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 Sr₂FeMoO_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 (T/T_C > 0.85), the model fails.

nanoparticles, magnetization, strontium ferromolybdate

Strontium ferromolybdate (Sr₂FeMoO_6-δ – SFMO) is the most studied ferrimagnetic double perovskite. SFMO double perovskites are promising candidates for magnetic electrode materials for room-temperature spintronics 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 low-field magnetoresistance [1]. However, a wide application of SFMO is still missing because of the low reproducibility of its magnetic properties originating in ceramic processing issues and its aging in contact with air and moisture.

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 [2].

Magnetization characterizes the density of permanent or induced magnetic dipole moments in a magnetic material. In granular magnetic films, magnetization determines the magnetoresistance [3] which is the key property for SFMO application in magnetic memories and magnetic sensors.

Up to now, there is no satisfactory analytical expression for the relative magnetization m (T), i.e. the magnetization M scaled to the saturation magnetization M_s, except for the two limiting cases, t → 0 and t → 1, where t = T/T_C with T_C the Curie temperature is the reduced temperature. In the first case, a function

(1)

proposed for ferromagnetic metals and alloys [4] is suitable, where the term

(2)

corresponds to Bloch´s 3/2 power law for non-interacting spin waves (magnons) at low temperatures [5]. Here, s and p in Eq. (1) are fit parameters, s > 0 and p > 3/2. The adjustable parameter p was found to be p = 5/2 for Co, Ni, Gd, YCo₅, Y₂Fe₁₇, GdZn, while for Fe p = 4. The slope factor s can be estimated also by means of semi-classical linear spin wave theory yielding [4, 6–8]

(3)

with µ_B the Bohr magneton, M_s the saturation magnetization, k the Boltzmann constant, and D the effective spin wave stiffness coefficient, and the function

(4)

representing the polylogarithm function Li_p (ξ) with an argument of ξ = exp(–x). In the low field limit (x → 1), Z_3/2(x) with x = µ_BB/kT reduces to the Riemann zeta function ζ(3/2). For SFMO, D at low temperatures (0.4 to 10 K) amounts to about 1.4 . 10^–21 eVm² [9]. Quoting M_s in terms of the number of Bohr magnetrons per formula unit and taking M_s ≈ 4 µ_B/f.u. and T_C = 415 K [1], we obtain s = 0.477 for B → 0. On the other hand, fitting Eq. (2) to experimental data of SFMO [1, 10, 11] yields s = 0.675.

For SFMO, Eqs. (1) and (2) coincides up to about 120 K while reproducing the experimental m (T) behavior [1, 10, 11] very well up to about 50 K (Fig. 1). At higher temperature, the slope dm (T)/dT becomes smaller than the experimental one, while the slope of Eq. (1) near T_C exceeds the experimental value significantly. The last term in Eq. (1) represents for p = 5/2 a higher order correction to the Bloch formula arising from the discreteness of the lattice [6]. This last term may be replaced also by a T² term attributed to collective electron behavior [12]. Here, a distinction between T² and T^5/2 terms can hardly be made since both terms equally well describe the experimental data.

Figure 1.

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 [13] yields

(5)

This relationship provides for SFMO an initial starting point for m (T) calculation (cf. Fig. 1). In this case, the inclusion of higher order terms makes the approximation worse.

For T → T_C, the low field limit of m (T) is given by [14]

(6)

where b_SFMO = 1/2 [10]. This results in c ≈

An empirical interpolation formula of m (T) for iron whiskers was presented [15]:

(7)

which matches the experimental behavior m (T) = (1 – At^3/2) at t → 0 with m (T) ∝ (1 – t)^β at t → 1. The t^7/2 term was chosen to improve the fit to the experimental data. Thereby, the coefficient C was determined by matching the interpolation formula and the t → 0 behavior for t → 1. Both, Eq. (6) and Eq. (7) are not suitable approximations for SFMO.

Figures 1 compares values of m (T) calculated by means of the discussed above equations with experimental data [1, 10, 11] and the model of this work for pure ferrimagnetic behavior. In order to compare samples with different T_C, a reduced temperature scale (T_C–T)/T_C was applied. Only Eq. (5) and the model of this work provide satisfactory results.

Recently, an inhomogeneous magnetic state was obtained in SFMO nanoparticles fabricated by solid-state reaction from partially reduced SrFeO_3-х and SrMoO₄ precursors studying the temperature dependences of the magnetization measured in the field-cooling (FC) and zero-field-cooling (ZFC) modes and small-angle neutron scattering [16]. This state was attributed to the frustration of the exchange bonds and the realization 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 by [17], the Mössbauer spectrum of SFMO fine particles of about 30 nm size consisting of small traces of SrMoO₄ 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_xZn_1-xFe₂O₄ nanoparticles by Mössbauer spectroscopy and curve fitting of the magnetic field dependence of the magnetization [18]. It was attributed to the distribution of particle size in the samples. Other magnetically inhomogeneous systems with ferroelectric and superparamagnetic components include, for instance, composites of Со–Р alloy with ultradispersed corundum and detonation nanodiamond [19].

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 M/M_s when measured in the FC mode was approximated by an equation consisting of a Bloch-law spin-wave term, a higher order spin-wave correction, and a superparamagnetic term including the Langevin function L [19]

(8)

where m_FM is the reduced ferrimagnetic magnetization, m_SPM = N_SPMµ_eff/M_s the reduced superparamagnetic magnetization of N_SPM particles, and µ_eff the effective magnetic moment of the superparamagnetic phase which is a fitting parameter in the order of 3 . 10⁴ µ_B [20]. Contrarily to [19], we do not fit the coefficients E and F to experimental data, but calculate them by means of spin-wave theory. In the long wavelength approximation, the coefficients E and F are given [6, 12, 21]

(9)

where g is the Landé spitting factor, and V the volume of a unit cell given by

(10)

with M the molar mass, ρ the density and N_A the Avogadro constant, and

(11)

where 〈r²〉 is a the range of exchange interaction amounting for only nearest-neighbor exchange 〈r²〉 = a² = V^2/3 with a the lattice parameter of one unit cell. Here, the saturation magnetization M_s is quoted in terms of the number of Bohr magnetrons per formula unit. The temperature dependence of the spin-wave stiffness constant is given by [21, 22]:

(12)

Also here, the long wavelength approximation was considered. The decrease of D (T) with temperature increases the coefficients E and F. In manganites close in the vicinity of T_C, Eq. (12) overestimates D (T) significantly [23] limiting the temperature range of the model to t ≤ 0.85. Note that the coefficients of the T^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 T_C = 420 K, the saturated magnetization to M_s = 3.75 µB/f.u., the the Landé spitting factor g to g ≈ 2, the low temperature spin-wave stiffness constant to D (0) = 1.4 . 10^–21 eVm², the range of exchange interaction to 〈r²〉 = a², the magnetic flux density to B = 10 mT, and the effective magnetic moment of the superparamagnetic phase to µ_eff = 3 . 10⁴ µ_B. The latter corresponds 1.18 . 10⁴ spins in a particle of a volume of about 1460 nm³. Figure 2 illustrates the fractional change of magnetization ∆M/M_s of ferromagnetic SFMO induced by the spin-wave T^3/2 and T^5/2 terms of Eq. (8). Above 200 K, the values of ∆M/M_s are probably slightly overestimated since the appearance of magnetic disorder in SFMO induces a pronounced extrinsic damping of spin waves [24]. On the other hand, above 360 K in the vicinity of the Curie temperature, ∆M/M_s will be significantly larger than predicted as a result of an incorrect description of D (T), Eq. (12), in this temperature range.

Figure 2.

Fractional change of magnetization ∆M/M_s induced by the spin-wave T^3/2 and T^5/2 terms of Eq. (8).

Figure 3 compares values of m (T) calculated by means of Eq. (8) for different fractions ξ = FM (FM + SPM). For sake of simplicity, we have assumed M_s^SPM = M_s^FM (1 – ξ). The m (T) behavior changes from a curve with increasing with temperature slope to a curve with decreasing with temperature slope. This change in the curve shape is in qualitative agreement with the one obtained in [17]. A quantitative comparison is not possible since the difference in the values of M_s^FM and M_s^SPM is not known. For ξ = 1 the model describes ferrimagnetic SFMO without any fitting parameter (cf. Fig. 1). Here, the agreement with experiment can be improved by using more reliable physical parameters, taking into account the damping of spin waves by magnetic disorder and improving modeling of the spin-wave stiffness constant D (T) in the vicinity of the Curie temperature.

Figure 3.

m (T) calculated by means of Eq. (8) for different fractions ξ.

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 [24]. In the vicinity of the Curie temperature (T/T_C > 0.85), the model fails. The model provides a base for the design of spintronic devices, magnetic sensors as well as for medical application of magnetic particles, e.g., contrast agents in clinical MRI magnetic resonance imaging or local heating using strong magnetic AC-fields (hyperthermia).

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.