3.1. Structure of Cubic Spinels
In the spinel structure, with the space group Fdm (No. 227), divalent anions X2− (X = O, S, Se, Te) form a close-packed face-centered cubic lattice (f.c.c.). One elementary unit cell contains 56 atoms (8 formula units) and a primitive tetrahedral cell contains 14 atoms—two formula units. Between the 32 anions in the elementary cell, there are 64 tetrahedrally coordinated interstitial positions and 32 octahedrally coordinated interstitial sites. Divalent A2+ cations are in one-eighth of the possible tetrahedral A-sites, and the trivalent B3+ cations occupy one-half of the possible octahedrally coordinated B-sites.
The spinel is called normal (A)[B
2]O
4 when A
2+ occupies only tetrahedral positions and B
3+ are positioned in octahedral sites. The anions are not in the ideal f.c.c. positions, depending on the difference in the effective ionic radii of the A- and B-cations, except that the spinel structures can be partially or completely inverse, with the general formula (A
1−xB
x)[B
2−xA
x]O
4, where
x is a degree of inversion; i.e., a part of the cations from the octahedral positions migrated to the tetrahedral sites. An alternative way to view the spinel structure is to observe the overall cation sublattice that possesses the same structure as the cubic MgCu
2 C15 Laves phase and the same length of unit cell as the f.c.c. anion lattice [
33]. For a precise definition of the spinel unit cell, except the length of the unit cell, a, it is also necessary to know the anion (or deformation) parameter
u.
Description of the atomic positions in the spinel structure is dependent on the choice of setting for the origin in the
Fdm space group. Two different Wyckoff positions with point symmetries
3m and
m are possible choices for the unit-cell origin. Moreover, the origin can be assigned to either a vacant site or an occupied lattice site. If the origin is an occupied A-site, i.e., A
2+ cation (setting 1), its Wyckoff position will be
8a (0, 0, 0), and then the cation in the B-site has Wyckoff position
16d (5/8, 5/8, 5/8) and the Wyckoff position of the anions is
32e (point symmetry
3m), with fractional coordinates (u, u, u). For an ideal close-packed face-centered cubic anion arrangement,
u = 3⁄8 (=0.375) for the
3m origin. The given fractional coordinates are along the body diagonal of the unit cell. On the basis of crystal symmetry, programs for Rietveld refinement generate coordinates of all other atoms in the unit cell and corresponding diffraction pattern. If the origin is on inversion center at the octahedral vacancy (setting 2) with point symmetry
m, all the lattice sites are translated by the vector [−1⁄8
a,−1⁄8
a,−1⁄8
a] and the fractional coordinates of the symmetry points will be changed: the Wyckoff position of the A
2+ cation will be
8a (1/8, 1/8, 1/8), the cation in the B-site has Wyckoff position
16d (1/2, 1/2, 1/2), and the fractional coordinates of the anions in
32e are u = 1/4 (=0.250) [
27]. Programs for the crystallographic calculation
PowderCell 2.4 [
34] and
MAUD (Material Analysis Using Diffraction) [
35,
36] support the first configuration, while
FullProf Suite [
32,
37,
38] supports the second one.
The VESTA (
Visualization for
Electronic and
STructural
Analysis) program [
39] can display both variants of the spinel unit cell (knowing the corresponding “cif”-files) with ions as spheres or with cations in the tetrahedral and octahedral voids, for example. Nevertheless, the schematic representation of the unit cell is often used in papers because it allows for easier observation of the geometrical relationships within the cell.
A conventional schematic representation of the elementary unit cell of the cubic spinel structure with the origin in an occupied A-site (
Fdm: setting 1) is presented in
Figure 1, left. This simple sketch of the atomic arrangement (with no fully drawn octahedral bonds) makes it easy to see the connection between the lattice constant and the octahedral and tetrahedral bond lengths. The tetrahedrally coordinated A-cations form a diamond cubic sublattice with a repeat motif equal to the lattice parameter
a. Two different “building blocks” with side length
a/4 are placed alternately in the octants of the A-cation unit cell. The contours of the primitive cell are plotted within the elementary cell in the middle of
Figure 1. Along the body diagonal of the primitive and elementary cell, there are the following sites: the A-cation in Wyckoff position
8a (0, 0, 0), the octahedral vacancy at
16c (1/8, 1/8, 1/8), the A-cation in position
8a (1/4, 1/4, 1/4), the anion in
32e (3/8, 3/8, 3/8), the tetrahedral vacancy at
8b (1/2, 1/2, 1/2), the B-cation in Wyckoff position
16d (5/8, 5/8, 5/8), the tetrahedral vacancy at
8b (3/4, 3/4, 3/4), and the anion in
32e (7/8, 7/8, 7/8).
Cobalt ferrite has a predominantly inverse spinel structure with the general formula (Co
1−xFe
x)[Fe
2−xCo
x]O
4. The mean ionic radii of the tetrahedral and octahedral sites for the partially inverse cobalt ferrite are:
The cation-to-anion distance in the tetrahedron or octahedron, i.e., bond length, is the sum of the corresponding mean ionic radius and oxygen anion radius: Rtet,oct = .
Consider the bottom row of the “building elements” with side lengths
a/4 (
Figure 1). The space between them is
a/4 and corresponds to the bond in the octahedron (plotted only in the bottom row in the added sketch of the octahedron). The value of the lattice parameter can be calculated using the bond lengths of the adjacent octahedron and tetrahedron as 3·
a/4 = 2·
Roct + 2·
Rtet/
. From this relation follows:
The magnitude of the ionic radius in a crystal depends on the chemical surroundings in which the ion is located. According to the magnetic measurements and the determination of the cation distribution (by Rietveld analysis, neutron diffraction, XPS measurements, or Mössbauer spectroscopy), the Co
2+ cations and Fe
3+ cations are in the high-spin state in cobalt ferrite [
40]. The effective ionic radii in the tetrahedral and octahedral coordination and oxygen anion radius are given in the Shannon comprehensive review [
41]. The effective radius of the Co
2+ cation with the high-spin (HS) electron arrangement in the tetrahedral position is
r(Co
2+)
A = 0.58 Å and in the octahedral site radius is
r(Co
2+)
B = 0.745 Å. For Fe
3+ in the tetrahedral position, the effective radius is
r(Fe
3+)
A = 0.485 Å (HS), and in the octahedral position, the site radius is
r(Fe
3+)
B = 0.645 Å. The oxygen radius is
rO = 1.38 Å (the oxygen anions have a four-fold coordination, formed by three B-cations and one A-cation (32e)). The mean ionic radii of the tetrahedral and octahedral sites can be calculated as follows:
Relation (4) shows that the cobalt ferrite with the maximal degree of the inversion will have the smallest lattice constant.
On the other side, the cation-to-anion distances,
Rtet,oct = <
rA,B> + <
rO>, for the case in
Figure 1 when the unit cell origin is the A-cation and the ideal
= 3/8 are given by:
When the unit cell origin is the B—vacancy and the ideal
= 1/4, the cation-to-anion distances are given by:
is converted to
and Equations (5) and (6) in (7) and (8), respectively, when the value of
u from Equations (5) and (6) is replaced by
u + 1⁄8 (because
) [
33]. The degree of inversion
x can be calculated using the equations for the tetrahedral and octahedral bond lengths expressed over experimentally determined
a and u together with the relations
Rtet,oct =
and the equations for the mean ionic radii of the tetrahedral and octahedral site, Equations (1) and (2).
The cation-to-cation distances, the hopping lengths, in the spinel elementary cell can be easily deduced from
Figure 1. The smallest distance is between the cations in the B-sites:
Tetrahedral A-cations are the furthest from each other:
The mutual distance of the cations at the adjacent A and B sites is:
The distances between the cations depend only on the lattice constant
a. The cation–ation distances, cation–anion bonds, and interbond angles are usually denoted as in
Figure 2.
All formulas for calculating cation–cation and cation–anion distances and angles between bonds (including formulas already given by Equations (5), (6), (9)–(11)) are shown in
Table 1.
As parameter
u increases, the oxygen anions in the tetrahedrons move along the 〈111〉 direction, changing the volumes of the tetrahedrons but not affecting the
3
m symmetry. Due to the increase in the tetrahedral interstices, the octahedral B-sites become correspondingly smaller and more deformed. Variation in
u is reflected in the bond-angle variations. These structural variations have important consequences on the properties of the material, especially the magnetic properties of ferrites. For instance, the angles
θ1 = <AOB,
θ3 = <BOB, and
θ5 = <AOA are the coupling angles for the most important super-exchange interactions. The values of the angles in the ideal f.c.c. spinel lattice are θ
1 = 126° (θ
1 = 125.264° ≈ 125.3° for oxide ferrimagnetic lattice [
33]), θ
2 = 154°, θ
3 = 90°, θ
4 = 125°, and θ
5 = 79° [
42]. The hopping lengths
e and
f are too large and together with the corresponding interbond angles
θ2 and
θ4, respectively, do not allow the creation of significant super-exchange interactions. Later, in the discussion of magnetic properties, we will calculate the values of these distances and angles for various coefficients of the cation inversion in the examined cobalt ferrite samples and evaluate their influence on the super-exchange interactions.
3.2. Structure Investigation of Cobalt Ferrite Nanoparticles by XRD
The X-ray diffractograms of five cobalt ferrite samples obtained with the various synthesis methods are presented in
Figure 3, along with the Bragg positions and differences in calculated and experimental intensities. All investigated samples are well crystalized in the single-phase spinel structure with characteristic Bragg reflections, corroborating JCPDS PDF 22-1086. The diffractograms were thoroughly analyzed by means of the
FullProf Suite program [
32]. To make it easier to see the details of the XRD patterns, a small interval of 2
θ (from 29° to 38°), with the reflections (220), (311), and (222) for all patterns, is shown on the left side of
Figure 3. The peaks were practically Lorentzian. There was a noticeable difference in the 2
θ positions and broadening of the corresponding reflections in the different samples. The auxiliary vertical lines coincided with the Bragg reflections of the sample with a maximal value of 2
θ, i.e., the smallest lattice constant. The sample obtained by US-CO had the smallest lattice constant, followed by the samples prepared by ME, CO, MW-HT, and MC-CO, respectively. The sample ME had the smallest peak width (
Figure 3).
Although a broadening of reflections increases with 2
θ, the crystallite sizes can be well evaluated from the strongest peak (311) with Debye–Scherrer’s formula:
D =
K λCu/(
w311·cos
θ311), where
K is a shape factor equal to 0.94 for spherical particles of cubic symmetry,
λCu is a wavelength of the Cu
kα radiation used, and
whkl is the FWHM (full width at half maximum) of a diffraction peak at the 2
θhkl position. The instrumental broadening of the peak is neglected because the measured width of the Bragg reflections for the nanoparticles is about ten times greater. Structural parameters,
a—lattice constant,
u—oxygen positional parameter,
Beff—effective temperature factor, and
x—coefficient of inversion derived by Rietveld profile refinement [
43], are listed in
Table 2, together with the average crystallite size (
D) and internal strain (
estr) obtained with the Williamson–Hall (W-H) plot:
The strain was assumed to be uniform in all crystallographic directions. From the linear fit to the data (where 4·sinθhkl is on the abscissa and the ordinate is whkl·cosθhkl), the crystalline size was estimated from the ordinate intercept and the strain estr from the slope of the fit.
The crystallite sizes estimated from the strongest (311) peak with Debye–Scherrer’s formula are mostly smaller than the size obtained with the
W-H plot (
Table 2,
D311 in brackets). In Debye–Scherrer’s formula, a microstrain broadening is neglected. On the other hand, the Williamson–Hall analysis is a method where both main causes of broadening—size-induced and strain-induced broadening—are deconvoluted by considering the peak width as a function of
θ [
44,
45].
The expected decrease in microstrain with an increase in the size of the nanoparticles is observed. The microstrain parameter of sample CO was slightly lower in comparison to the parameters obtained for the other investigated samples. In the case of the MC-CO sample, the microstrain value increased noticeably due to the internal stresses.
It can be noticed that the obtained lattice parameters of the studied nanocrystals were smaller than the calculated ones. This is common for nanomaterials since a lot of structural imperfections exist on the surface of nanoparticles.
The volume density obtained by Rietveld refinement,
ρXRD, is a theoretical density for an obtained lattice constant,
ρXRD = 8·
M/(
NA·
aexp3), where
M is the molecular mass and
NA is Avogadro’s number. The
ρXRD value is higher than that of the bulk material (
ρb = 3.24 g cm
−3). The specific surface area,
Am, is calculated under the approximation that the nanoparticles are spherical,
Am = 6/(
ρXRD·
D). The
Am value decreases with the increase in the nanoparticle size. Although the Rietveld profile-refined diffractograms look satisfactorily good, the quality of fit,
χ2, remains insufficiently sensitive to the changes in the cation distribution
x. Therefore, the obtained values for
x should be taken with caution. Generally, this is a problem with cations whose atomic factors are of similar magnitudes. The difference in diffraction intensities between the normal and inverse CoFe
2O
4 is less than the line thickness, so the diffractograms look identical (see
Supplementary Materials).
As already mentioned, the experimentally obtained lattice constants for the nanocrystalline CoFe
2O
4 samples were significantly smaller than those calculated via Shannon effective ion radii [
41] (Equation (4)) (
Table 2). As the crystal dimension decreases, the surface/volume ratio increases and the amount of disturbed surface structures with broken bonds and vacancies increases. The expected difference in the amount of internal stress between the largest and smallest nanoparticles is noticeable (
Table 2). The magnitude of the internal stress and the lattice constant of the nanocrystals depends mostly on the synthesis conditions. In the tested samples, there is no connection between the nanoparticles’ size and the size of the lattice constant. It is obvious that in the examined range of nanoparticle sizes, cationic inversion has an incomparably more significant influence on the lattice constant value.
At the same time, the contraction of the crystal lattice is accompanied by the decrease in the oxygen parameter value. The value of the oxygen positional parameter depends on the difference in the size of the cations and their redistribution by the tetrahedral and octahedral sites. The huge specific surface area of the nanoparticles with incomplete bonds, vacancies, and other imperfections significantly contributed to the effective value of the lattice constant and oxygen parameter values.
It is shown that the intensity of the Bragg diffractions does not change sufficiently with variation in the inversion coefficient of the cations with close atomic scattering factors (see
Supplementary Materials). Unlike cobalt ferrite, where changes in cation inversion do not significantly affect the Bragg reflection intensities (making it difficult to accurately determine the inversion parameter
x through profile spectra refinement), the lattice constant strongly influences the positions of the Bragg reflections and can be measured with high precision. Starting from the values of the lattice constant,
aexp, and the oxygen parameter,
u, obtained by XRD refinement and assuming that the cation radii were not changed significantly in the nanoparticles, the cation–anion distances in the tetrahedrons and octahedrons,
Rtet and
Roct, were calculated using Equations (7) and (8).
Rtet,oct = <
rA,B> +
ro, via Equations (1) and (2), is the estimated coefficient of the cation inversion,
x, and average oxygen radius, <
ro>, in the cobalt ferrite nanocrystals produced with the various synthesis methods. The obtained values are given in
Table 3.
The results listed in
Table 3 are graphically presented in
Figure 4a–c.
Figure 4c also shows the inversion coefficients obtained from the Rietveld profile refinement,
x-Rietv, together with x-struct, obtained by subsequent structural analysis. It is seen that the
x-Rietv deviates within the limits of ±0.05 related to the x-struct, determined with greater accuracy.
At the same time, the structural analysis was used to determine the average oxygen radius <ro> in the cobalt ferrite nanocrystals. The nanoparticles have a high proportion of surface defects. Such defects include incomplete tetrahedrons and octahedrons, which cause the contraction of cation–anion bonds. This contraction directly leads to a decrease in the average radius of the oxygen anion (<ro>). The lowest values of <ro> were observed for the samples with the smallest lattice constants, i.e., for the powders obtained by US-CO and ME. Both samples had a high cation inversion coefficient x, confirming that for the nanoparticles in the investigated range, the size of x had a decisive influence on the lattice constant value.
In that case, the vacancy parameter
β, defined as the relative volume of the missing ions at the nodal points of the spinel structure, is not a reliable indicator of the number of vacancies in the sample. The vacancy parameter can be evaluated as
β = [(
acalc3 −
aexp3)/
acalc3] ·100%, where
acalc is the theoretical value of the crystal cell parameter calculated by Equation (4) for the supposed coefficient of the inversion
x. For larger nanoparticles, the influence of the cation inversion coefficient on the lattice constant is not dominant, and
β has the expected trend; i.e., it is a good indicator of the number of vacancies in the sample. A previous study clearly showed that
β decreased with an increase in the size of the nanoparticles from 27 to 53 nm [
46].
The calculated vacancy parameters for the studied samples are given in
Table 3. The vacancy parameters decreased with the increasing value of the oxygen radius and the increase in the lattice constant value, as expected. However, it does not show any dependence on the size of the nanoparticles. Larger nanoparticles must have a less disordered crystal structure due to a smaller surface-to-volume ratio and therefore a lower
β value.
The structural XRD analysis based on the measured values of the lattice constant (aexp) and oxygen parameter (u) of the nanoparticles gives a more reliable value of the cation inversion. However, it should be emphasized that it is the average cation inversion of the material, which represents a “mixture” of crystalline cores with the undisturbed structure and shells of the structurally altered material since a noticeable difference in the size of aexp and acalc, as well as an almost ten times greater slope of the function aexp = f(x), occurred. In the case of the relatively small nanoparticles which were examined in this work (<20 nm), not only the cation inversion affects the value of the lattice constant. Therefore, it is necessary to compare the data obtained by XRD analysis with the magnetic measurements, which can separate the part of the predominantly regular structure and the contribution of the incomplete tetrahedrons and octahedrons in the surface layers.
3.3. Magnetic Properties of Cobalt Ferrite Nanoparticles
The magnetizations of the cobalt ferrite samples obtained with the various synthesis methods are shown in
Figure 5 as a function of temperature and magnetic field. The temperature dependencies of the magnetizations are measured by zero-field-cooled (ZFC) and field-cooled (FC) protocols at the applied field of 100 Oe. An irreversible magnetic behavior is shown by the splitting between the ZFC and FC curves. The irreversibility arises from the competition between the energy required for the magnetic moment reorientation in field direction vs. the energy barrier associated with the magnetoelectricity and the crystalline anisotropy.
The FC magnetization shows less temperature dependence than the ZFC magnetization. This indicates that the particle assemblies retain their magnetic history. The CO and MC-CO samples exhibit slight magnetization decreases above 50 K, becoming nearly constant at 300 K. In contrast, the samples obtained with the ME and MW-HT methods maintain almost constant FC magnetization up to about 200 K and then slightly increase near 300 K. At 5 K, the FC magnetization values are notably higher for US-CO, CO, and MC-CO samples (MUS-CO = 4.1 emu g−1, MCO = 4 emu g−1, MMC-CO = 3.5 emu g−1) compared to the MW-HT and ME samples (1.75 emu g−1 and 1.2 emu g−1, respectively), suggesting easier magnetic moment rotation in the field direction for the former group. This aligns with their lower coercivities.
The shape of the FC curves in the irreversibility region indicates the interparticle interaction intensity. The FC magnetization increases with the rising of temperature (for
T <
TB) when the dipolar interaction is significant [
46]. Vargas et al. showed that an increase in the dipolar interaction simultaneously affects a decrease in the relative height of the FC magnetization with respect to the maximum of the ZFC curve [
47]. Increasing dipolar interaction affects the relative height of the FC magnetization compared to the ZFC curve maximum. The MW-HT and ME samples show signs of greater dipolar interaction than the US-CO, CO, and MC-CO samples. Initial FC magnetization values gradually decrease and the slope of the FC curves changes as the nanoparticles’ sizes increase (
Figure 5). A reduction in the initial FC magnetization value for the US-CO sample and a slight increase in the magnetization for the ME sample as temperature rises was observed.
The characteristics of the FC/ZFC curves indicate moderately wide distributions of particle sizes and therefore similar distributions of blocking temperatures in the particle assemblies. In addition, it is clearly seen that the temperature of the maximum of the ZFC curves,
Tmax, is over 300 K. The blocking temperatures can be determined,
TB (
TB ≈
Tx crossover temperature) according to Livsey et al. in the inflection point of the ZFC magnetization curves, i.e., in the maximum of
, where Δ
M =
MFC −
MZFC, [
48]. Other authors imply that
TB corresponds to the maximum of the
[
49,
50]. Micha et al. state that “the blocking temperature,
TB, is determined by the FC and ZFC junction” [
51]. In the following text, the term “blocking temperature” will mean the maximum of
. To estimate the
TB of the nanocrystalline CoFe
2O
4 materials obtained with the different synthesis methods, a modified Stoner–Wohlfarth two-stage model can be used [
48,
52]. This model implies the existence of non-interacting, spherical, uniaxial nanoparticles.
As can be seen with SEM and TEM [
31], the CoFe
2O
4 nanoparticles obtained with the investigated methods are agglomerated and have different shapes and sizes. Defined crystal planes are seen on the largest particles, while smaller particles are generally spherical. The nanoparticles with sizes up to 20 nm usually grow spherical due to the large surface tension originating from the large surface-to-volume ratio [
27] in the investigated nanoparticles.
Therefore, the first condition for applying the modified Stoner–Wohlfarth model was not met, but the nanoparticles are supposed to be mostly spherical. The presence of a single magnetic domain (uniaxial structure) was evaluated by observing a consistent rise in coercivity as the nanoparticle size increased. This steady increase is a clear sign of a single-domain configuration. In
Figure 5, it is clearly visible that the investigated nanoparticles have almost linear growth of coercivity. At a characteristic critical nanoparticle size, a multidomain structure appears in order to lower the internal energy of the crystallites, and with a further increase in the size of the nanoparticles, the coercivity begins to decrease, reaching the coercivity of the bulk material (~230 Oe) [
30]. According to Maaz and Kim, the single-domain limit for CoFe
2O
4 is about 28 nm [
30].
One of the significant features in the
M(H) loops of the nanoparticles is that the magnetization does not saturate, even at high applied field. This is generally attributed to the canted or disordered spins at the surface of the nanoparticles that are difficult to align along the field direction. The numerical results of measuring the dependence of the magnetization of the cobalt ferrite samples on the applied magnetic field (
Figure 5) are given in
Table 4.
Due to the asymptotic increase in magnetization for high fields, the maximal saturation magnetization
Ms(calc) can be obtained from extrapolating the magnetization function
M = f(1/
H) to 1/
H = 0. The measured saturation magnetization
Ms at 90 kOe and extrapolated saturation magnetization
Ms(calc) are shown in
Table 4. In the following text,
Ms instead of
Ms(calc) will denote the maximum possible magnetization in the sample.
The values of the slope of the magnetization in the near-zero field region (dM/dH)H=0 show that smaller nanoparticles, obtained with different coprecipitation methods, follow changes in the magnetic field faster. The slope of the magnetization in the high-field region (dM/dH)H>>Hc implies that samples with a smaller slope (US-CO and MC-CO) have nanoparticles with a relatively small amount of disturbed spins and probably thin surface layers.
Coercivity,
Hc, increases remarkably with the size of the magnetic domains. The magnetic remanence
Mr and “squareness”
Mr/
Ms for the obtained samples are also given in
Table 4. The presence of super-paramagnetic nanoparticles in each sample, smaller than average size measured by XRD, reduces the value of “squareness” and the magnetization curves become less steep. It can be seen that “squareness” increases with the average nanoparticle size (see
Table 2 and
Table 4), which is an indication of a smaller influence of the disturbed surface layer in bigger nanoparticles.
The maximal difference in the average sizes of the CoFe
2O
4 nanoparticles of the different samples is less than ±1.6 nm (<19%), but some of the physical quantities, shown in
Table 4, vary considerably more. That is a direct consequence of the synthesis method. For small nanoparticles (<28 nm), the value of the saturation magnetization depends primarily on the coefficient of the cation inversion, but structural factors also play a significant role. The synthesis conditions, average nanoparticle size, distribution, and morphology of the nanoparticles crucially affect the magnetic remanence and coercivity. The maximum difference in
Mr between the investigated samples is 79%, and for coercivity, the maximal difference is even 150%.
3.4. Estimation of Blocking Temperature from ZFC/FC Curves by Fitting Procedure
A modified Stoner–Wohlfarth model with certain limitations, as previously seen, can be used to estimate the blocking temperature in the tested CoFe
2O
4 samples. We have mostly spherical, uniaxial monodomain nanoparticles, but their interaction is not negligible. In the Stoner–Wohlfarth model, the particles are considered to be completely blocked as long as the temperature is below their blocking temperature
TB, with the magnetization in the ZFC (heating) regime equal to the initial magnetization
MZFC (
t = 0) =
MZFC (
T→0) =
Mb. The blocked magnetic moment is
Mb =
μ0μ2H/(3
K), where
μ is the magnetic dipole moment (
μ =
N·μat =
Ms·
v).
N is the density and number of atomic moments
μat per unit volume, and
v is the volume of the nanoparticle.
Ms is a saturation magnetization (per unit volume)—the maximum magnetization value that the system can reach, which corresponds to the perfect aligning of all magnetic moments to the external magnetic field
H used in measurement.
K is the magnetic anisotropy energy (MAE),
K =
Keff·
v, and
Keff is the effective anisotropy constant, i.e., the density of anisotropic energy. At
T >
TB, the nanoparticles become perfectly superparamagnetic (at thermodynamic equilibrium) with magnetization
Meq =
μ0·
μ2·
H/(3·
kB·
T). The crossover between the two regimes is supposed to be abrupt (
TB =
Tx) and occurs at a transition temperature related to the magnetic anisotropic energy. The Stoner–Wohlfarth two-stage model is modified by introducing the size distribution of the nanoparticles. This approximation is quite satisfactory for assemblies of magnetic nanoparticles with a wide size distribution [
48].
A range of particle sizes implies a range of blocking temperatures in the particle assemblies. The average Néel time,
τ, for a particle to flip from one well to another is given by the Néel–Arrhenius law [
53]:
where
τo is a characteristic relaxation time, typically taken as 10
−9–10
−12 s [
54,
55]. If
τ is the measurement time,
τm, we can define the blocking temperature for a given experimental conditions as:
The value of
α = ln(
τm/
τo) for usual conditions in magnetic measurement, with
τm = 100 s, is 25–32. If
τm = 10
−8 s, like in Mössbauer measurements, α is 2–9 [
56]. Changes in the Mössbauer spectra related to the transition to the superparamagnetic state appear at higher temperatures compared to the magnetic measurements [
57].
TB depends on the measurement conditions, so the relevant time frame (
τm), temperature, and size of the nanoparticles (that affect
τo) must be specified. The blocking temperature
TB is equal to the crossover temperature
Tx in the approximation that
α = ln(
τm/
τo) is not dependent on
v [
50].
The distribution of the particle volumes gives rise to a distribution of blocking temperatures
TB(
v). So, the ZFC magnetization and the FC magnetization become, respectively [
54]:
The first integral in Equations (15) and (16) represents the contribution of the superparamagnetic particles, while the second corresponds to the blocked ones. The expected value (mean, average) of the particle volume is va, and vm(T) is the limit (maximum) volume of the nanoparticles in the superparamagnetic state at a given temperature, vm(T) = α·kB·T/Keff. The probability density function f(v) can be the volume log-normal or normal Gaussian distribution with mean value νa and “width”, i.e., standard deviation, σ.
The contribution of the superparamagnetic particles is the same for the
MFC and
MZFC curves and is canceled in the difference between
MFC and
MZFC magnetization. The difference Δ
M =
MFC −
MZFC can be used to simplify the estimation of different quantities such as
TB,
Keff,
va, and the standard deviation
σ.
Instead of a distribution of particle volumes, the distribution of the reduced volumes,
y, which is the same as the distribution of the
T/TB temperature,
y =
v/
va =
T/TB, due to the linear relation
α·
kB·
TB =
Keff·va can be used. In that case, the contribution of the blocked particles becomes [
58,
59,
60,
61]:
If the distribution of particle volumes f(v) is log-normal, the same is f(y) .
The probability that
y has values higher than
ym is:
A similar result is obtained for the normal distribution.
The contribution from the blocked particles in Δ
M =
MFC −
MZFC can be written as:
Now, we can introduce the distribution of the blocking temperatures and directly fit the function ΔM(T) = MFC − MZFC as a function of temperature. For the normal distribution, it is x = (T/TB − 1), and for the log-normal distribution x = ·ln(T/TBn). Here, TBn is the median on the natural scale, TBn = exp(μ), and the mode, the global maximum of the T-distribution, is TB = exp(μ − σ2).
The program for the fitting of the blocking temperature was tested on the data for the Fe
3O
4 samples with all
TB-
s in the measurement interval. The data were used from [
47]. Vargas et al. made samples from the same nanomaterial diluted in paraffin, with Fe
3O
4 concentrations of 0.05%, 0.5%, 5%, 45%, and 100% [
47]. Obviously, all the nanoparticles had the same size distribution and the same anisotropy, but with increasing nanoparticle concentration, the dipole interaction increased in the samples, leading to an apparent increase in measured
TB and a change in the appearance of the FC and ZFC curves.
A dipolar field reduces the ordering of the magnetic moments and has a role similar to the effect of the temperature. The corresponding temperature Td = Ed/kB, where Ed is dipolar energy, can be introduced in the presented model by writing TB + Td instead of TB (=TBeff) and fitting Td as a new parameter (the Vogel–Fulcher law). Indeed, this is provided that the true value of TB is known. The excellent fits of the diluted and powder Fe3O4 data (fits that give the same TB as the maximum of the experimental ) with the same temperature distributions were obtained, as it was expected.
Due to the lack of samples of diluted nanoparticles in which the dipole interaction is negligible, the fitting results could only be the type and width of the distribution of the effective blocking temperatures.
Figure 6 shows the result of the Δ
M(
T) =
MFC −
MZFC fitting, as well as the obtained distribution functions whose maxima correspond to the blocking temperature. The “blocking temperatures” obtained by the fitting were surprisingly low. This means that the actual blocking temperatures of the prepared nanomaterials could be even lower. Accurate determination of the real
TB would be important for the application of nanomaterials in the form of ferrofluids, particularly for biomedical application.
From the papers concerning the
TB of the CoFe
2O
4 nanomaterials diluted in organic [
62] or inorganic [
23,
58] matrices, one cannot determine unambiguously the impact of the dipolar interaction on
TB and estimate the real
TB (not effective) for a given size of nanoparticles. We did not find articles in which the authors systematically dealt with the dipole interaction in CoFe
2O
4 nanoparticle ensembles.
Babić-Stojić at al. determined the temperature of the maximum of the ZFC magnetization,
Tmax = 156 K, for the organic liquid suspension of 0.84% vol. fraction CoFe
2O
4 nanoparticles with a size of about 5–6 nm [
62]. The ZFC-FC magnetization was recorded under
H = 100 Oe. Unfortunately, no ZFC-FC measurement of the powder sample was performed under the same field. Only under the field of 1000 Oe were both measurements performed. The maximal temperatures of the corresponding ZFC magnetizations for the suspension and for the nanopowder are observed to be matched at approximately 164 K. Probably, a sufficiently strong magnetic field partially compensates for the effects of the dipolar interaction. However, we used the ZFC/FC magnetization data for the suspension measured at 100 Oe to determine the exact
TB (and test the program again). By the fitting of the Δ
M(
T) =
MFC −
MZFC data, we obtained
TB = 105 K and
σ = 0.33. Almost the same blocking temperature and the same Gaussian distribution were obtained by direct differentiation of the experimental Δ
M(
T) in the
Origin Lab program.
To the best of our knowledge, there are no relevant data in the literature to determine the relationship between the dimensions of the CoFe2O4 nanoparticles and their exact TB-s. In this study, the effective blocking temperatures TBeff and the standard deviation (“width”) of their distributions were estimated as is usual for solid materials.
In all cases, including the presented CoFe
2O
4 samples and the data utilized from Babić-Stojić et al. [
62] and Vargas et al. [
47], the distributions of the blocking temperatures are found to conform to a normal (Gaussian) distribution. The effective blocking temperatures and relative widths of the distributions shown in
Figure 6 are also given in
Table 5.
According to the Stoner–Wohlfarth model, blocking temperature is directly dependent on magnetic anisotropic energy
Keff·
v. As can be seen in
Figure 6, a fairly uniform increase in
TB with augmentation in the nanoparticle size was achieved, consistent with the model [
48,
59].
3.5. The Effective Anisotropy Constant, Keff
The effective anisotropy constant
Keff can be calculated from the blocking magnetization
MZFC(
T→0) =
Mb as
Keff =
μo·
H·
Msv2/(3·
Mbv). The label “
v” emphasizes that the magnetization must be calculated per unit volume. The saturation magnetization at 300 K is known from the measurement of magnetization as a function of the applied magnetic field (
Table 4). However, in the equation for the blocking magnetization at
T→0,
Mb, it should be the
Ms-value obtained at
T→0. Chatterjee et al. [
63] and Ananthramaiah and Joy [
64] obtained magnetizations at 5 K which were 1.25 to 1.7 times larger than at 300 K. Based on the aforementioned literature, it could be assumed that the magnetization of smaller nanoparticles increases more with decreasing temperature. For nanoparticles with the size 15–20 nm, the magnetization at
T→0 should be about 40% higher than the
Ms measured at 300 K. The calculated values of the anisotropy constants
Keff with the adjusted value of
Ms are rather close to the value of the anisotropy constant for single crystals at 5 K obtained by Shenker [
65]. According to Shenker, the temperature dependence of the effective anisotropy (precisely the principal anisotropy constant
K1) for bulk CoFe
2O
4 with cubic symmetry follows the empirical relationship:
It can easily be calculated that the anisotropy constant K1 at 5 K for the bulk is 19.6·105 [J m−3] and at 300 K is 3.5·105 [J m−3].
In
Table 5 are given the values of
Keff at 5 K (marked by “*”) obtained from the measured initial blocking magnetizations
Mb and with
Ms values 40% higher than the values measured at 300 K, which is the expected increase in
Ms at
T→0 K [
56,
60]. The low-temperature values for
Keff obtained with the specified
Ms correction are somewhat lower than Shenker’s value obtained on a single crystal [
60].
Considering that we do not have the measured values of the magnetization of the examined nanopowders at 5 K, we will determine Keff at 300 K using the “law of approach to saturation” on high magnetic fields. The values of the effective anisotropy constants estimated on the basis of the “law of approach to saturation” on high magnetic fields are more reliable because all the quantities required for the calculation are already determined at the same temperature.
According to Brown’s theory [
66], internal forces lead to laws of the form
M/
Ms = 1 − 1/
Hn/2, where
n = 1, 2, 3, for the point, line, and plane concentrations of forces, respectively, or
n = 4 for forces uniform throughout an extended volume. A lattice distortion concentrated in a small region will produce a deviation from perfect saturation over a much larger region. When two or more of these types of force distributions are present and there is no correlation between them, their contributions to
M(H) must be added. The empirical relation for
H>>Hc, known as the “law of approach to saturation”, is consistent with the following theory:
Brown showed that the coefficient A is proportional to the number of dislocations and therefore to the plastic strain. As the investigated nanoparticles are monodomain, the term A/H can be neglected. Coefficient B is related to the effective anisotropy constant as:
where
K2eff is the sum of the squares of all the contributions to the anisotropy–shape anisotropy
Ksh, stress anisotropy
Kst, surface anisotropy
Ksu, and magnetocrystalline anisotropy
Kcry [
63].
In the case of uniform anisotropy, the effective anisotropy constant is:
For randomly oriented spherical polycrystalline nanomaterials with cubic symmetry and dominant magnetocrystalline anisotropy, the effective anisotropy constant will be approximately equal to the principal anisotropy constant
K1 [
63,
67]:
In the case of nanomaterials, the linear term in the equation for the “law of approach to saturation” indicates the effect of canted spins on the surface of nanoparticles. These spins begin to contribute to the total magnetization only at very strong magnetic fields (χp·H—“forced magnetization term”).
The anisotropy at 300 K calculated by Shenker’s formula [
65] for a cobalt ferrite single crystal is 3.5 × 10
5 J m
−3. Fitting the “law of approach to saturation”
M(H) of sintered polycrystalline cobalt ferrite gives roughly similar values: (2–4) × 10
5 J m
−3, depending on the synthesis method [
64]. The effective anisotropy constants for the investigated CoFe
2O
4 nanopowders also obtained by fitting the “law of approach” are shown in
Table 5 as
Keff (300 K). The obtained values for the nanopowders (4–5.5) × 10
5 J m
−3 are slightly higher than for sintered polycrystalline cobalt ferrite. The observed difference can be attributed to the surface anisotropy of
Ksu. It is noticeable that the anisotropy constants
Keff(300K) for various samples have similar a trend as the corresponding
TB.
The fitting of the magnetization
M(H) in the range of
H>>Hc is illustrated in
Figure 7. It can be seen that the values of the effective anisotropy constants in the case of ME and MW-HT samples are higher than for nanomaterials obtained by the coprecipitation methods. (The shapes of the ZFC and FC magnetization curves also confirm this.) The obtained almost linear dependence of
Keff on
Hc verifies the direct proportionality of these two physical quantities. Both quantities increase expectedly with the size of the nanoparticles.
Values of anisotropy constants in CoFe
2O
4 are almost ten times higher than in other ferrites [
24] due to significant spin–orbit interactions of the Co
2+ ions. It has been observed that the single-ion magnetocrystalline anisotropy for Co
2+ located in the octahedral site is much larger when related to the tetrahedral site [
68]. This implies that the higher degree of the inversion in bulk (when the influence of nanoparticle’s size has no effect) leads to a higher value of anisotropy. For small nanoparticles, the influence of the size is dominant.
The slope of the
M(H) curves and the difference in the magnitude of the slope between the samples obtained with the different synthesis routes can be better observed in
Figure 7 than in
Figure 5. Achieving magnetic saturation in nanoparticle systems is complicated by several factors. The surface of the nanoparticles often exhibits disordered structures and magnetic moments, which interferes with uniform magnetic alignment. Additionally, nanoparticle samples typically contain a distribution of particle sizes, including both larger particles and smaller superparamagnetic ones. This size variation leads to different magnetic behaviors within the same sample. The combination of surface disorder and size distribution makes it challenging for the entire nanoparticle system to reach a state of complete magnetic saturation. Nevertheless, the fact that samples MC-CO and US-CO, with the smallest average size of nanoparticles, at the same time have the smallest slopes of the linear part of the function
M(H), for
H>>Hc clearly shows that the methods of their synthesis yielded nanoparticles with the narrowest size distribution and probably the thinnest disordered surface layers.
The size of the magnetic cores, i.e., the central part of the nanoparticles with undisturbed magnetic moments, were estimated from Equation (14):
TB =
Keff·v/(
kB·α) as
Dm =
, with
Keff (300 K). The obtained values of the magnetic cores show that the thicknesses of the “dead” layers,
t = (
DXRD −
Dm)/2, are from 4.3 nm for MC-CO to 5.8 nm for ME. This trend corresponds to the difference in slopes, Δ
M/Δ
H at
H>>
Hc (
Table 4), but the obtained values seem to be too high. In the next section, a different estimation of the size of the magnetic cores (
Dc) will be presented. With the approximation used, more realistic values for the “dead” layers are obtained at
t ≈ 1 nm.
3.6. Average Magnetic Moment of Co and Fe Cations at 300 K
The magnetic properties of the cobalt ferrite originate from the spin magnetic moment of the unpaired 3d electrons of the transition metals. Metal cations in the tetrahedral and octahedral coordination of oxygen anions have a relatively small crystal-field splitting Δ, which leads to a high-spin (HS) arrangement.
Theoretically, the contribution to the magnetic moment from the free spins in the Fe
3+ ion (with
n = 5 unpaired spins) or in the Co
2+ ion (with
n = 3) is
, which gives 5.92
μB and 3.87
μB, respectively. Although it is theoretically possible that Fe
3+ and Co
2+ have the stated values of the magnetic moments, such high values indicate perfect ordering and are not realistic in the case of nanoparticles. Commonly used values for low-temperature magnetic moments in nanomaterials are 5
μB for Fe
3+ magnetic cations and 3
μB (or 3.5
μB) for Co
2+ magnetic cations [
62,
69,
70].
Neutron diffraction measurements on CoFe
2O
4 confirm that at low temperatures, the magnetic moments of the A- and B-cations are antiparallel [
71]. In that case, Néel’s model of collinear ferrimagnetism (which stands for many inverse spinel ferrites) can be applied [
72]. Therefore, the resulting magnetic moment per formula unit of cobalt ferrite (Co
1−xFe
x)
A[Fe
2−xCo
x]
BO
4 will be:
where
μFe and
μCo are the magnetic moments of the Fe and Co cations, respectively, expressed by the number of Bohr magnetons (
μB = 9.27 × 10
−24 J T
−1).
In the inverse cobalt ferrite (
x = 1), the value of the magnetization is equal to the magnetic moment of Co
2+ (the magnetic moment of the Fe
3+ from the tetrahedral site compensates the same antiparallel moment from the octahedral site). Guillot et al. observed a spontaneous magnetization of 3.95
μB in the inverse single crystal. The obtained value is close to the saturation magnetization of 4
μB. The experimental result is higher than the theoretical prediction for Co
2+ ions. The discrepancy suggests that there is an additional orbital contribution to the magnetic moment of cobalt [
26].
Based on the measured values of magnetization
Ms at
T = 300 K (
Table 4) and the inversion coefficients obtained by XRD structural analysis (
Table 3), it can be estimated that the average magnetic moments
μFe and
μCo are equal to 3.25
μB and 2.5
μB, respectively. Both values are about 1.7 times smaller compared to the theoretical value or 1.4 times smaller compared to the commonly used magnetic moments in the CoFe
2O
4 nanomaterials for
T→0 K, which is in agreement with the experimental data for nanoparticles with a size of 15–20 nm. As already mentioned, in the case of nanoparticles, the experimental magnetic moments of the cations in nanostructures are smaller than the theoretical ones [
62,
73].
Different interplanar distances, characteristic of the FCC crystal structure, were previously observed and identified by high-resolution transmission microscopy (HR-TEM) [
74,
75,
76,
77], thus confirming that the central part of the nanoparticles has an undisturbed structure such as that in the bulk. This magnetic core, with undisturbed magnetic moments (
Mc), is surrounded by a layer with a disturbed structure and much lower magnetic moments (unoriented, canted, or both). If we assume for the sake of simplicity that this layer is “magnetically dead”, then the total magnetization of the nanoparticle would originate only from the core. With this approximation, we can easily estimate the possible diameter of the core.
If we calculate the magnetic moment using the Néel model, Equation (26), with the inversion coefficient obtained by XRD analysis and the values of
μFe = 5
μB and
μCo = 3.36
μB as they are in single crystals at room temperature [
26], we should obtain an enlarged value (
Ms*):
For the MC-CO sample, for instance, with the degree of inversion
x = 0.58 and
Ms = 3.12
μB (74.2 emu g
−1), the enlarged magnetization would be
Ms* = 4.74
μB (113 emu g
−1) at room temperature. To recall, the saturation magnetization is defined as the maximum possible magnetization of the sample per unit volume. Common values of
Ms in units of emu g
−1 must be divided by the average volume of the nanoparticles (π/6)·
D3. Then, by comparing
Ms and
Ms*, the ratio of the respective nanoparticle diameters is obtained. This means that the diameter of the core,
Dc, is:
Diameter
D is obtained by Rietveld analysis of the X-diffractograms and Williamson–Hall plot (
Table 2). In the case of the MC-CO sample, the magnetic core diameter should be
Dc = 14.32 nm and the thickness of the “dead” layer
t = (
D −
Dc)/2 = 1.07 nm.
The calculated values of the magnetic core diameters
Dc for all the investigated CoFe
2O
4 samples are given in
Table 5. The thickness of the disturbed surface layer obtained in this way is in better accordance with the literature data. Along with
Dc, the volume percentages (
Vshell) of the disturbed structures in the corresponding samples are given. This evaluation shows that the investigated nanomaterials produced with different synthesis methods have a thickness of “dead” layers (t) of about 1 nm, independent of the average particle size. Indeed, the total amount of disturbed surface structures and superparamagnetic particles is higher in the samples with smaller nanoparticles. The exception is the “US-CO” sample, obtained by ultrasonically assisted coprecipitation, which has a narrow distribution of nanoparticle sizes (characteristic for this synthesis method [
78]) and therefore a small number of superparamagnetic particles.
3.7. Exchange Interaction in Investigated Nanoparticles of CoFe2O4
According to spin–wave theory, the unpaired 3
d electrons in transition metals are coupled by the super-exchange interaction via oxygen ions separating the magnetic ions. Electrons are shared between the 3
d orbitals of the metal ions and the 2
p orbitals of the oxygen ions. The interaction strength is proportional to the overlap between these orbitals and directly depends on the bond length and angle between the metal ions and oxygen ion [
79,
80,
81]. A strong antiferromagnetic structure is made when the magnetic ion (with partially filled d-shells), ligand, and neighboring magnetic ion tend to form an angle of ~180° and a weak ferromagnetism or mediate antiferromagnetism when the angle is ~90°. In spinel ferrites, there are three types of significant super-exchange interactions. (The remaining two super-exchange interactions are negligible due to the large distance between the cations.) The strongest is J
AB (A-O-B), with angle
θ1; then J
BB (B-O-B), with angle
θ3; and weak J
AA (A-O-A), with angle
θ5 (see
Figure 2). A and B refer to the ions on the tetrahedral and octahedral sites, respectively. The ideal spinel value of A-O-B angle
θ1 is 125.3°. Since there can be two types of ions in the A and B positions, the actual number of interactions is 10 for partially inverse ferrite or 6 for inverse ferrite. The J
AA interaction could be expected to be the weakest due to the largest distance between the tetrahedral A–A-cations in the crystal lattice of the spinel ferrite; see Equations (9)–(11). At the same time, the angle for the overlap of the A-O-A orbitals is much less than 90° and unfavorable compared to the angle between the B-cations. Any alteration in cation distribution changes the lattice constant and the oxygen parameter, thereby altering the spin interactions that determine the super-exchange interactions. Ferrimagnetic ordering occurs because the negative (antiferromagnetic) exchange interaction J
AB between the magnetic cations occupying the tetrahedral A-sites and the octahedral B-sites dominates the also antiferromagnetic intra-sublattice exchange interactions J
AA and J
BB, causing each of the two magnetic sublattices on the A-sites and on the B-sites to be aligned. Apparently, there is a complex magnetic structure within the sublattices. The substructure of the A-sublattice was studied on the example of Co
3O
4, whose crystal field determines the low-spin configuration of the Co
3+ cations in the octahedral sublattice (
S = 0), so the antiferromagnetic arrangement that exists up to 40 K originates only from the magnetic moments of the Co
2+ cations from the tetrahedral A-sub-sublattices [
82]. The substructure of the B-sublattice was investigated in the normal spinels ZnFe
2O
4 and CdFe
2O
4 with nonmagnetic cations in the tetrahedral A-sublattice. Antiferromagnetic ordering in these ferrites up to 9 K originates from the ordering of the Fe
3+ cations with
S = 5/2 on the B-sub-sublattices [
83]. Relatively high
TN in Co
3O
4 related to ZnFe
2O
4 and CdFe
2O
4 (although weaker exchange interaction J
AA would be expected due to the greater distance between tetrahedral cations) originates from the additional spin–orbit coupling in the Co
2+ cations of Co
3O
4.
The exchange constants are determined experimentally based on the temperature dependence of spontaneous magnetization and magnetic susceptibility. The main difficulty is to determine a set of exchange constants which simultaneously satisfy the thermal variation in the spontaneous magnetization and the high-temperature change of the magnetic susceptibility. Various experimentally obtained values for interaction constants can be found in the literature, which the authors show to be in agreement with Anderson spin–wave theory [
84,
85]. The reliable values given by Srivastava et al. are listed as J
AA = −15 K (Fe
tet−Fe
tet), J
AB′ = −22.7 K (Fe
tet−Co
oct), J
AB’’ = − 26 K (Fe
tet−Fe
oct), J
B′B′ = +46.9 K (Co
oct−Co
oct), J
B′B″ = − 18.5 K (Co
oct−Fe
oct), and J
B″B″ = −7.5 K (Fe
oct−Fe
oct) [
85]. All interactions are antiferromagnetic, except Co−Co at octahedral sites. Sawatzky et al. used the molecular-field theory and the analysis of Mössbauer spectra to show that the interaction constants are slightly smaller, but the difference in J
AB′ and J
AB″ is greater in favor of the Fe
tet−Fe
oct interaction [
86]. The values of J
AB″ and J
AB′ were found to be −20.07 K and −13.7 K, respectively.
The change in the strength of the super-exchange interactions in cobalt ferrite with the change in the inversion coefficient can only be roughly illustrated here using the structural data obtained by XRD analysis. X-ray diffraction provides “average data” for the investigated material, so the mean distances between cations and the mean angles between different pairs of cations represent a measure of the strength of the “average” super-exchange interactions.
As the lattice constant increases and the inversion coefficient decreases, all hopping distances (b, c, d, e, and f) increase because they depend only on
aexp (see
Table 1). We have already seen that the octahedral bonds
Roct =
p decrease, while the tetrahedral bonds
Rtet =
q increase (
Table 3). Other cation–anion distances,
r and
s, also grow. It can be seen that with a decrease in the degree of inversion
x (increased number of Co cations in tetrahedral sites), deviations from the ideal values for spinel ferrites increase (
Table 6). Enlarging the distances between the magnetic cations and increasing the deviation of the value of the interbond angles with decreasing
x lead to a decrease in the “average” strength of the super-exchange interactions. This also leads to a decrease in the magnetic ordering temperature [
87].
In the ideal case, the completely inverse structure of cobalt ferrite, with
x = 1, (Fe)
A[FeCo]
BO
4, each Fe
3+ cation in the A-site would be surrounded by 12 cations from the B-site: 6 Fe
3+ cations and 6 Co
2+ cations. At the same time, J(Fe
tet − O − Fe
oct) > J(Fe
tet − O − Co
oct). The cation Fe
3+ in the B-site would be surrounded by 6 Fe
3+ cations from the A-site [
86,
88]. This is the highest possible exchange energy that the Fe
3+ in the B-site can have (and the smallest that the Fe
3+ in the A-site can have). Obviously, when the inversion coefficient
x decreases, when Co
2+ cations migrate from the B- to the A-site, the total super-exchange energy in the B-site will decrease and, in the A-site, energy will increase by the same amount. Since the tetrahedral A-cation has a larger number of B-neighbors, it will have a higher total exchange energy and will therefore be less affected by the changes in cation inversion. In contrast, the B-cation undergoes much more significant changes. At first glance, it seems that the total energy does not change; however, as we have already shown by XRD structural analysis (
Table 6), the migration of the larger Co
2+ cations from the B-sites to the A-sites, i.e., decreasing of the inversion
x, leads to an increase in the lattice constants and all hopping distances and a decrease in the average strength of the super-exchange interactions (and also to a decrease in the magnetic ordering temperature).