1P.L.Kapitza Institute for Physical Problems, RAS, Moscow, 119334 Russia
2National Research University Higher School of Economics, Moscow, 101000
Russia 3Voronezh State Technical University, Voronezh, 394026 Russia 4National
Research Centre Kurchatov Institute, Moscow, 123182 Russia 5Kotelnikov
Institute of Radio Engineering and Electronics, Fryazino Branch, RAS,
Fryazino, Moscow oblast, 141190 Russia

A. B.Drovosekov1 N. M.Kreines1 D. A.Ziganurov12 A. V.Sitnikov34 S. N.Nikolaev4
V. V.Rylkov45

Specific features of $g\approx 4.3$ EPR line …

# Specific features of $g\approx 4.3$ EPR line behavior in magnetic
nanogranular composites

###### Abstract

Films of metal-insulator nanogranular composites MxD100-x with different
composition and percentage of metal and dielectric phases (M = Fe, Co, CoFeB;
D = Al2O3, SiO2, LiNbO3; $x\approx 15{-}70$ at. %) are investigated by
magnetic resonance in a wide range of frequencies ($f=7{-}37$ GHz) and
temperatures ($T=4.2{-}360$ K). In addition to the usual ferromagnetic
resonance signal from an array of nanogranules, the experimental spectra
contain an additional absorption peak, which we associate with the electron
paramagnetic resonance (EPR) of Fe and Co ions dispersed in the insulating
space between the granules. In contrast to the traditional EPR of Fe and Co
ions in weakly doped non-magnetic matrices, the observed peak demonstrates a
number of unusual properties, which we explain by the presence of magnetic
interactions between ions and granules.

##  1 Introduction

Magnetic nanoparticles and nanogranular systems have long been a subject of
intensive research, due to unusual physical properties of these objects, as
well as the wide possibilities of applications [1, 2, 3].

Magnetic metal-insulator nanogranular composites (nanocomposites) are an array
of ferromagnetic (FM) nanogranules randomly distributed in a solid dielectric
medium (matrix). In our previous works [4, 5, 6, 7, 8], we studied
nanocomposite films of composition MxD100-x based on the metal FM alloy M =
Co40Fe40B20 (hereinafter CoFeB for brevity) and dielectrics D = Al2O3 and
LiNbO3. The value $x$ in the structural formula MxD100-x reflects the nominal
content of the metal phase in the nanocomposite, a significant part of which
forms FM nanogranules. At the same time, a specific feature of the studied
films was the high concentration of paramagnetic (PM) ions Fe and Co dispersed
in the insulating space between CoFeB granules [9, 10, 11]. It has been shown
that the presence of such ions in the dielectric gaps between the granules
leads to the manifestation of unusual electrical, magnetic and
magnetoresonance properties of nanocomposites, due to the increase of
intergranular electron tunneling and exchange interaction [12, 13, 4, 11, 14,
15, 5, 6].

In the works [7, 8] nanocomposite films (CoFeB)x(Al2O3)100-x and
(CoFeB)x(LiNbO3)100-x ($x\approx 30{-}55$ at. %) were studied by magnetic
resonance in a wide range of frequencies ($f=7{-}37$ GHz) and temperatures
($T=4.2{-}360$ K). It was found that in addition to the usual ferromagnetic
resonance (FMR) signal from an ensemble of FM granules, the experimental
spectra of the films contained an additional much weaker absorption peak. The
frequency-field dependence $f(H)$ for this peak proved to be linear in the
high frequency region and characterized by an effective $g$-factor $g\approx
4.3$ typical of electron paramagnetic resonance (EPR) of Fe3+ ions in
amorphous solids [16, 17, 18]. It should be noted that such resonance is often
observed while studying the iron-based nanoparticles in various media [19, 20,
21, 22, 23]. However, unlike the traditional EPR of Fe3+ ions in weakly doped
matrices, in our case the observed peak demonstrates a number of unusual
properties:

  * •

its intensity proves to be approximately the same in the transverse and
longitudinal geometry of the resonance excitation;

  * •

the dependence $f(H)$ demonstrates the presence of a finite frequency in the
zero field, which increases with the growth of the FM phase content;

  * •

the position of the peak depends on the orientation of the magnetic field with
respect to the film plane;

  * •

as temperature decreases, the peak shifts towards weaker fields and decreases
in intensity, disappearing at $T\lesssim 60$ K.

Nanogranular films have previously been studied by magnetic resonance in a
number of works [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41]. Often, besides the main FMR line, the authors observed additional
absorption peaks, which were associated either with the inhomogeneity of the
samples [30, 31, 32], or with the excitation of inhomogeneous oscillations in
the films [33, 34], in particular spin-wave and surface modes [35, 36, 37, 38,
39, 40, 41]. Observation of several resonance peaks caused by the excitation
of inhomogeneous modes is also possible in ordered arrays of magnetic
nanoparticles [42, 43, 44, 45].

In our case, the behavior of the additional peak does not agree with the
described scenarios. It proves to be more productive to assume that this peak
is associated with EPR of Fe3+ ($g\approx 4.3$) ions dispersed in the
insulating space between FM granules. As shown in [7], the frequency and
orientational dependencies of the resonance field for the additional peak are
well described taking into account the shift of the EPR frequency due to
dipole-dipole and exchange interactions of the Fe3+ ions with the ensemble of
FM granules. However, the proposed model does not entirely explain the non-
standard excitation conditions of this peak and its anomalous behavior with
temperature [8].

In order to further clarify the reasons for the anomalous behavior of the EPR
peak, in this work, we study a wider set of nanocomposites with various
compositions MxD100-x. Besides the systems based on the CoFeB alloy, we
investigate films in which FM granules are formed of pure iron or cobalt in
different insulating matrices D = Al2O3, SiO2, LiNbO3. Furthermore, the range
of the FM phase content ($x\approx 15{-}70$ at. %) is significantly expanded
as compared with previous works.

##  2 Samples, their preliminary characterization, methods

Nanocomposite films MxD100-x with a thickness of $\approx 1{-}3$ $\mu$m were
synthesized by ion beam sputtering on glass-ceramic substrates using composite
targets [46, 47, 48]. The target represents a plate of FM metal Fe, Co or an
alloy Co40Fe40B20 (CoFeB), on which a number of rectangular strips of oxides
Al2O3, SiO2 or LiNbO3 are placed. The uneven arrangement of dielectric strips
on the target surface allows the formation of a nanocomposite film MxD100-x
with a smooth controlled change in the concentration $x$ along the substrate
in a wide range $\Delta x\approx 30{-}40$ at. %. Further studies are carried
out on individual pieces of the grown film with a size of $5\times 5$ mm2, so
that the change of $x$ within one sample is less than 1 at. %. The content of
the metal phase in the films was determined by energy dispersive X-ray
microanalysis. The following series of samples were studied:

(CoFeB)x(Al2O3)100-x, $x\approx 15{-}56$ at. %;

Fex(Al2O3)100-x, $x\approx 31{-}58$ at. %;

(CoFeB)x(SiO2)100-x, $x\approx 20{-}67$ at. %;

Cox(SiO2)100-x, $x\approx 24{-}67$ at. %;

(CoFeB)x(LiNbO3)100-x, $x\approx 30{-}48$ at. %;

Cox(LiNbO3)100-x, $x\approx 33{-}41$ at. %;

According to transmission electron microscopy data, the obtained composites
consist of crystalline FM nanogranules randomly distributed inside the
amorphous oxide matrix [12, 48, 49, 50]. The granules usually have
approximately spherical shape with diameter $\approx 2{-}8$ nm depending on
the composition and concentration $x$. However, in the case of the LiNbO3
matrix, the granules tend to stretch in the direction of film growth up to
$\approx 20$ nm [13, 14]. The structures Cox(LiNbO3)100-x are characterized by
strongly inhomogeneous distribution of the shape and size of granules over the
film thickness [14].

Depending on the composition, the percolation threshold $x_{p}$ of
nanocomposites differs within the range $x_{p}\approx 45{-}60$ at. % [12, 13,
14, 28, 47, 48]. Slightly below this threshold at $x_{c}<x<x_{p}$
($x_{p}-x_{c}\approx 5{-}10$ at. %), the films demonstrate an interesting
logarithmic temperature dependence of the conductivity [11, 12, 13, 14],
typical of granular systems with ‘‘strong tunnel coupling’’ between the
granules [51]. According to magnetic data, approximately in this concentration
range, the samples demonstrate a transition from superparamagnetic to
ferromagnetic behavior [5, 15, 26, 27, 28, 29, 38]. At the same time, a sharp
increase in the magnetization of the films is observed in the low temperature
region, which is explained by a large number of magnetic ions Fe and Co
dispersed in the insulating matrix [9, 10, 11, 12].

In this work, the nanocomposite samples are studied by magnetic resonance in a
wide range of frequencies ($f=7{-}37$ GHz) and temperatures ($T=4.2{-}360$ K)
using a laboratory transmission-type spectrometer based on rectangular and
tunable cylindrical resonators [7]. The experiments are carried out at
different orientations of the external field $\mathbf{H}$ (up to 17 kOe) with
respect to the film plane. Note that in the case of in-plane field, it is
possible to realize both transverse ($\mathbf{h}\perp\mathbf{H}$) and
longitudinal ($\mathbf{h}\parallel\mathbf{H}$) geometry of resonance
excitation by the microwave field $\mathbf{h}$ [7].

##  3 Magnetic resonance spectra and their discussion

###  3.1 The case of in-plane field (room temperature)

Almost for all samples, regardless of composition, the qualitative behavior of
the magnetic resonance spectra looks identical (Fig. 1). In usual transverse
resonance excitation geometry ($\mathbf{h}\perp\mathbf{H}$), one intensive FMR
peak is observed as a rule. The width of this peak varies for different
compositions. In the case when the FMR peak is sufficiently narrow, as for the
system (CoFeB)x(Al2O3)100-x, in lower fields it is possible to resolve a
second, less intensive absorption peak, which we associate with EPR of
magnetic ions dispersed in the insulating matrix [7]. When the resonance
excitation geometry changes to a longitudinal one
($\mathbf{h}\parallel\mathbf{H}$), the intensity of the FMR peak drops
significantly, which is natural, but the amplitude of the EPR peak remains
approximately the same. As a result, the EPR peak is much better manifested in
the geometry $\mathbf{h}\parallel\mathbf{H}$ and is reproduced almost for all
compositions of the films, with the exception of Cox(LiNbO3)100-x [7].

Figure 1: Room temperature spectra for nanocomposite films MxD100-x ($x\approx
30{-}40$ at.%) with different compositions of metal and dielectric phase (M
and D). Spectra are obtained in magnetic field applied in the film plane at
frequency $f\approx 25$ GHz in transverse ($\mathbf{h}\perp\mathbf{H}$) and
longitudinal ($\mathbf{h}\parallel\mathbf{H}$) geometries of resonance
excitation. Figure 2: Spectra for films (CoFeB)x(Al2O3)100-x with different
content of the FM phase $x$. Spectra are obtained in magnetic field applied in
the film plane in longitudinal geometry of resonance excitation
($\mathbf{h}\parallel\mathbf{H}$) at frequency $f\approx 21$ GHz ($T=296$ K).
Spectra are normalized on the FMR peak amplitude. Figure 3: Frequency-field
dependencies $f(H)$ for FMR and EPR peaks in films (CoFeB)x(Al2O3)100-x with
different content of the FM phase $x$. Magnetic field is applied in the film
plane ($T=296$ K). Points are the experimental data, solid lines correspond to
Kittel formula (1), dashed lines are linear dependencies (2).

The position and intensity of the EPR peak depends on the content of the metal
FM phase $x$ in the films. Interestingly, it is best manifested at
concentrations well below the percolation threshold $x_{p}$. When approaching
this threshold, the intensity of the EPR peak decreases, and completely
vanishes beyond $x_{p}$.

The detailed concentration dependence of the magnetic resonance spectra was
obtained for the system (CoFeB)x(Al2O3)100-x (Fig. 2). In the limit of low
concentrations $x<25$ at. % the FMR line approaches the nominal position
corresponding to $g$-factor $g\approx 2.1$, which is characteristic of bulk FM
metals Fe, Co and their alloys. This peak is obviously associated with the
resonance from the array of weakly interacting FM nanoparticles. Note that
instead of the term ‘‘FMR’’, the term ‘‘superparamagnetic resonance’’ can be
used in this case [21].

The position of the second peak at $x<25$ at. % corresponds to the effective
$g$-factor $g\approx 4.3$, characteristic of EPR for Fe3+ ions in amorphous
solids. Note that the obtained type of magnetic resonance spectra for
nanocomposites in the limit of low FM phase concentrations is quite consistent
with the results of other authors for iron-based nanoparticles in various
media [19, 20, 21, 22, 23]. In our case, however, there is an interesting
trend that the EPR peak with $g\approx 4.3$ is better manifested in the
unusual longitudinal geometry of resonance excitation
($\mathbf{h}\parallel\mathbf{H}$).

Note another not quite expected result, that in our case the EPR peak
($g\approx 4.3$) is observed not only for systems based on iron-containing
granules Fe and CoFeB, but also for the cobalt-based nanocomposite
Cox(SiO2)100-x (Fig. 1). The theory of EPR predicts the possibility of
effective $g$-factor $g\approx 4.3$ for Co2+ ions in the case of octahedral
ligand field [52]. Experimentally, such a line is also sometimes observed in
some cubic crystals or nanocrystallites with Co ions [53, 54, 55]. Perhaps a
similar scenario is realized in our case. It is also possible that the
observed lines with $g\approx 4.3$ are caused by the excitation of
‘‘forbidden’’ transitions between spin states of PM centers with the change of
the spin projection $\Delta m_{S}=\pm 2$. Note that transitions of this type
can be excited by both transverse and longitudinal microwave magnetic field
[52].

An increase of the FM phase content in the films leads to the growth of
magnetodipole interactions in the system and the appearance of significant
demagnetizing fields. In this situation, the FMR line shifts towards weaker
fields (Fig. 2). At the same time, a similar shift of the resonance field is
also observed for the EPR peak.

Figure 3 shows the frequency-field dependencies $f(H)$ for both absorption
peaks in films (CoFeB)x(Al2O3)100-x. At low FM phase content $x<25$ at. %
these dependencies are close to linear with effective $g$-factors $g\approx
2.1$ and $g\approx 4.3$ for FMR and EPR peaks, respectively. At higher
concentrations $x$, the dependence $f(H)$ for the FMR peak is described by the
Kittel formula

$f=\gamma_{\mathrm{FMR}}\sqrt{H(H+4\pi M)},$ (1)

where the gyromagnetic ratio $\gamma_{\mathrm{FMR}}\approx 2.92$ GHz/kOe
corresponds to $g$-factor $g\approx 2.1$, and the effective demagnetizing
field of the film $4\pi M$ increases with the growth of $x$.

For the EPR peak, the dependence $f(H)$ in the high frequency region is
described by a linear function

$f=\gamma(H+\delta H),$ (2)

where the gyromagnetic ratio $\gamma\approx 6.0$ GHz/kOe corresponds to
$g$-factor $g\approx 4.3$, and the line shift $\delta H$ increases with the
growth of $x$.

According to a simple model proposed in [7], the line shift $\delta H$ in
formula (2) arises due to the interaction of PM ions with FM granules and is
determined by the effective field

$\delta H=JM,$ (3)

which acts on the PM ion from the ensemble of FM granules ($M$ is average
magnetization of the array of FM granules, $J$ is the effective field
constant).

In [7, 8] this field was associated with the exchange interaction between ions
and granules. In this case, the dimensionless constant $J$ has the meaning of
the effective exchange field parameter. As will be shown below, an alternative
explanation of the effective field $JM$ of a dipole-dipole nature is possible.

###  3.2 The diagram $\delta H-4\pi M$. Manifestation of the ‘‘Lorentz
field’’?

Equations (1)$-$(3) qualitatively explain the correlated shift of FMR and EPR
lines to low fields with an increase of the concentration $x$. Indeed, in both
cases, the shift of the absorption peak is determined by the average
magnetization of the film $M$. The values $4\pi M$ and $\delta H$ can be
determined experimentally from the positions of the FMR and EPR peaks,
respectively. In this case, the effective constant $J$ is determined by the
ratio $J/4\pi=\delta H/4\pi M$.

It should be noted that the effective field $4\pi M$ in formula (1), generally
speaking, may depend on the shape and anisotropy of the FM granules and differ
from the static value $4\pi M$ of the film. The equivalence of these values
can be expected in the case of spherical granules in the absence of any
preferred anisotropy axis [56, 57]. In our case, this condition seems to be
fulfilled for most structures, with the exception of nanocomposites based on
the LiNbO3 insulating matrix, where the granules have a shape elongated in the
direction of film growth [4, 9, 14].

Assuming the exchange nature of the effective field $\delta H$, one would
expect that the constant $J$ in equation (3) should depend on many factors:
the chemical composition of the granules and the dielectric matrix, the
content of the FM phase in the nanocomposite, temperature. However, this
constant proves to be quite universal.

Figure 4 shows a summary diagram $\delta H-4\pi M$ for all the films studied.
It can be seen that, regardless of the specific composition of the films, the
experimental points lie near some universal linear dependence. This
universalism means that the shift of the EPR field $\delta H$ is mainly
determined by the average magnetization of the film, and therefore has a
dipole-dipole origin, similar to the demagnetization field $4\pi M$. The ratio
$\delta H/4\pi M$ is found to be about 1/3, that is, $\delta H\approx 4\pi
M/3$, and the constant $J$ approximately corresponds to the demagnetizing
factor of the sphere $J\approx 4\pi/3$.

Figure 4: Summary diagram $\delta H-4\pi M$ for all investigated samples.
Points are the experimental data obtained in the present work and in previous
works [7, 8]. Dashed line corresponds to the dependence $\delta H=4\pi M/3$.

This result suggests that the EPR peak shift $\delta H$ is associated with the
so-called ‘‘Lorentz field’’ of dipole nature, which acts on PM ions from the
ensemble of FM granules. The concept of the Lorentz field arises in the
problem of calculating local fields inside a (quasi-) continuous medium,
taking into account its real inhomogeneity at the microscopic level. This
concept is better known from general physics courses, where it is applied to
the calculation of local electric fields in dielectric crystals [58, 59], as
well as to the determination of local magnetic fields in magnetically ordered
media by NMR [60] and muon spectroscopy [61]. Its application to the magnetism
of nanostructures is sometimes discussed in theoretical works [56, 57, 62]. A
possible experimental manifestation of the Lorentz field in such systems was
considered in [63, 64] while studying the temperature dependence of the
susceptibility of nanogranular films.

The Lorentz method of calculating the local field $\mathbf{H}_{loc}$ at some
selected point inside an inhomogeneous medium suggests the decomposition of
this field into several components:

$\mathbf{H}_{loc}=\mathbf{H}+\mathbf{H}_{dem}+\mathbf{H}_{L}+\sum\mathbf{H}_{dip}.$
(4)

Here $\mathbf{H}$ is the external magnetic field, $\mathbf{H}_{dem}$ is the
demagnetization field associated with the shape of the sample. For a thin
film, this term has the form

$\mathbf{H}_{dem}=-4\pi\mathbf{M}_{\perp},$

where $\mathbf{M}_{\perp}$ is the vector component of magnetization normal to
the film plane. The third term in (4) $\mathbf{H}_{L}$ is the Lorentz field
created by a fictitious spherical cavity (‘‘Lorentz sphere’’) cut out in a
magnetized medium around the selected point. It is determined by the well-
known formula

$\mathbf{H}_{L}=\frac{4\pi}{3}\mathbf{M}.$

Finally, the last term in (4) is the sum of the dipole fields
$\mathbf{H}_{dip}$ created at the selected point by magnetic dipoles (FM
granules) located inside the Lorentz sphere.

The EPR frequency of magnetic ions is determined by the local fields on each
of them, taking into account these four contributions:
$f=\gamma|\mathbf{H}_{loc}|$. Note, that the last term in (4) has random
nature due to the disorder in the granules arrangement inside the Lorentz
sphere. It can be assumed that the average value of this contribution over all
PM centers is zero. This assumption seems to be reasonable in the case of
isotropic distribution of spherical granules in the dielectric matrix [59,
65]. In this situation, the last term in (4) leads only to a broadening of the
resulting EPR line, while the shift of the absorption peak is determined by
the first three terms:

$f=\gamma|\mathbf{H}-4\pi\mathbf{M}_{\perp}+4\pi\mathbf{M}/3|.$ (5)

In the case of in-plane orientation of the magnetic field, the demagnetizing
field is absent ($\mathbf{M}_{\perp}=0$). At the same time the situation
$\mathbf{H}\parallel\mathbf{M}$ is realized, and equation (5) transforms to

$f=\gamma(H+4\pi M/3).$ (6)

Thus, the presence of a Lorentz field in a granular medium can explain the
shift of the EPR peak $\delta H=4\pi M/3$. Experimentally observed deviations
from this value, which are seen in Fig. 4 as a scatter of points relative to
the theoretical straight line, can be caused by various reasons: not quite
accurate account for the contribution of the fourth term in equation (4), the
non-spherical shape of the granules, the presence of additional exchange
interactions.

Note that the best agreement with the model is achieved for nanocomposite
films (CoFeB)x(Al2O3)100-x, which are characterized by a close to spherical
shape of granules with small diameters $2{-}4$ nm [9, 12]. Films of this
composition are also distinguished by the narrowest resonance peaks (Fig. 1),
indicating a higher degree of homogeneity of the system.

Note that the presence of random magnetic interactions in the system leads to
deviations of local fields at PM centers from the direction of the external
magnetic field. In this situation, EPR can be excited not only by transverse,
but also by longitudinal microwave field, which behavior is observed
experimentally.

###  3.3 The case of out-of-plane field (room temperature)

According to equation (5), in the case of deviation of the magnetic field from
the film plane by an arbitrary angle $\theta_{H}$, it is necessary to take
into account the additional contribution to the EPR frequency associated with
the appearance of the demagnetizing field $4\pi\mathbf{M}_{\perp}$. In the
particular case of normal orientation of the field, when
$\mathbf{H}\parallel\mathbf{M}=\mathbf{M}_{\perp}$, the frequency-field
dependence takes the form

$f=\gamma(H-8\pi M/3).$ (7)

Thus, comparing with the in-plane geometry, the EPR peak is shifted towards
stronger fields. This effect is observed experimentally (Fig. 5). Note that
when the magnetic field deviates from the film plane, the FMR line also shifts
towards stronger fields. In a normal field, the position of the FMR peak is
determined by the well-known Kittel formula:

$f=\gamma_{\mathrm{FMR}}(H-4\pi M).$ (8)

As it was shown in [7], the angular dependence of the EPR field
$H_{res}(\theta_{H})$ can be calculated analytically, neglecting the
dependence of the film magnetization on the magnetic field. Such approximation
can be considered adequate for films with sufficiently high content of the FM
phase in not too weak magnetic fields, when the effects of superparamagnetism
can be disregarded. In the work [7], a good agreement between the calculated
and experimental angular dependencies $H_{res}(\theta_{H})$ was demonstrated.

Let us show that taking into account the superparamagnetism of films and the
presence of the Lorentz field, the frequency-field dependencies for the EPR
peak can be well described in cases of in-plane and normal field, using
formulas (6), (7).

Figure 6a shows the experimental dependencies $f(H)$ for FMR and EPR peaks
obtained in two geometries for one of the samples (CoFeB)x(Al2O3)100-x. The
dashed lines correspond to the calculation according to Kittel equations (1),
(8) for the FMR peak and (6), (7) for the EPR peak under the assumption of
constant value $4\pi M=4\pi M_{S}$ (‘‘ideal’’ FM film). It can be seen that in
this case the formulas work only in the region of high frequencies (strong
fields). Deviations in weak fields are explained by a decrease of $4\pi M$ as
compared to the saturation value $4\pi M_{S}$.

Figure 5: Magnetic resonance spectra in nanocomposite (CoFeB)47(Al2O3)53 for
different orientations of the magnetic field with respect to the film plane
($\theta_{H}$). Spectra are obtained at room temperature at frequency
$f\approx 25$ GHz in transverse geometry of resonance excitation
($\mathbf{h}\perp\mathbf{H}$). Figure 6: (a) Frequency-field dependencies
$f(H)$ for FMR and EPR peaks in the film (CoFeB)48(Al2O3)52 in the cases of
in-plane and normal field ($T=296$ K). (b) Field dependencies of $4\pi M$
value. Points are the experimental data, lines are calculations. The dashed
lines correspond to the case of ‘‘ideal’’ FM film with $4\pi M=4\pi M_{S}$,
the solid lines are obtained taking into account the filed dependence of $4\pi
M$ value (see text).

In Fig. 6b, the field dependencies $4\pi M(H)$ for in-plane and normal
geometries are recalculated from experimental FMR data using Kittel formulas
(1), (8). In the case of in-plane geometry, the resulting dependence $4\pi
M(H)$ can be well described by the usual Langevin function $L(x)$ for
superparamagnets

$4\pi M=4\pi M_{S}\cdot L\left(\frac{\mu H}{k_{B}T}\right),$

where $\mu$ is the magnetic moment of FM granules, $k_{B}$ is the Boltzmann
constant (for the considered nanocomposite $\mu\approx 10^{4}\mu_{B}$ Bohr
magneton). In normal geometry, the experimental dependence $4\pi M(H)$ can be
well approximated by a similar equation with the replacement of $H$ by the
‘‘internal’’ field $H_{in}=H-4\pi M$:

$4\pi M=4\pi M_{S}\cdot L\left(\frac{\mu(H-4\pi M)}{k_{B}T}\right)$

with the same parameters $\mu$ and $4\pi M_{S}$. In this case, the function
$4\pi M(H)$ is not expressed explicitly, but can be defined parametrically
considering the value $H_{in}$ as a parameter.

Taking into account the calculated field dependencies of $4\pi M$ in two
geometries, the resulting dependencies $f(H)$ obtained by Kittel formulas (1),
(8) for the FMR peak and (6), (7) for the EPR peak are plotted in Fig. 6a by
solid lines. It can be seen that taking into account the Lorentz field in
equations (6), (7) provides a close to perfect approximation of the
experimental behavior for the EPR peak.

###  3.4 Temperature evolution of the spectra

In the work [8], we investigated the temperature evolution of magnetic
resonance spectra in nanocomposite films (CoFeB)x(Al2O3)100-x with FM phase
content $x\approx 47{-}51$ at. % and (CoFeB)x(LiNbO3)100-x with $x\approx
30{-}40$ at. %. Qualitatively, the samples showed identical behavior. In the
case of magnetic field applied in the film plane, temperature decrease from
360 to 4.2 K initiates a monotonic shift of the FMR peak towards weaker
fields, which behavior is explained by an increase of the film magnetization
and, as a consequence, the demagnetizing field $4\pi M$. At the same time, the
EPR peak ($g\approx 4.3$) also shifts to weaker fields in accordance with the
formula $\delta H\approx JM$, where $J\approx 4\pi/3$ (see Fig. 4).

An unusual feature in the behavior of the EPR peak was that its intensity
reduced with decreasing temperature, and below $T\approx 60$ K it disappeared.
In the present work, it is found that the similar behavior of the EPR line
($g\approx 4.3$) is reproduced for films with different compositions MxD100-x
in the case of sufficiently high content of the FM phase $x\gtrsim 30$ at. %.
The observed reduction of the EPR peak intensity with temperature decrease
contradicts the typical situation for systems weakly doped with Fe and Co
ions, where the weakening of the EPR line occurs, on the contrary, with
temperature increase [19, 20, 21, 22, 53, 54, 55].

It turned out that for the studied nanocomposites, the transition to the limit
of low FM phase contents $x\lesssim 30$ at. % leads to a more traditional
temperature behavior of the EPR peak. Figure 7 shows the magnetic resonance
spectra obtained at different temperatures for the film (CoFeB)25(Al2O3)75 in
magnetic field applied in the film plane. As can be seen from the figure, with
temperature decreasing from 296 to 60 K, the intensity of the EPR peak
increases significantly. A similar, but less pronounced increase in the
intensity is observed for the FMR peak (‘‘superparamagnetic’’ resonance peak
in other terms). In this regard, the evolution of the spectra in the
temperature range $60{-}296$ K looks quite natural.

Figure 7: Spectra of the film (CoFeB)25(Al2O3)75 at different temperatures
$T=4.2{-}296$ K in the case of in-plane field at frequency $f\approx 25$ GHz
for (a) transverse and (b) longitudinal geometries of resonance excitation. In
the case of longitudinal geometry, the vertical scale of the spectra is
enlarged 20 times with respect to the transverse geometry.

However, with a further decrease of temperature from 60 to 4.2 K, the form of
the spectra changes significantly. In the case of transverse resonance
excitation geometry ($\mathbf{h}\perp\mathbf{H}$), it is seen that the FMR
peak strongly broadens and shifts to weak fields. The EPR peak observed in the
longitudinal geometry ($\mathbf{h}\parallel\mathbf{H}$) vanishes. At the same
time, a strong absorption appears in the vicinity of $H=0$. A small peak,
manifested in weak fields at $T=4.2$ K, is associated with PM impurities in
the substrate [8].

The observed transformation of the spectra in the low temperature region is
probably explained by a decrease of thermal fluctuations in the system of PM
ions and FM nanogranules and the formation of larger magnetically ordered
clusters coupled by exchange and magnetodipole interactions. This is
accompanied by suppression of EPR of individual ions and the formation of
collective oscillation modes with a wide frequency spectrum due to the strong
inhomogeneity of the system and fluctuations of local anisotropy. This
scenario explains the observed disappearance of the EPR peak and the strong
broadening of the FMR line in the low temperature region.

Further, it can be assumed that with an increase of the FM phase content in
the nanocomposite, the formation of macroscopic magnetically ordered clusters
begins at higher temperatures. This explains the expansion of the temperature
range with ‘‘anomalous’’ behavior of the EPR peak, as well as the
disappearance of this peak in samples with the FM phase content above the
percolation threshold.

It can also be expected that due to the high degree of disorder, the low-
temperature state of the system has the features of spin (cluster) glass. This
state is characterized by a high density of local energy minima corresponding
to various magnetic configurations of the system. The density of these minima
(quasi-equilibrium states) decreases with increasing magnetic field when the
system approaches the saturation. Thus, the absorption in the vicinity of
$H=0$ observed in low-temperature spectra at $\mathbf{h}\parallel\mathbf{H}$
can be associated with the excitation of transitions between various quasi-
equilibrium states of the system.

##  4 Conclusion

Nanogranular metal-insulator films MxD100-x with different compositions (M =
Fe, Co, CoFeB; D = Al2O3, SiO2, LiNbO3) and various FM metal contents $x$ were
studied by magnetic resonance. The experimental spectra of the films contain
the FMR line from the ensemble of FM granules, as well as an additional
absorption peak with effective $g$-factor $g\approx 4.3$, which we associate
with resonance at PM centers present in the insulating space between FM
granules. Fe3+ and Co2+ ions dispersed in the insulating matrix during the
deposition of the films can serve as such centers.

With an increase of the FM phase content, the observed EPR line ($g\approx
4.3$) demonstrates an additional shift depending on the orientation of the
magnetic field with respect to the film plane. The correlation of this shift
with the demagnetization field of the film $4\pi M$ has been experimentally
established. When magnetic field is applied in the film plane, the EPR line is
shifted to weaker fields by an amount of $\approx 4\pi M/3$ relative to its
position in the limit of low FM phase contents. On the contrary, for the
normal orientation of the field, the EPR peak is shifted to stronger fields by
$\approx 8\pi M/3$. This behavior can be explained by magnetodipole fields
acting on PM centers from the ensemble of FM granules: the demagnetizing field
$-4\pi M$, which arises in a normally magnetized film, and the Lorentz field
$4\pi M/3$, which is independent of the external field orientation.

Due to fluctuations of the local dipole and exchange fields at PM centers, the
EPR peak is manifested not only in the usual transverse, but also in the
longitudinal geometry of the resonance excitation.

The presence of magnetic interactions in the system of PM ions and FM granules
also leads to a peculiar temperature dependence of the EPR peak amplitude.
When cooling from the high temperature region, the intensity of the EPR line
first rises due to an increase in the susceptibility of PM ions. However, at
low temperatures, the weakening of thermal fluctuations leads to the formation
of macroscopic coupled clusters in the system of PM ions and FM granules. In
this situation, the EPR peak from individual ions decreases until its complete
disappearance, when the magnetically ordered state extends to the entire film.

Thus, in this paper we have shown that PM ions dispersed in an insulating
matrix in metal-insulator nanogranular composites can serve as markers of
magnetic interactions present in the system. These interactions are manifested
while studying the behavior of the EPR line of the dispersed ions.

## Funding

The work was carried out within the framework of a state assignment and was
financially supported by the Russian Science Foundation (project no.
22-29-00392).

## Conflict of interest

The authors declare that they have no conflicts of interest.

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