Magnetization of Magnetically Inhomogeneous Sr 2 FeMoO 6-δ Nanoparticles

: In this work, we describe the magnetization of nanosized SFMO particles with a narrow size distribution around ca. 70 nm fabricated by the citrate-gel technique. The single-phase composition and superstructure ordering degree were proved by X-ray diffraction, the superparamagnetic behavior by magnetization measurements using zero-ﬁeld cooled and ﬁeld-cooled protocols, as well as by electron magnetic resonance. Different contributions to the magnetic anisotropy constant and the temperature dependence of the magnetocrystalline anisotropy are discussed.


Introduction
Strontium ferromolybdate (Sr 2 FeMoO 6-δ , SFMO) is a widely studied ferrimagnetic double perovskite.SFMO is a promising candidate for magnetic electrode materials for room-temperature spintronic applications, because they present a half-metallic character (with theoretically 100% polarization), a high Curie temperature (T C ) of about 415 K (ferromagnets should be operated in their ordered magnetic state below T C ), and a lowfield magnetoresistance [1].The same is valid for its derivatives such as (SrBa)FeMoO 6-δ or (SrLa)FeMoO 6-δ .However, wide application of SFMO is still missing, because of the low reproducibility of its magnetic properties originating in ceramic processing issues and its ageing upon contact with air and moisture.SFMO is a material which properties are very sensitive to disordering at the B-site, i.e., Fe Mo and Mo Fe antisite defect formation, nonstoichiometry of the magnetic Fe ions, and, with a lesser extent, to oxygen-and Sr-deficit [2].
Recently, an inhomogeneous magnetic state was obtained in SFMO nanoparticles fabricated by solid-state reactions from partially reduced SrFeO 3−x and SrMoO 4 precursors by studying the temperature dependences of the magnetization measured in the field-cooling (FC) and zero-field-cooling (ZFC) modes and small-angle neutron scattering [3].This state was attributed to the frustration of the exchange bonds and simultaneous occurrence of various magnetic states: antiferromagnetic, ferrimagnetic, and superparamagnetic when the spin inversion does not change the energy of the system in a wide range of temperatures.In another report [4], the Mössbauer spectrum of SFMO fine particles of about 30 nm size, consisting of small traces of SrMoO 4 , revealed a paramagnetic doublet above a blocking temperature T B = 45 K, while the spectrum of a similar sample with a size of 197 nm taken at 77 K included superparamagnetic, ferromagnetic and surface contributions.Diverse magnetic behavior -superparamagnetic, ferromagnetic and paramagnetic-was detected in single-phase Mg x Zn 1−x Fe 2 O 4 nanoparticles by Mössbauer spectroscopy and curve fitting of the magnetic-field dependence of the magnetization [5].This was attributed to the distribution of particle sizes in the samples.
In this work, we determine the magnetization behavior of nanosized magnetically inhomogeneous SFMO particles by means of ZFC-FC magnetization and electron magnetic resonance measurements.Furthermore, we discuss different contributions to magnetic anisotropy and evaluate the temperature dependence of the uniaxial magnetocrystalline anisotropy constant.

Materials and Methods
The citrate-gel technique was used for the synthesis of SFMO nanoparticles using ultra-high purity Sr(NO 3 ) 2 , Fe(NO 3 ) 3 •9H 2 O, (NH 4 ) 6 Mo 7 O 24 and citric acid monohydrate C 6 H 8 O 7 H 2 O as initial reagents.The solid foam resulting from the synthesis was crushed and annealed at T = 770 K and p O2 = 2.1×10 4 Pa for 10 h.After that, the powders were annealed in a 5% H 2 /Ar gas mixture flow at 1170 K for 5 h, followed by quenching at room temperature.Synthesis details are described in Ref. [6] (cf.Supplemental Materials, Figure S1).Since the SFMO powders were agglomerated, ultrasonic dispersion as used to obtain highly dispersed particles with an average size smaller than 100 nm.Ethyl alcohol was chosen as a liquid medium since there is neither chemical interaction with SFMO nor SFMO decomposition.
To optimize the process of SFMO dispersion, the effects of time and radiation power of the ultrasonic homogenizer on the dispersion of the particles were studied in a suspension of 0.1 g of powder per 25 mL of C 2 H 5 OH.The power of ultrasonic exposure was varied from 1% to 100% of the nominal power (P n ) with a step of 25% at a frequency of 22 kHz and an exposure time ranging from 15 to 60 min with a step of 15 min.To prevent the process of coagulation of SFMO grains, different surfactants were used.The surfactant was selected to achieve a high Z-potential.Cationic hexadecyltrimethylammonium bromide, anionic sodium tripolyphosphate, nonionic polyoxyethylene (20) sorbitan monolaurate Tween20 TM , and nonionic polycarboxylate/sodium salt Dolapix PC 67 were tested.The most suitable was the nonionic Tween20 TM surfactant providing the main results described in this work.In this case, the Zeta potential of SFMO nanopowder has negative values.The highest values of the Zeta potential (Z = −63.2mV) in 25 mL (C 2 H 5 OH) + 0.01 g SFMO + x g Tween 20 TM subjected to dispersion for 45 min was achieved at x = 0.05 g Tween20 TM .
X-ray diffraction (XRD) patterns of samples with a weight of ca.60 mg were recorded at room temperature at a rate of 60 • /h in the range of angles Θ = 10-90 • using a DRON-3 diffractometer (Bourevestnik, Saint Petersburg, Russia) in CuK α radiation and analyzed by means of the ICSD-PDF2 database (Release 2000) and PowderCell software.The microstructure, morphology, and elemental composition of grains of annealed SFMO powders were investigated by field-emission scanning electron microscope JEOL JSM-7000F (JEOL Ltd., Tokyo, Japan).The grain sizes were estimated by an NT-206 atomic force microscope (Microtestmachines, Gomel, Belarus).The particle size distribution was obtained by dynamic light scattering (DLS) analysis using a Zetasizer Nanoparticles analyzer (Malvern Nano ZS90, Malvern, UK).
Magnetization measurements of the samples were performed at magnetic fields up to 4 T in a temperature range from 4.2 K to 320 K using a Liquid Helium-Free High-Field Measurement System vibrating sample magnetometer (Cryogenic Ltd., London, UK).ZFC-FC curves were acquired in a magnetic field of 0.01 T.
The electron magnetic resonance (EMR) measurements of samples with a weight of ca.20 mg were performed in the X-band (9.45 GHz) at a field modulation frequency of 100 kHz using an EleXsys E-500 electron paramagnetic resonance spectrometer (Bruker, Billerica, MA, USA) equipped with a nitrogen-flow cryostat.The first field derivative of the microwave absorption, dP mw /dB ∝ dχ /dB, with P mw the microwave power and χ the imaginary part of the magnetic AC susceptibility, was registered.The magnetic field was swept from high to low values.

Results
Single-phase SFMO powders were obtained with crystal lattice parameters a = b = 0.55629 nm, c = 0.78936 nm and with a B-site superstructural ordering degree of 88% (cf.Supplemental Materials, Figures S2 and S3).After ultrasonic treatment, the sizes of agglomerates in a suspension of 25 mL (C 2 H 5 OH) + 0.01 g (SFMO) + 0.05 g (Tween 20), subjected to sonication for 60 min, smoothly decrease from 230 nm to about 70 nm (Figure 1).At an ultrasonic power of P = 25% P n , a release of dissolved gases begins in the rarefaction regions being at low pressure, which is accompanied by the formation of a durable bubble.The latter is stabilized by a monolayer of SFMO particles and oscillates linearly with the frequency of the ultrasonic wave relatively to its equilibrium position.Here, dispersion is less effective.A further increase of the power up to P = 75% P n leads to a violation of linearity of oscillations of the bubble walls.The stage of stable cavitation begins.The bubble itself becomes a source of ultrasonic higher-harmonic vibrations.Waves, microcurrents and electric discharges appear on its surface [7,8].This increases the dispersion of SFMO particles.Thus, at sonication times up to 45 min, particles are effectively dispersed in the entire suspension volume, and processes of coagulation and sedimentation of particles are prevented.The ultrasonic wave passing through the liquid creates zones of compression and depression, which alternate in each half-period of the wave.For this reason, the particles of the medium vibrate with small amplitudes (in the sub-µm range) and huge accelerations [7,8].A further increase of p beyond 75% P n leads to unstable cavitation and the formation of a cavitation cloud on the surface of the alcohol solution, which hinders the transmission of ultrasonic vibrations into the volume [7,8].This state of the alcohol suspension is characterized by the formation of rapidly growing vapor-gas bubbles, which instantly contract in volume during the compression phase and collapse.Because of this, the average particle size increases again up to 450 nm as a result of ultrasonic treatment with P = P n for times up to 60 min.This indicates intensive coagulation prevailing over the dispersion process, which can be explained by the formation of a cavitation cloud (a large number of air bubbles) on the surface of the working tool hindering the transmission of ultrasonic vibrations into the volume and, thus, reducing the grinding efficiency.Saturation of particle size was reached after 80 min ultrasonic treatment.Figure 2 illustrates the increase of the volume fraction of fine SFMO particles with sonication time.
Electron.Mater.2021, 2, FOR PEER REVIEW 3 the imaginary part of the magnetic AC susceptibility, was registered.The magnetic field was swept from high to low values.

Results
Single-phase SFMO powders were obtained with crystal lattice parameters a = b = 0.55629 nm, c = 0.78936 nm and with a B-site superstructural ordering degree of 88% (cf.Supplemental Material, Figures S2 and S3).After ultrasonic treatment, the sizes of agglomerates in a suspension of 25 mL (C2H5OH) + 0.01 g (SFMO) + 0.05 g (Tween 20), subjected to sonication for 60 min, smoothly decrease from 230 nm to about 70 nm (Figure 1).At an ultrasonic power of P = 25% Pn, a release of dissolved gases begins in the rarefaction regions being at low pressure, which is accompanied by the formation of a durable bubble.The latter is stabilized by a monolayer of SFMO particles and oscillates linearly with the frequency of the ultrasonic wave relatively to its equilibrium position.Here, dispersion is less effective.A further increase of the power up to P = 75% Pn leads to a violation of linearity of oscillations of the bubble walls.The stage of stable cavitation begins.The bubble itself becomes a source of ultrasonic higher-harmonic vibrations.Waves, microcurrents and electric discharges appear on its surface [7,8].This increases the dispersion of SFMO particles.Thus, at sonication times up to 45 min, particles are effectively dispersed in the entire suspension volume, and processes of coagulation and sedimentation of particles are prevented.The ultrasonic wave passing through the liquid creates zones of compression and depression, which alternate in each half-period of the wave.For this reason, the particles of the medium vibrate with small amplitudes (in the sub-µ m range) and huge accelerations [7,8].A further increase of p beyond 75% Pn leads to unstable cavitation and the formation of a cavitation cloud on the surface of the alcohol solution, which hinders the transmission of ultrasonic vibrations into the volume [7,8].This state of the alcohol suspension is characterized by the formation of rapidly growing vapor-gas bubbles, which instantly contract in volume during the compression phase and collapse.Because of this, the average particle size increases again up to 450 nm as a result of ultrasonic treatment with P = Pn for times up to 60 min.This indicates intensive coagulation prevailing over the dispersion process, which can be explained by the formation of a cavitation cloud (a large number of air bubbles) on the surface of the working tool hindering the transmission of ultrasonic vibrations into the volume and, thus, reducing the grinding efficiency.Saturation of particle size was reached after 80 min ultrasonic treatment.Figure 2 illustrates the increase of the volume fraction of fine SFMO particles with sonication time.
where dav and σ are the average diameter and dispersion of the particle size distribution, respectively [9].The calculated average particle diameter for the powder subjected to sonication for 60 min amounts to dav = 67.8nm and the dispersion to σ = 0.214 nm (Figure 3). Figure 4 shows a scanning electron microscopy image of the as-fabricated, agglomerated SFMO nanoparticles (Figure 4a) and an atomic force microscopy (AFM) scan of the nanoparticles on a substrate (Figure 4b).The figures demonstrate a narrow size distribu- where d av and σ are the average diameter and dispersion of the particle size distribution, respectively [9].The calculated average particle diameter for the powder subjected to sonication for 60 min amounts to d av = 67.8nm and the dispersion to σ = 0.214 nm (Figure 3).
where dav and σ are the average diameter and dispersion of the particle size distribution, respectively [9].The calculated average particle diameter for the powder subjected to sonication for 60 min amounts to dav = 67.8nm and the dispersion to σ = 0.214 nm (Figure 3). Figure 4 shows a scanning electron microscopy image of the as-fabricated, agglomerated SFMO nanoparticles (Figure 4a) and an atomic force microscopy (AFM) scan of the nanoparticles on a substrate (Figure 4b).The figures demonstrate a narrow size distribu- Figure 4 shows a scanning electron microscopy image of the as-fabricated, agglomerated SFMO nanoparticles (Figure 4a) and an atomic force microscopy (AFM) scan of the nanoparticles on a substrate (Figure 4b).The figures demonstrate a narrow size distribution of particles with a size in the order of 70 nm and the de-agglomeration after ultrasonic treatment.The height of the AFM features corresponds approximately to the average particle size obtained by DLS.Note that features with a size of about 10 nm could not be resolved by AFM.
tion of particles with a size in the order of 70 nm and the de-agglomeration after ultrasonic treatment.The height of the AFM features corresponds approximately to the average particle size obtained by DLS.Note that features with a size of about 10 nm could not be resolved by AFM.Measurements of the temperature dependence of the magnetization following the ZFC-FC protocol revealed the presence of at least superparamagnetic and metastable blocked behavior (Figure 5) (cf.[4]).The maximum of the ZFC curve is located at a maximum temperature Tmax of ca.20 K.The low blocking temperature, as well as a large increment of magnetization with decreasing temperature in the FC data, reveal that the magnetic interaction between the nanoparticles is quite weak.With regard to the lognormal size distribution, the position of Tmax is not the same as the average blocking temperature TB.The latter is shifted down to smaller values by a correction factor of 1.5 [10] giving rise to TB ≈ 13 K.The corresponding K1V value amounts to 5.25 × 10 -21 J.Note that the difference between the maximum position of the ZFC curve at 20 K and the bifurcation point between the ZFC and FC curves at ca. 170 K is another measure of the dispersion of grain sizes [11]: the biggest particles are deblocked at the latter temperature.Also, we have to take into account that the value of TB depends on the applied magnetic field.Following a model proposed in [12], we found that in our case TB is underestimated by about 7%.Measurements of the temperature dependence of the magnetization following the ZFC-FC protocol revealed the presence of at least superparamagnetic and metastable blocked behavior (Figure 5) (cf.[4]).The maximum of the ZFC curve is located at a maximum temperature T max of ca.20 K.The low blocking temperature, as well as a large increment of magnetization with decreasing temperature in the FC data, reveal that the magnetic interaction between the nanoparticles is quite weak.With regard to the lognormal size distribution, the position of T max is not the same as the average blocking temperature T B .The latter is shifted down to smaller values by a correction factor of 1.5 [10] giving rise to T B ≈ 13 K.The corresponding K 1 V value amounts to 5.25 × 10 −21 J.Note that the difference between the maximum position of the ZFC curve at 20 K and the bifurcation point between the ZFC and FC curves at ca. 170 K is another measure of the dispersion of grain sizes [11]: the biggest particles are deblocked at the latter temperature.Also, we have to take into account that the value of T B depends on the applied magnetic field.Following a model proposed in [12], we found that in our case T B is underestimated by about 7%.
In Figure 6a, EMR spectra of a similar SFMO sample with a B-site superstructure ordering degree of 81% are plotted for measurement temperatures ranging from 120 K to 480 K. Figure 6b shows the corresponding resonant magnetic field B r , defined as the field for which the derivative dP mw /dB goes through zero, and the linewidth ∆B pp defined as the field interval between the positive and negative peaks of the resonance curve.The signal reduces rapidly at temperatures above 400 K, however, the line is still observable at 440 K and completely disappears at 460 K. Thus, the Curie temperature T C is slightly above 400 K (the FMR is usually observed even at the paramagnetic side of T C [13]).The resonance field B r is always below that of the paramagnetic resonance value corresponding to g = 2 and approaches the latter with increasing temperature.As the temperature decreases below T C , the spectra are broadened and shift towards lower magnetic fields.The latter effect is typical of predominantly cubic magnetic anisotropy.
that the difference between the maximum position of the ZFC curve at 20 K and the bifurcation point between the ZFC and FC curves at ca. 170 K is another measure of the dispersion of grain sizes [11]: the biggest particles are deblocked at the latter temperature.Also, we have to take into account that the value of TB depends on the applied magnetic field.Following a model proposed in [12], we found that in our case TB is underestimated by about 7%.In Figure 6a, EMR spectra of a similar SFMO sample with a B-site superstructure ordering degree of 81% are plotted for measurement temperatures ranging from 120 K to 480 K. Figure 6b shows the corresponding resonant magnetic field Br, defined as the field for which the derivative dPmw/dB goes through zero, and the linewidth Bpp defined as the field interval between the positive and negative peaks of the resonance curve.The signal reduces rapidly at temperatures above 400 K, however, the line is still observable at 440 K and completely disappears at 460 K. Thus, the Curie temperature TC is slightly above 400 K (the FMR is usually observed even at the paramagnetic side of TC [13]).The resonance field Br is always below that of the paramagnetic resonance value corresponding to g = 2 and approaches the latter with increasing temperature.As the temperature decreases below TC, the spectra are broadened and shift towards lower magnetic fields.The latter effect is typical of predominantly cubic magnetic anisotropy.
Since in magnetic resonance measurements in the X-band the characteristic measurement time τm amounts to 10 −10 s, the characteristic blocking temperature is roughly given by kTB ≈ K1V [14].Considering ZFC magnetization measurements, this yields TB EMR ≈ 330 K. Unfortunately, passing through 330 K produces no clear effect on the curves in Figure 6b.In [14], the TB value has been obtained by plotting the total intensity (second integral) of the experimental EMR absorption band versus temperature.However, this method is rather imprecise when a part of the resonance band is not recorded as is the case at low temperatures.
The linewidth is determined by the spread of the resonance fields of the individual nanoparticles caused by the random distribution of their easy magnetization axes and by the precession damping of the magnetic moments.The spontaneous magnetization can contribute to the linewidth of individual grains via its site-to-site distributions concerning both the Zeeman and the magnetostatic energy terms [14].With lowering temperature, the thermal fluctuation energy becomes smaller than the anisotropy energy.Here, the random thermal fluctuation field Bf ~ kT/µeff (with µeff being the effective for which the derivative dPmw/dB goes through zero, and the linewidth Bpp defined as the field interval between the positive and negative peaks of the resonance curve.The signal reduces rapidly at temperatures above 400 K, however, the line is still observable at 440 K and completely disappears at 460 K. Thus, the Curie temperature TC is slightly above 400 K (the FMR is usually observed even at the paramagnetic side of TC [13]).The resonance field Br is always below that of the paramagnetic resonance value corresponding to g = 2 and approaches the latter with increasing temperature.As the temperature decreases below TC, the spectra are broadened and shift towards lower magnetic fields.The latter effect is typical of predominantly cubic magnetic anisotropy.
Since in magnetic resonance measurements in the X-band the characteristic measurement time τm amounts to 10 −10 s, the characteristic blocking temperature is roughly given by kTB ≈ K1V [14].Considering ZFC magnetization measurements, this yields TB EMR ≈ 330 K. Unfortunately, passing through 330 K produces no clear effect on the curves in Figure 6b.In [14], the TB value has been obtained by plotting the total intensity (second integral) of the experimental EMR absorption band versus temperature.However, this method is rather imprecise when a part of the resonance band is not recorded as is the case at low temperatures.
The linewidth is determined by the spread of the resonance fields of the individual nanoparticles caused by the random distribution of their easy magnetization axes and by the precession damping of the magnetic moments.The spontaneous magnetization can contribute to the linewidth of individual grains via its site-to-site distributions concerning both the Zeeman and the magnetostatic energy terms [14].With lowering temperature, the thermal fluctuation energy becomes smaller than the anisotropy energy.Here, the random thermal fluctuation field Bf ~ kT/µeff (with µeff being the effective magnetic moment of the superparamagnetic particle) diminishes [15], thus broadening the linewidth.Another reason for the temperature-dependent broadening is the variation of the anisotropy energy with temperature.At very low temperatures, the spin-wave spectrum of nanoparticles is quantized with a dominating uniform mode n = 0 [16].This results in a linear temperature dependence of magnetization: Since in magnetic resonance measurements in the X-band the characteristic measurement time τ m amounts to 10 −10 s, the characteristic blocking temperature is roughly given by kT B ≈ K 1 V [14].Considering ZFC magnetization measurements, this yields T B EMR ≈ 330 K. Unfortunately, passing through 330 K produces no clear effect on the curves in Figure 6b.In [14], the T B value has been obtained by plotting the total intensity (second integral) of the experimental EMR absorption band versus temperature.However, this method is rather imprecise when a part of the resonance band is not recorded as is the case at low temperatures.
The linewidth is determined by the spread of the resonance fields of the individual nanoparticles caused by the random distribution of their easy magnetization axes and by the precession damping of the magnetic moments.The spontaneous magnetization can contribute to the linewidth of individual grains via its site-to-site distributions concerning both the Zeeman and the magnetostatic energy terms [14].With lowering temperature, the thermal fluctuation energy becomes smaller than the anisotropy energy.Here, the random thermal fluctuation field B f ~kT/µ eff (with µ eff being the effective magnetic moment of the superparamagnetic particle) diminishes [15], thus broadening the linewidth.Another reason for the temperature-dependent broadening is the variation of the anisotropy energy with temperature.
At very low temperatures, the spin-wave spectrum of nanoparticles is quantized with a dominating uniform mode n = 0 [16].This results in a linear temperature dependence of magnetization: Fitting the asymptote of the low-temperature part of M(T) to Equation (2), we obtain K 1 V ≈ 5.0 × 10 −21 J in good agreement with the value derived above from the blocking temperature.Divided by the superparamagnetic particle volume this yields K 1 '≈ 5.0 × 10 3 J/m 3 .
In our case, we have to take into account that the size of our particles is above the superparamagnetic limit of about 10.5 nm, Equation ( 6).Here, a correction factor (1 − H/H c ) 2 appears for the K 1 V value [17].For K 1 ≈ 1.7 × 10 5 J/m 3 (cf.Section 4), we find µ 0 H c ≈ 12 mT (the measuring field was µ 0 H = 10 mT) in accordance with the experimental value of µ 0 H c ≈ 12-14 mT at 4.2 K of similar samples, but pressed into pellets and long-term annealed at 1420 K in a flux of a gas mixture of 5% H 2 /Ar [18].

Discussion
The remanent magnetization of uniaxial, single-domain nanoparticles which are fully magnetized along the easy axis vanishes after removing the magnetic field as: where M s is the saturation magnetization, t the time after removal of the field.The value τ denotes the Néel-Brown relaxation time for an energy barrier ∆E = K 1 V [19,20]: with τ 0 a time constant in the order of 10 −9 s [21], K 1 the uniaxial anisotropy constant, V the particle volume, k the Boltzmann constant and T the temperature.Note that K 1 is also temperature dependent [22].Superparamagnetic behavior is obtained using an instrument with a characteristic measuring time τ m ≥ τ.For τ m < τ, the magnetic moments remain in a fixed direction during the measurement.This leads to a remanent magnetization and appearance of a coercive field, thus, a metastable ferromagnetic (ferrimagnetic) state is detected.The condition τ = τ m defines the so-called blocking temperature [23]: Here, τ m amounts to ca. 100 s in static magnetometry [24], and to τ m ≈ 10 −8 s in Mössbauer spectrometry [23].Simultaneous ferromagnetic and SPM behavior is obtained when τ is of the order of τ m .Thus, particles at a temperature T > T B reveal a dominating SPM behavior, while particles at T < T B show a predominant ferromagnetic behavior.In the latter case thermal excitations are not sufficient to overcome the barrier K 1 V.The easiest way to implement such conditions is to use a mixture of particles of different sizes slightly below and slightly above a critical size, which for spherical particles amounts to: where the parameter q = π/6•ln(τ m /τ 0 ) amounts in magnetometry to ca.48 and in Mössbauer spectrometry to ca. 4.4.
In the simplest form of uniaxial anisotropy, the total free energy U of a particle is given by [24]: where ϕ is the angle between µ and the symmetry axis of the particle.Experimental values of K 1 for SFMO are scarce.EMR measurements on SFMO thin films grown by pulsed laser deposition onto (100) SrTiO 3 substrates with a thickness of 400 nm revealed for M s = 3.8 µ B an in-plane uniaxial magnetocrystalline anisotropy constant of ca. 3 × 10 4 J/m 3 at room temperature [25].On the other hand, the estimated zero-temperature value of K 1 is much higher and amounts to 1.7 × 10 5 J/m 3 [26].This is the low-temperature value used in the present work.The shape-anisotropy constant calculated employing demagnetization factors γ in the c and a directions [27] is given by: Here, µ 0 is the vacuum permeability.Optical inspection of the particle shape revealed that the latter is well approximated by oblate spheroids with a c/a ratio in the range from 0.85 to 1.0.With M s = 282 kA/m (50 Am 2 /kg), this results in a negligible value of K sh ≤ 3.3 × 10 3 J/m 3 .
Inhomogeneous surface conditions during synthesis yield an effective anisotropy constant of [28,29]: where K V is the anisotropy constant per unit volume, K S the surface anisotropy constant per unit of surface area S, and d the particle size.For a sphere or cube, the total surfacearea-to-volume ratio amounts to 6/d.Surface anisotropy arises due to symmetry breaking and reduction in the nearest-neighbor coordination at the surface.It can significantly enhance the total magnetic anisotropy of nanoparticles.For instance, K eff of γ-Fe 2 O 3 nanoparticles with a mean size of about 8.6 nm dispersed in a polymer exceeds K V by about two orders of magnitude [30].However, in our case, the surface conditions after a long-time ultrasonic treatment are mostly balanced.On the other hand, strains relax in homogeneous, twin-free nanoparticles, that have no shell or grain boundary.Therefore, only the particle size needs to be considered as the origin of the simultaneous observation of ferromagnetic and superparamagnetic behavior.As a result, we have only to account for the magnetocrystalline anisotropy constant.Replotting B r in Figure 6b using a reduced temperature ϑ = (1 − T/T C ), we obtain a slope of dB r /dϑ = −1.825T (cf.Supplemental Materials, Figure S4).The shift of B r is determined by the temperature dependence of the magnetic anisotropy field: where α is an averaged factor, which depends only on the angles between the applied field and the magnetic axes of the particles.A satisfactory approximation of the temperature dependence of the magnetization derived from Monte-Carlo and Landau-Lifshitz-Gilbert simulations is M(T) = M(0)•ϑ 1/3 [31].With regard to the linear temperature dependence of B r , we conclude K 1 (T) = K 1 (0) • (1 − T/T C ) 4/3 (11) Taking into account dB r /dϑ = −1.825T, for M s = 3.8 µ B /f.u.we arrive at 2αK 1 (0) = 5.2 × 10 5 J/m 3 , which is of the order of the values considered above.For K 1 (0 K) = 1.7 × 10 5 J/m 3 [26], Equation (11) yields a room-temperature value of K 1 (300 K) = 2.35 × 10 4 J/m 3 comparable to the in-plane uniaxial anisotropy constant of SFMO thin films amounting to ~3 × 10 4 J/m 3 at room temperature [25].
A lower limit of the critical diameter of a single-domain particle amounts to [32]:

Figure 1 .
Figure 1.Dependence of the SFMO agglomerate size on sonication time.Figure 1. Dependence of the SFMO agglomerate size on sonication time.

Figure 1 .
Figure 1.Dependence of the SFMO agglomerate size on sonication time.Figure 1. Dependence of the SFMO agglomerate size on sonication time.

Figure 2 .
Figure 2. Volume fraction of particles with size d for sonication times tus of (a) 10 min, (b) 15 min, and (c) 60 min.The nanoparticle size distribution was analyzed by means of DLS.It is most widely represented by the lognormal distribution:

Figure 3 .
Figure 3. Size distribution of the volume fraction of an ensemble of SFMO nanoparticles after ultrasonic dispersion for 60 min determined by DLS analysis (black line) and its approximation by the log-normalized function ƒ(d) (red line).

Figure 2 .
Figure 2. Volume fraction of particles with size d for sonication times t us of (a) 10 min, (b) 15 min, and (c) 60 min.The nanoparticle size distribution was analyzed by means of DLS.It is most widely represented by the lognormal distribution:

Figure 2 .
Figure 2. Volume fraction of particles with size d for sonication times tus of (a) 10 min, (b) 15 min, and (c) 60 min.The nanoparticle size distribution was analyzed by means of DLS.It is most widely represented by the lognormal distribution:

Figure 3 .
Figure 3. Size distribution of the volume fraction of an ensemble of SFMO nanoparticles after ultrasonic dispersion for 60 min determined by DLS analysis (black line) and its approximation by the log-normalized function ƒ(d) (red line).

Figure 3 .
Figure 3. Size distribution of the volume fraction of an ensemble of SFMO nanoparticles after ultrasonic dispersion for 60 min determined by DLS analysis (black line) and its approximation by the log-normalized function ƒ(d) (red line).

Figure 4 .
Figure 4. Scanning electron microscopy image as the as-fabricated, agglomerated SFMO nano-particles (a) and an atomic force microscopy scan of the nanoparticles on a substrate (b).

Figure 4 .
Figure 4. Scanning electron microscopy image as the as-fabricated, agglomerated SFMO nanoparticles (a) and an atomic force microscopy scan of the nanoparticles on a substrate (b).

Figure 5 .
Figure 5. Magnetization of the fabricated SFMO nanoparticles in the FC mode (blue dots) and in the ZFC mode (magenta squares).

Figure 5 .
Figure 5. Magnetization of the fabricated SFMO nanoparticles in the FC mode (blue dots) and in the ZFC mode (magenta squares).

Figure 6 .
Figure 6.Temperature dependence of the EMR spectra (a), resonance field Br and peak-to-peak linewidth ΔBpp (b).

Figure 6 .
Figure 6.Temperature dependence of the EMR spectra (a), resonance field B r and peak-to-peak linewidth ∆B pp (b).