In this section, we will review some recent experimental studies of ferromagnetic resonance and its applications to nanoparticles. Particular attention is given to effects such as temperature (and annealing), doping, and the effects of nanoparticle synthesis techniques and their oxidation.
5.1. Effect of Temperature
The effect of temperature is a critical factor in the understanding of the magnetic properties of ferromagnetic nanoparticles. Indeed, the thermal energy of the system can influence the magnetic moments of the particles, leading to changes in the magnetic properties. In addition, annealing, which is commonly used in the preparation of nanoparticles, can alter the magnetic behavior of the material. Therefore, it is essential to investigate the effects of temperature and annealing on the magnetic properties of ferromagnetic nanoparticles to fully understand their magnetic behavior.
An illustration of these property changes through annealing is shown in
Figure 8. The observed increase in the annealing temperature from 1000 °C to 1250 °C allows the formation of different phases (
,
and
) for
nanoparticles [
62]. In particular, at 1200 °C only the
particle phase is detected. All samples showed a natural ferromagnetic resonance measured by terahertz spectroscopy. This increases from 161 GHz to 170 GHz as the size of iron oxide nanoparticles increases (from 7 to 38 nm) due to annealing, while the half width at half maximum (FWHM represented by factor
in
Figure 9) decreases monotonically.
To understand and explain the origin of this variation, it is important to remember that the characteristics of the FMR line are affected by many factors, such as the saturation magnetization, the magnetocrystalline anisotropy, the particles morphology, the thermal fluctuation of magnetization, the demagnetizing field, the porosity of the material, etc.
For these samples, the influence of impurities (
and
) can be neglected since their anisotropy fields are much lower than that of the
[
63] and the low spontaneous magnetization leads to very weak demagnetizing fields (less than 100 Oe). Furthermore, the presence of the impurities should not affect the FMR line in the aspect of the porosity of the material as the self-demagnetizing field of the
particles is weak (0.9 kOe) with respect to its high anisotropy field (≈40 kOe). Thus, the observed dependence of
(FWHM) on the annealing temperature should be attributed to the variation of the magnetocrystalline anisotropy constant and size of the
particles.
Nickel ferrite magnetic nanoparticles annealed at 600 °C, 900 °C, and 1100 °C were studied by FMR in order to investigate the magnetic anisotropy [
64]. The samples were prepared by the sol-gel technique and then isothermally treated at different temperatures for 8 h.
As shown in
Figure 10, XRD patterns confirm that the crystalline structure of Ni ferrite particles increases proportionally with the annealing temperature. The nanostructured Ni ferrite particles are
-based structures and contain different chemical phases such as NiO and
.
Figure 11 displays the FMR spectra of the three Ni ferrite annealed samples with their theoretical fittings. The model applied to analyze the FMR line was presented by Pessoa et al. in [
64,
65].
The FMR line fitting results are summarized in
Table 1. They confirm the presence of the large four-fold cubic magnetic anisotropy as revealed by the XRD data. The FMR line also reveals easy and hard anisotropy axis along the [100] and [111] directions, respectively, for the Ni ferrite sample annealed at 600 °C, while the easy axis is along [111] directions resulting in a negative magnetic anisotropy field for samples annealed at 900 °C and 1100 °C (
Table 1). This change in the direction of the easy axes is the consequence of the annealing and thus the increase in the crystalline fraction, as shown by XRD analysis. These changes in the magnetic anisotropy field and the directions of the easy axis can be explained by the redistribution of Ni/Fe ions in both crystalline sites, i.e., the reorganization of the
ions in the tetragonal and octahedral sites of the spinel structure, evolving to the more stable configuration, which possesses the negative crystalline anisotropy [
66]. Moreover, the decrease in the FMR linewidth as a function of the annealing temperature once again confirms the evolution of the system to its more stable and higher crystalline configuration.
Another similar study concerning the effect of annealing time on the oxidation of Ni nanoparticles has been investigated by Chhaganlal
et al. [
67]. In this work, a series of Ni/NiO core-shell nanoparticles were synthesized at 300 °C under an ambient atmosphere at different annealing times, indicated as
, which varies from 10 min to 600 min. From the XRD measurements, the variation of Ni and NiO grain size as a function of
was observed. As shown in
Figure 12, the Ni grain size stays constant (≈30 nm) below 200 min and drops abruptly from 200 min to 600 min whereas the particle size of NiO slowly varies from 8.6 nm to 13.6 nm, corresponding to an annealing time from 10 to 600 min. The annealing effect on the lattice constants for Ni and NiO is also clearly observed from the lattice constants in
Figure 12, which shows lattice contraction for
min. Finally, it was concluded from the XRD results that the expansion of the NiO shell coincides with the reduction in the Ni core.
Figure 13a represents the FMR absorption spectra for annealed samples from 10 to 600 min at 300 K. The spectra show both decreasing intensity and line-width with increasing
. To fit these experimental results, a model was used based on a deconvolution system using the sum of three Gaussian functions [
67]:
where
,
and
are amplitude,
,
and
resonance field and
,
and
line-width of
,
and
peaks, respectively, as shown in
Figure 13b for the sample annealed at 600 min. The use of the deconvolution method with the sum of Gaussian functions allowed a best fit for these samples to be obtained. However, interpreting the results of these fits is also not an easy task. For this, the calculated values of the g-factor and linewidth were plotted as a function of
. From
Figure 13c and for samples with
less than 180 min,
1.95 can be attributed to the free spins [
68]. For samples at 180 min, peak
with
2.7 and peak
with
8, Ni nanoparticles and inter-particle interactions are assigned to them, respectively [
69]. The decrease in
and
with increasing
, suggests the reduction in magnetic inhomogeneity.
For a more in-depth study, the temperature (T) dependencies of (
,
) and (
,
) for three typical samples with
= 10, 120 and 600 min were measured as displayed in
Figure 14. These T dependencies were then analyzed by dividing the data into three regions:
(330 K–200 K),
(200 K–130 K), and
(130 K–90 K). The separation between these regions is delimited by the vertical dotted lines as shown in
Figure 11.
In regions and , both and increase with the decrease in T, which indicates an increase of magnetic interactions. Moreover, the dependence of with respect to temperature overlaps for the three samples, while the intensity of increases as increases, indicating that the interaction between free spins gets stronger for large size Ni nanoparticles. On the other hand, in region , reaches a peak at ≈130 K with = 600 min. The observed effects may be linked to a rise in effective anisotropy resulting from the emergence of short-range antiferromagnetic (AF) correlations in the NiO particles. This leads to exchange-bias (EB) coupling near the interface of Ni and NiO. Additionally, in region , there is a significant increase in at 600 min, which could be attributed to the long-range AF ordering of NiO. However, shows a sudden decline at around 130 K for all the three different values, suggesting again changes in effective anisotropy due to EB coupling.
As the temperature decreases, the linewidth shows a similar behavior to that of the g-factor by increasing in both regions
and
. The
values of free spins overlap in region
, then begin to separate in region
. At temperatures below 200 K, an increase in
is observed, which is associated with increased anisotropy at low temperatures. To obtain the change of spin relaxation rate, the plot of
versus (1/T) is used, as shown in the inset of
Figure 14d. The slope of the curve
(1/T) changes at 208 K, 175 K, and 160 K with respect to
= 600, 120, and 10 min, which are approximately the same transition temperatures where
starts to increase in region
. The observed change in spin relaxation rate is associated with the EB effect at the interface of Ni and NiO.
In order to determine the spin relaxation rate changes, the graph of
as a function of (1/T) is used, as displayed in the inset of
Figure 14d. The slope of the
(1/T) curve varies at temperatures of 208 K, 175 K, and 160 K for different values of
(600, 120, and 10 min), which approximately correspond to the same transition temperatures at which
begins to increase in region
. The authors concluded that the observed alteration in spin relaxation rate is linked to the EB effect at the interface of Ni and NiO.
The behavior of magnetic nanoparticles is also determined by their magnetic anisotropy. For this purpose, P. Hernandez-Gomez et al. [
70] studied Li ferrite nanoparticles by FMR at different annealing temperatures. These nanoparticles were prepared by the sol-gel technique and then annealed for 4 h in a temperature range from 400 °C to 1000 °C. X-ray diffractograms indicated the particle size increases with annealing temperature, which is in good agreement with the higher crystallinity of the samples induced by the annealing. The
size varies from (30 ± 3) nm for samples annealed at 400 °C, to (720 ± 26) nm for those annealed at 1000 °C.
Figure 15 shows FMR spectra of Li ferrite nanoparticles at 400 °C annealed temperature with frequencies varying from 4 GHz to 8 GHz where inhomogeneous broadening behavior is observed. This inhomogeneity is common to nanoparticles and can have several origins, such as magnetic inhomogeneities in the sample, random orientations of magnetic anisotropy axis [
1], or interaggregate dipolar interaction. By plotting the extracted linewidth as a function of frequency, it is possible to study the damping of the resonance and then identify each damping mechanism contribution. We note that damping can result from both intrinsic and extrinsic contributions [
71].
Figure 16 shows the evolution of the linewidth of the sample annealed at 400 °C as a function of frequency. Note that if the Gilbert coefficient is dominant in damping, a linear increase in linewidth with the applied field is observed, which is not the case here. Therefore, it is concluded that in addition to the Gilbert damping parameter, there are dominant extrinsic magnetic relaxation contributions. These contributions would likely come from a high interparticle interaction induced by the aggregation of the nanoparticles; thus, in addition to the dipolar interactions, intracluster exchanges appear, which leads to the observed distorted resonance curve [
72].
In order to evaluate the magnetic anisotropy, the dependence of the frequency versus the magnetic resonance field is analyzed as illustrated on the left of
Figure 16. This linear dependence is fitted using the Kittel expression:
where
is the resonance frequency,
the magnetic field at resonance,
is the gyromagnetic ratio, and
the effective magnetic anisotropy field. The effective anisotropy values calculated for the samples show a variation, with a value of 1.02 kOe for the sample annealed at 400 °C, and values of 1.42 kOe and 1.92 kOe for the samples annealed at 600 °C and 800 °C, respectively. The anisotropy field is reduced for samples annealed at 1000 °C, which is in good agreement with other spinel ferrite nanoparticles [
73].
The saturation magnetization for samples annealed at 1000 °C (
) was measured in order to calculate the value of the effective anisotropy constant
, which is three times higher than the magnetocrystalline anisotropy reported for bulk lithium ferrites [
70]. In fact, it is well-known that in the case of nanoparticles, the magnetic anisotropy constant is not only determined by the magnetocrystalline anisotropy but also by other contributions such as surface, strain, shape anisotropy and anisotropy from interparticle interactions, which can explain this increase with respect to the bulk magnetocrystalline anisotropy. Moreover, this value is close to the calculated one (
) by Yang et al. [
74] who assumed that the nanoparticles annealed at lower temperatures have a core-shell configuration, whereas the larger particles exhibit multidomain behavior. Thus, this anisotropy may also indicate that the Li ferrite nanoparticles are in a core-shell configuration.
A complementary study was carried out by the same authors and by doping Mn in the Li ferrite nanoparticles [
75]. Mn-doped Li nanoparticles were prepared under the same conditions as the above-mentioned samples and also annealed in a temperature range between 400 °C and 1000 °C. The extracted FMR data are shown in
Figure 17, where the anisotropy magnetic field is obtained by fitting these data using Equation (
47). The anisotropy magnetic field lies in the range of 0.9 kOe for samples at 400 °C to 3.30 and 3.70 kOe for annealing temperatures at 800 °C and 1000 °C, respectively. The value of the effective anisotropy constant for sample annealed at 1000 °C was then deduced using
measurements (
), which gave
. This value is also very close to the calculated
) by Yang et al. [
74], suggesting a core-shell nanoparticle configuration.
To explain this result, it is important to remember that in polycrystals, Mn substitution reduces grain growth and porosity. This is confirmed by the fact that the average nanoparticle size in Mn-doped Li ferrite samples annealed at 1000 °C is lower than that of undoped Li ferrite samples annealed for the same temperature, with a size of 205 nm compared to 720 nm, respectively. Moreover, Mn cations can impede the presence of ferrous cations; thus,
and
are present, where it (
) can modify the local crystal fields, and consequently, the anisotropy constant [
76]. Furthermore, it is also well-established that Mn addition in Li ferrites significantly reduces the magnetostriction constants [
77].
FMR was also used to investigate the annealing effect on Nickel Cobal ferrite nanoparticles [
78].
nanoparticles were prepared using the solvothermal method, known as a simple and inexpensive technique for preparing ferrite nanoparticles at low temperatures [
79]. A portion of these samples was then annealed for 5 h at 1000 °C and will be referred to as an-NCF nanoparticles, whereas the as-prepared NiCo ferrite nanoparticles are referred to as ap-NCF.
FMR measurements of NCF nanoparticles were performed over a temperature range between 100 K and 300 K. These FMR curves are illustrated in
Figure 18 and show an asymmetric and inhomogeneous line shape. In addition, the FMR spectra indicate the presence of two distinct resonance lines that merge into a single broad resonance line at temperatures exceeding 250 K. Multipeaked spectra are common in FMR studies with nanoparticles and have been observed by other authors such as for
, illustrated in
Figure 19, which was measured by Dutta et al. [
80]. It should be noted that additional resonances can also be observed in FMR experiments for nanoparticles, typically due to surface effects and inhomogeneities in the assembly.
The FMR spectra in
Figure 18 were then fitted using Gaussian curve [
81] as displayed in the inset of
Figure 18a for a representative sample of ap-NCF at 300 K. The obtained resonance field (
), peak-to-peak linewidth (
) and g-factor are summarized in
Table 2. From
Table 2, it can be noted that the values of the g-factor and linewidth are higher for the ap-NCF samples than in an-NCF samples, unlike the resonance field, which is higher for an-NCF nanoparticles.
To discuss these findings, it should be remembered that the FMR condition resonance can be expressed as
, where
,
, and
represent the resonance frequency, gyromagnetic ratio, and effective magnetic field, respectively.
is composed of various contributions, such as magnetic anisotropy field (
) exchange interaction field (
), demagnetization field (
), and dipole–dipole interactions (see
Section 4.2). For strongly coercive magnets,
and dipole–dipole interaction effects can be ignored.
, depends on the exchange interaction and the magnetic anisotropy and can be represented by
. Thus, this expression indicates that the magnetic anisotropy broadens
while the exchange interaction narrows it. In addition,
is determined by the effective anisotropy constant,
, as
.
is calculated using the diameter D of a spherical nanoparticle and is expressed as
, where
and
represent the bulk and surface anisotropy constants, respectively. This means that anisotropy is enhanced for smaller nanoparticles due to the factor of 6/D. On the other hand, Mossbauer studies have shown that the exchange interactions in ap-NCF nanoparticles are weaker than those in an-NCF nanoparticles because of broken bonds at large surfaces [
82]. The disorder on the surface for ap-NCF nanoparticles results in a decrease in
, increasing
and thus broadening
. In contrast, an-NCF nanoparticles are octahedral and arrange magnetic domains along different facets to reduce surface anisotropy. Moreover, annealing at 1000 °C reduces cationic disorder (which refers to the randomness or irregularity in the arrangement of cations in a crystal structure) and enhances exchange interaction among crystallites, consequently resulting in a decrease in the FMR linewidth for an-NCF nanoparticles.
For the resonance field, it is noteworthy that the ap-NCF nanoparticles exhibit a significantly higher with a significantly smaller up to 300 K, as per the equation: , where is the Bohr magneton and = 9.27 GHz, the frequency used for recording FMR spectra.
The magnetic behavior of nanoparticles is significantly influenced by the temperature dependence of the magnetic anisotropy. Antoniak et al. [
49] conducted a study where the used both FMR and SQUID data to evaluate
and
in FePt nanoparticles from two different blocking temperatures for these methods. The analysis of the FMR data was performed using the Kittel equation:
where
is the effective anisotropy field including the uniaxial contribution resulting from minor deviations from a spherical shape, surface, and step anisotropies present at the particle surface that are not averaged. Averaging of the external magnetic field angles
gives:
where
gives g-factor as 2.054 ± 0.010. The blocking temperature was evaluated by analyzing the intensity versus temperature, as the intensity of the FMR line is proportional to the magnetization. However, the blocking temperature evaluated using this method is expected to be higher than that obtained from SQUID measurements due to the significant difference in the time windows of the two methods;
s and
s. By comparing the two blocking temperatures, as shown in
Figure 6 and applying the Arrhenius relationship, the expression of the effective anisotropy can be expressed in the form:
where
and
is the mean volume. From this equation and using
0.8 and
= 110 K, it was found that
= (
at 23 K (as shown in
Figure 20) and
s. The experimental data reveals that the values of
follow a Bloch law-like relationship with a power of 2.1, described as
proportional to
[
49]. The anisotropy in this sample was discovered to be roughly ten times higher than that observed in the bulk.
5.2. Doping Effect
The use of ferromagnetic resonance to study the effects of doping on the properties of the magnetic nanoparticles is of great importance due to the potential applications of these materials in various fields, such as biomedicine and data storage. Doping, which consists of the introduction of impurities, can alter, in addition to the electrical and optical properties, the magnetic properties of nanoparticles, including their magnetic moment, magnetic anisotropy, etc. Indeed, magnetization is strongly dependent on the distribution of different ions on the different sites of the crystal lattice. Thus, understanding how different dopants affect these properties can lead to the development of more efficient and versatile magnetic nanoparticles depending on the targeted applications.
The study of the influence of changes in chemical composition by introducing Zn content in
nanoparticles (x = 0.4, 0.5, 0.6, 0.7) were studied by Tsay et al. [
83]. These nanoparticles were synthesized using the hydrothermal method. XRD data allowed an estimation of the average nanocrystallite sizes
nm to
nm, which was consistent (9 nm to 10.3 nm) with data extracted from TEM images as shown in
Figure 21 (left).
The FMR spectra of the Mg-Zn ferrite nanoparticles are plotted in
Figure 21, from which the values of resonance field (
) g-factor and linewidth are obtained and given in
Table 3. For these nanoparticles, the g-factor and resonance field varied independently of the Zn content, unlike the linewidth, which increases as a function of Zn content up to x = 0.6 before significantly decreasing to 265 Oe for x = 0.7. On closer inspection, it is observed that the FMR line is not symmetrical (particularly for x = 0.4, 0.5, and 0.6), which could also distort the measurements of the linewidth. However, these linewidth values compared to other ferrite nanoparticles are narrower, suggesting a lower magnetic loss and a better magnetic field homogeneity [
84].
The doping of nonmagnetic ions in Mg-Zn ferrite nanoparticles such as scandium (Sc) was investigated for the first time by Angadi et al. [
85]. They synthesized
-doped
(x = 0.00, 0.01, 0.03, and 0.05) nanoparticles using solution combustion method.
Dynamic magnetic properties have been investigated using FMR at 9.1 GHz, where the obtained spectra are represented in
Figure 22. The value of
,
, and g-factor are summarized in
Table 4. While the resonance field shows no correlation with the Sc composition, the peak-to-peak linewidth decreases with increasing Sc. Note that the values of
for x = 0.00 and x = 0.01 are very close, as it is also the case for x = 0.03 and x = 0.05. Moreover, considering the FMR lines in
Figure 22, it appears that it is composed of two resonances very close to each other (x = 0.00 and x = 0.01), which merge from x = 0.03. This can also explain the drop of
between x = 0.01 and x = 0.03 and the broadening of the FMR line for these two samples.
FMR can also be employed to investigate how the use of multiple dopants affects the magnetic properties of nanoparticles. For this purpose Sharma et al. [
86] prepared a series of samples based on barium hexaferrite nanoparticles doped with cobalt (Co), samarium (Sm), and neodymium (Nd). The obtained samples are identified as S1 (
), S2 (
), S3 (
), and S4 (
).
FMR measurements were performed in field sweep mode as shown in
Figure 23a for sample S3. The extracted
values were then plotted as a function of frequency as illustrated in
Figure 23b, where it can be inferred that the Sm-doped hexaferrite (S4) exhibits the highest resonance field at a specific frequency, whereas sample S2 (without Nd or Sm substitution) shows the lowest resonance field. The experimental data in
Figure 23b can be fit using a spherical nanoparticle model where the effective magnetic field and the resonance field are given by:
where
is the interparticle interaction field and
the Larmor precession frequency. The author did not give more details on the fitting parameters, nor on the deduced values of
, for example. However, the measurement of the linewidth (shown in
Figure 24) has been well developed. The experimental data of
Vs. frequency was first fitted using the Landau–Lifshitz–Gilbert (LLG) damping model:
where
is the Gilbert damping parameter and
is the inhomogeneous contribution to the linewidth. The application of Equation (
53) yields a linear relationship between linewidth and frequency (dotted lines in
Figure 24). However, upon careful examination of experimental
values, it was observed that the high-frequency values exhibit a downward curvature. In order to explain this behavior in the high-frequency region, a nonlinear model proposed by Bastrukov et al. [
87] was used to fit the data points (solid lines in
Figure 24). Thus, the modified LLG model:
where
is the intrinsic contribution to damping (the Gilbert parameter), and
represents the overall extrinsic contributions to damping such as defects or inhomogeneities in the nanoparticles system.
Table 5 shows the obtained FMR parameters after fitting data in
Figure 24 by Equation (
53) and (
54).
From the
Table 5, the LLG model reveals that the highest
was measured for sample S4, where
increases from
for sample S2 to
for sample S4. Indeed, the steepness of the line for S4 in
Figure 24 is due to the slope of (
) being 0.026 for S4, whereas it is half this value (0.012) for S2. The value of
remains unchanged by doping. In the same way, using the modified LLG model,
Gilbert damping was found to increase from
for sample S2 to
for sample S4, while the extrinsic damping coefficient (
) was highest for sample S3 (
) and lowest for the sample S2 (
). These observations allow us to conclude that Nd or Sm ion substitution in Barium hexaferrite induces an increase in the Gilbert damping parameter.
In another study, Pessoa et al. [
14] used different synthesis techniques to produce iron-based mixed cubic ferrites
(x = 0.0, 0.4 and 1.0). These nanoparticles were synthesized by hydrothermal (HT), forced hydrolysis (FH), and combustion (C) methods [
14]. From the X-ray measurements, a large difference in the average grain size was observed depending on the synthesis method. The FH method, for example, resulted in nanoparticles with a lower degree of crystallinity and average grain size below 10 nm as shown in
Table 6.
FMR measurements were carried out at room temperature for all the samples in order to investigate their magnetic anisotropy. The measured FMR spectra are shown in
Figure 25, where the solid lines represent experimental data whereas the dotted lines represent the fits using the model proposed by Tsay et al. [
65].
Table 6 summarizes the resulting values of the peak-to-peak linewidth (
), the magnetic anisotropy field (
), and the g-factor. For the same synthesis method,
Figure 25 shows an increase in the FMR spectra symmetry as a function of increasing x, which allows a better fit.
Table 6 highlights a noteworthy trend, where the peak-to-peak linewidth decreases with the increase in nickel content (x). This is in good agreement with a better fit of the FMR lines for x = 1. It also aligns with the fact that nickel ferrites (x = 1) have a smaller absolute value of crystalline anisotropy than cobalt ferrites (x = 0), as shown by the extracted magnetocrystalline anisotropy field in
Table 6. Indeed, as the magnetocrystalline anisotropy field value increases, the external magnetic field becomes stronger and able to sweep the anisotropy from the [111] direction to the [100] direction, which corresponds to the larger (positive) and smaller (negative) crystalline anisotropy, respectively [
64]. It should also be noted that the FH method gives samples with the lowest
and
values.
g-factor was also extracted from FMR line fitting, and did not show any dependence on x, but gave the lowest values with the FH technique. However, the g-factor values shown in
Table 6 are very unusual. Indeed, the model proposed by Tsay et al. only considers the magnetocrystalline anisotropy field [
65]. Consequently, other contributions to the magnetic effective field, such as shape anisotropy, dipolar, and stress are included in the g-factor, which is employed as a fitting parameter, responsible for horizontally shifting the fitting curve.
5.3. Other Effects
Self-assembly (regular arrays) of monodisperse Co FCC nanocrystals, with a diameter of 12 nm, were studied by Spasova et al. [
88] using FMR. The effective magnetic resonance field was measured as a function of the orientation of the externally applied field with in-plane (azimuthal) and out-of-plane (polar) directions. For the interpretation of the angular dependencies of the resonance field (
), the authors considered the effective magnetic anisotropy of the individual particles (shape, volume, and surface contributions) and the shape anisotropy caused by the stripes and the magnetic anisotropy due to the particle interaction inside the stripe.
The in-plane resonance field could be obtained by introducing a cubic symmetry, in-plane uniaxial anisotropy, and the small in-plane shape anisotropy (
) as commonly used in thin films:
where
is the angle of the applied external magnetic field with respect to the axis of in-plane uniaxial anisotropy.
and
are the effective uniaxial and four-fold in-plane magnetic anisotropy, respectively. The effective anisotropy field can be written as
, where
is the perpendicular anisotropy energy and
the volumetric filling factor of the obtained stripes.
The fit of the experimental data using Equation (
55) is shown (solid line) in
Figure 26a and is in good agreement with the experiment. This fit gave
T,
(only uniaxial anisotropy is present) and
T.
For the polar dependence of the applied magnetic field, the equation for the resonance gives:
the fit of the experimental data with the above equation (see
Figure 26b) gives, once again,
T and
T, without the requirement of incorporating fourth-order contributions. This particular uniaxial anisotropy is attributed to the shape anisotropy of the stripes and the potential alignment of the crystalline anisotropy axes of the individual crystals.
The FMR technique has also made it possible to study the effect of the dipole–dipole interaction in superparamagnetic nanoparticles. Slay et al. [
23] have experimentally investigated iron-oxide (
) nanoparticles with an average diameter of (10 ± 1) nm. They developed a theoretical model that allowed them to fit the obtained FMR spectra in order to interpret these experimental results. For this, they considered contributions from the externally applied magnetic field, magnetocrystalline anisotropy, and shape anisotropy, which includes dipole–dipole interactions. The angle dependence of the resonance field at fixed frequency gives [
23]:
where
is the cubic magnetocrystalline anisotropy field and
is the effective field due to dipolar interaction, with
is the effective demagnetizing tensor. Indeed, as it is not possible to explicitly calculate the dipole–dipole interactions for a group of particles, the authors represented the particle interactions as an effective anisotropy field called
.
The resonance field
was then computed from Equation (
57) to produce individual FMR spectra for all
and
directions. This procedure was then repeated for a distribution of the particle sizes. The shape of each FMR spectrum can be described as a Gaussian derivative, which corresponds to the derivative of the imaginary part of the susceptibility
:
In this context, the weight function
is used to project the contribution of each resonance onto the field axis. It is assumed that the particle axes are uniformly distributed in orientation. Additionally,
represents the linewidth distribution, and the parameter
scales the contribution of each particle size, which is assumed to follow a normal distribution:
where
= 10 nm is the average particle diameter, and
represents the particle size distribution, i.e., the deviation from
. More details on the theoretical model and the expression of the linewidth distribution are available on [
23].
FMR measurements of suspended (in toluene) and dried commercial
nanoparticles have been carried out as shown in
Figure 27. A difference is clearly visible between the FMR spectra for the particles in suspension and after drying. In suspension, the spectrum is asymmetric with a resonance field of
= 323 mT. Additionally, sharp paramagnetic resonances are observed at 170 and 338 mT corresponding to g ≈ 4 and 2, respectively, in addition to the broad ferromagnetic spectrum. However, these features disappear upon drying, and the spectrum becomes symmetric with a resonance field of =
294 mT. Slay et al. [
23] deducted that these sharp features are due to surface effects caused by surfactants (specifically, oleic acid) because they disappear once the surfactants are neutralized. Thus, the peak with g ≈ 4 was attributed to
in a rhombohedrally distorted ligand field, and the peak with g ≈ 2 to
radicals, as has already been demonstrated for a similar sample [
89].
The fits with the developed model, yielded to = 63 mT and = −3 mT (using g = 2.11) for the nanoparticles in suspension, and = −50 mT and = −4 mT (using g = 2.24) for dried nanoparticles. Note that in comparing the two fits, with or without the distribution (), the parameters (g, , and ) varies only about between these two situations. Therefore, it is possible to fit these spectra with or without taking the distribution into consideration. However, by taking the distribution into account, the fit was significantly improved with high fidelity.
The
= 63 mT for the suspension, indicated an easy-axis symmetry with
= 0.11, which corresponds to an average elongation of c/a ≈ 1.3 for the prolate ellipsoid. Despite the uniformly shaped particles with nearly spherical symmetry as observed in the TEM images [
23], the value of
is not a result of particle shape, but rather dipolar interactions between the particles. This assumption is supported by the change of sign of the dipolar field upon drying (from positive to negative), as it is the configuration of the nanoparticles that changes and not their shape. Indeed, in the dried state, when the nanoparticles cluster together, the anisotropy changes to easy-plane (negative uniaxial field). Thus, this modification in the magnetic properties illustrates the impact of dipole–dipole interactions in short timeframes since it is measurable within the FMR experiments with a
s time window.