process of abstaining from nanoparticles, as a result of physico‐chemical methods of preparation, or when the nanoparticles are surfacted or embedded in different solid matrices, elastic stresses can occur which induces an additional magnetic anisotropy (stress anisotropy) compared to those above. And this anisotropy, in the case of nanoparticles, can become large or high compared to magnetocrystalline anisotropy, and it must be taken into account when it appears. Example Coey (Coey and Khalafalla 1972) obtains a value of 1.2 × 105 J m−3 for nanoparticles of 6.5 nm in diameter and Vassiliou et al. (1993) obtains the value of the anisotropy constant of 4.4 × 105 J m−3, values that are approximately twice higher in magnitude than the magnetocrystalline anisotropy constant of the α‐Fe2O3 massive ferrite (K1 = 4.6 × 103 J m−3).
To conclude, in the case of magnetic nanoparticles, a magnetic anisotropy determined by magnetocrystalline anisotropy, shape anisotropy, surface anisotropy, and induced anisotropy must be considered:
(1.20)
and an effective magnetic anisotropy constant
(1.21)
respectively.
Typically, in the case of magnetic nanoparticles, this effective anisotropy constant increases when the magnetic nanoparticles become smaller, generally below 15–20 nm, depending on the nature of the material (Figure 1.12).
Figure 1.12 (a) Schematic view of the general spin canting geometry (the core@shell model). (b) Theoretical Ms (blue solid line) and Keff (orange solid line) versus magnetite nanoparticle diameter D at 300 K. Horizontal dashed–dotted lines exhibit Msb and Kb, respectively. Theoretical data are compared with experimental ones. Blue diamonds from Abbas et al. (2013), blue hexagons from Goya et al. (2003), orange triangle from Guardia et al. (2007), orange squares from Vargas et al. (2008), orange pentagon from Ferguson et al. (2011), and orange circle from Park et al. (2004). Note: 1 J (m−3T)−1 = 1 A m−1.
Source: Wu et al. (2017). Reprinted by permission of IOP Publishing.
The Figure 1.12b also shows the variation of the saturation magnetization (see Section 1.1.4) when the diameter of the nanoparticles decreases, this becoming smaller when the size of the nanoparticles decreases.
Moreover, in the case of magnetic core‐shell nanoparticles, the presence of a unidirectional anisotropy (Nogues et al. 2005; Caizer 2019) has recently been highlighted as a result of the coupling between neighboring layers (surface‐layer core) with different magnetic orders of magnetic moments in the network: ferromagnetic core (FM) and antiferomagnetic shell (AFM) (Figure 1.12). Also, in Ref. (Berkowitz and Kodama 2006) a review of the unidirectional (exchange) anisotropy for different FM‐AFM nanostructures may be found. In the case of CoFe2O4/NiO ferrimagnetic/antiferromagnetic nanocomposites, a similar behavior was found (Peddis et al. 2009) (Figure 1.13).
Figure 1.13 (a) Schematic drawing of a core‐shell structure and (b) transmission electron microscopy (TEM) image of an oxidized Co particle.
Source: Reprinted from Nogues et al. (2005), with permission from Elsevier.
Such situations may occur frequently in the case of different more complex magnetic nanostructures which are currently developed in nanotechnology and bionanotechnology for various applications.
1.1.6 Magnetic Behavior in External Magnetic Field
The magnetization of the bulk magnetic material in the external magnetic field (Cullity and Graham 2009), between two maximum values corresponding to the magnetic saturation, is generally with hysteresis (Figure 1.14a and b), due to the existence of a phase shift between the magnetization of a material and the applied magnetic field. Magnetization of the magnetic material, represented by the type of magnetization curve in Figure 1.14c, takes place both by processes of displacement of the walls of the magnetic domain (in low fields) and by processes of rotations of spontaneous magnetization (in high fields and near saturation), processes that are both reversible and irreversible. The basic macroscopic magnetic quantities characteristic of the hysteresis cycle, which are determined experimentally, are the saturation magnetization (Msat), the remanent magnetization (Mr), the rectangular ratio, r = Mr/Msat, and the coercive field (Hc) (Figure 1.14a). Depending on their applications, certain values and different shapes of the hysteresis curve are targeted. For example in the case of use of magnetic materials in high‐frequency fields, those materials are used that have the hysteresis cycle as narrow as possible with Hc and r as small as possible, close to 0 (preferably r < 0.1) (Figure 1.14b curve (2)); and in the case of applications for memory information, there magnetic materials are used that have the hysteresis cycle as rectangular as possible, with Hc and r as large as possible, theoretically close to the value 1 (preferably r > 0.9) (Figure 1.14b curve (3)), compared to the general case (curve (1)).
Figure 1.14 (a) Typical hysteresis loop for ferromagnetic materials.
Source: Reprinted from Sung and Rudowicz (2003), with permission of Elsevier.
(b) A typical hysteresis loop such as that obtained for soft and hard ferromagnetic materials.
Source: Mody et al. (2013). Reproduced with permission from Walter de Gruyter GmbH;
(c) Magnetization curves of iron, cobalt, and nickel at room temperature (H‐axis schematic). The SI values for saturation magnetization in A m−1 are 103 times the cgs values in emu cm−3.
Source: Cullity and Graham (2009). Reproduced with permission from John Wiley & Sons.
For magnetization of the bulk magnetic material in the external field (Figure 1.14c), there is no universal function,