\rm SD}\)) is the size at which NPs spontaneously separate into multiple domains, causing a decrease in coercivity when the size of NPs increases. However, size is not a conclusive parameter that dictates the magnetic properties. Depending on the size, geometry, crystallinity, surface properties, and synthetic methods, the superparamagnetic-ferrimagnetic transition in Fe\({}_{3}\)O\({}_{4}\) has been identified at approximately 20 nm [52], or even at \(\sim\)30 nm [53]. Similar to the superparamagnetic-ferrimagnetic transition, the single-domain to multidomain transition occurs at different values and depends on the overall size, degree of crystallinity, and surface properties of NPs. Thus, the relationship between the size and magnetic properties of Fe\({}_{3}\)O\({}_{4}\) NPs is not easy to predict due to various contributing factors. Figure 3: (**a**) Magnetization versus applied field (M–H) curves for superparamagnetic (SPM) (green color) and ferrimagnetic (FM) (orange) Fe\({}_{3}\)O\({}_{4}\) nanoparticles and (**b**) relations between size, coercivity, and magnetic behavior. In general, the magnetic properties of Fe\({}_{3}\)O\({}_{4}\) NPs such as saturation magnetization (M\({}_{\text{S}}\)) and coercivity (H\({}_{\text{C}}\)) are strongly affected by four main factors including finite size effects, surface effects, magnetic anisotropy, and the degree of crystallinity [2; 5; 42; 45; 46]. These factors are correlated to each other and strongly influenced by the size and geometry of the NPs. The finite size effects are typically related to special behaviors of a substance in a finite nanoscale size, such as quantum confinement of electrons [2]. In nano-magnetism, the single-domain limit and the superparamagnetic limit (presented in Figure 3) are the most studied finite size effects, which typically dictate the magnetic behavior of particles for ferrimagnetic and superparamagnetic responses, respectively [2]. While the superparamagnetic limit has been discussed in the previous paragraph, the single-domain limit is driven by balancing magnetostatic energy and domain wall energy, which induces the formation of magnetic domains [2]. Thus, the finite size effect of the single-domain limit regulates the change in ferrimagnetic properties, such as the change in coercivity in ferrimagnetic NPs when the size of the NPs increases. As illustrated in Figure 3b, increasing the size of NPs boosts H\({}_{\text{C}}\) until the specific size limit called the \"single-domain limitation\" is reached, where the separation of multiple domains begins and induces the decrease in H\({}_{\text{C}}\). The surface effect has been used to rationalize the decrease in M\({}_{\text{S}}\) of smaller Fe\({}_{3}\)O\({}_{4}\) NPs due to a surface-disordered spin layer [45]. For small NPs, the ratio of surface atoms to bulk atoms increases, giving rise to more significant contributions of surface spins to magnetization. The detrimental role of the surface effect on magnetization can be rationalized by various contributions, such as canted spins, magnetically dead layers, and spin glass-like behavior of surface spins [2]. As a consequence, the surfactants coated on the NPs also alter the magnetic properties of NPs, which can either positively or negatively affect the surface spin. Thus, the size of NPs can help to predict the change in M\({}_{\text{S}}\); however, it is not the conclusive factor. In addition, as different geometries of NPs have different ratios of surface atoms, Fe\({}_{3}\)O\({}_{4}\) NPs with comparable sizes but having different shapes can exhibit different magnetic properties. Magnetic anisotropy describes the directional dependence of the magnetic moment in materials.