_{2}FeMoO

_{6-δ}nanoparticles

Corresponding author: Gunnar Suchaneck (Gunnar.Suchaneck@tu-dresden.de)

Magnetization is a key property of magnetic materials. Nevertheless, a satisfactory, analytical description of the temperature dependence of magnetization in double perovskites such as strontium ferromolybdate is still missing. In this work, we develop, for the very first time, a model of the magnetization of nanosized, magnetically inhomogeneous Sr_{2}FeMoO_{6-δ} nanoparticles. The temperature dependence of magnetization was approximated by an equation consisting of a Bloch-law spin wave term, a higher order spin wave correction, both taking into account the temperature dependence of the spin-wave stiffness, and a superparamagnetic term including the Langevin function. In the limit of pure ferromagnetic behavior, the model is applicable also to SFMO ceramics. In the vicinity of the Curie temperature (_{C}

Strontium ferromolybdate (Sr_{2}FeMoO_{6-δ} – _{C}_{C}

Magnetic nanoparticles are building units of spintronic devices, magnetic sensors, radio-frequency and microwave devices, biomedical sensing and photonic systems, etc. Depending on particle, element or island size, magnetic properties change sufficiently. Below a certain size, the element first takes a single-domain state while in an ensemble of nanoparticles a superparamagnetic state appears at smaller sizes in dependence on temperature and observation (measurement) time. In the latter state, demagnetization occurs without coercivity since it is caused by thermal energy and not by the application of a magnetic field. Thus, the memory of the remanent state of the element is lost [

Magnetization characterizes the density of permanent or induced magnetic dipole moments in a magnetic material. In granular magnetic films, magnetization determines the magnetoresistance [

Up to now, there is no satisfactory analytical expression for the relative magnetization _{s}_{C}_{C}

proposed for ferromagnetic metals and alloys [

corresponds to Bloch´s 3/2 power law for non-interacting spin waves (magnons) at low temperatures [_{5}, Y_{2}Fe_{17}, GdZn, while for Fe

with µ_{B} the Bohr magneton, _{s}

representing the polylogarithm function Li_{p}_{3/2}(_{B}^{–21} eVm^{2} [_{s}_{s}_{B}/f.u. and _{C}

For SFMO, Eqs. (1) and (2) coincides up to about 120 K while reproducing the experimental _{C} exceeds the experimental value significantly. The last term in Eq. (1) represents for ^{2} term attributed to collective electron behavior [^{2} and ^{5/2} terms can hardly be made since both terms equally well describe the experimental data.

Temperature dependence of the reduced magnetization of SFMO according to Eq. (1), (2), (5), (6), and (7) in comparison with experimental data [1, 10, 11].

The simulation of the temperature dependent magnetization by Monte Carlo methods and Landau–Lifshitz–Gilbert atomistic spin models [

This relationship provides for SFMO an initial starting point for

For _{C}

where b_{SFMO} = 1/2 [

An empirical interpolation formula of

which matches the experimental behavior ^{3/2}) at ^{β} at ^{7/2} term was chosen to improve the fit to the experimental data. Thereby, the coefficient

Figures _{C}_{C}_{C}

Recently, an inhomogeneous magnetic state was obtained in SFMO nanoparticles fabricated by solid-state reaction from partially reduced SrFeO_{3-}_{х}_{4} precursors studying the temperature dependences of the magnetization measured in the field-cooling _{4} revealed a paramagnetic doublet above a blocking temperature of 45 K, while the spectrum of a similar sample with a size of 197 nm taken at 77 K included superparamagnetic, ferrimagnetic and surface contributions. The coexistence of different magnetic phases – superparamagnetic, ferromagnetic and paramagnetic – was revealed in single phase Mg_{x}_{1-}_{x}_{2}O_{4} nanoparticles by Mössbauer spectroscopy and curve fitting of the magnetic field dependence of the magnetization [

In this work, a model is developed which describes the temperature dependence of magnetization of nanosized and magnetically inhomogeneous SFMO nanoparticles.

2. Methods

The temperature dependence of the reduced magnetization _{s}

where _{FM}_{SPM}_{SPM}_{eff}/_{s}_{SPM}_{eff} the effective magnetic moment of the superparamagnetic phase which is a fitting parameter in the order of 3 . 10^{4} µ_{B} [

where

with _{A}

where 〈^{2}〉 is a the range of exchange interaction amounting for only nearest-neighbor exchange 〈^{2}〉 = ^{2} = ^{2/3} with _{s}

Also here, the long wavelength approximation was considered. The decrease of _{C}^{5/2} term in Eqs. (11) and (12) are identical except for the numerical factor of 3/4.

For all calculations, the Curie temperature was fixed to _{C}_{s}^{–21} eVm^{2}, the range of exchange interaction to 〈^{2}〉 = ^{2}, the magnetic flux density to _{eff} = 3 . 10^{4} µ_{B}. The latter corresponds 1.18 . 10^{4} spins in a particle of a volume of about 1460 nm^{3}. Figure _{s}^{3/2} and ^{5/2} terms of Eq. (8). Above 200 K, the values of ∆_{s}_{s}

Fractional change of magnetization ∆_{s}^{3/2} and ^{5/2} terms of Eq. (8).

Figure _{s}^{SPM}_{s}^{FM}_{s}^{FM}_{s}^{SPM}

Magnetically inhomogeneous nanoparticles can be analyzed by measuring the temperature dependence of the magnetization. A model for the determination of ferrimagnetic and superparamagnetic fractions of SFMO nanoparticles is presented in this work. In the limit of pure ferrimagnetic behavior, the model is applicable also to SFMO ceramics. However, it overestimates the magnetization change at higher temperatures (> 200 K) since the appearance of magnetic disorder in SFMO induces a pronounced extrinsic damping of spin waves [_{C}

This research was funded by the European Union within the scope of the European project H2020-MSCA-RISE-2017-778308–SPINMULTIFILM. The author has benefited from valuable discussions with N.A. Sobolev, N.A. Kalanda and E. Artiukh.

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_{6}double perovskite: Spin-glass behavior of the grain boundary.

^{3/2}law for magnetization of ferrometals: Ni, Fe, and Fe+3% Si.

_{2}FeMoO

_{6-δ}materials.

_{0.6}Fe

_{0.4}Fe

_{2}O

_{4}nanoparticles in ferrofluid at low temperatures.

^{th}International Confonference on Low Temperature Physics

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_{2}FeMoO

_{6}films.