Coercive Field Model for Description of Mono- and Multidomain

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C: Physical Processes in Nanomaterials and Nanostructures

Coercive Field Model for Description of Mono and Multi-Domain Magnetic Granular Systems Benjamim Zucolotto, Cristiani Campos Plá Cid, Fabrício L. Faita, Walter Sydney Dutra Folly, and André Avelino Pasa J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b11246 • Publication Date (Web): 14 May 2019 Downloaded from http://pubs.acs.org on May 14, 2019

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The Journal of Physical Chemistry

Coercive Field Model for Description of Mono and Multi-Domain Magnetic Granular Systems ∗,†,‡

Benjamim Zucolotto,

Cristiani C. Plá Cid,

§

Folly,

†Laboratório

†

¶

Fabrício L. Faita,

and André A. Pasa

Walter S. D.

†

de Filmes Finos e Superfícies, Departamento de Física, Universidade Federal de Santa Catarina, Florianópolis 88040-900, Brasil

‡Departamento

de Ciências Exatas e Engenharias, Universidade Regional do Noroeste do Estado do Rio Grande do Sul, Ijuí 98700-000, Brasil

¶Instituto

de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre 91501-970, Brasil

§Departamento

de Geologia, Universidade Federal de Sergipe, São Cristóvão 49100-000, Brasil

E-mail: [email protected]

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Abstract The study of nanoscopic magnetic systems of granular materials is an interesting subject for fundamental research in the eld of condensed matter physics that can be modeled through convenient theory. Here, we evaluate the attempt time τ0 and magnetic anisotropy energy barrier Ean of magnesium ferrite (MgFe2 O4 ) samples annealed at dierent temperatures using scaling plots from ac magnetic susceptibility data. The scaling of ln(τ ) versus 1/Tm00 is valid for a range of annealing temperatures. Particularly, we identify MgFe2 O4 nanoparticles with an increase in size and the emerging of agglomerates, which are also corroborated by data from XRD, TEM and magnetic measurements. By analyzing ZFC-FCW curves and working out hHC iT , we show that the model describing noninteracting mono-domain nanoparticles remains valid for samples without agglomerates, as long as the received thermal energy does not surpass a given threshold value. For increasing annealing energy, the magnetic anisotropy energy barrier is substantially increased while the magnetic relaxation time scale becomes much lower. We presented a generalized model for the description of magnetic coercivity as a function of temperature systems composed by magnetic mono and multi-domains. With this model, we described the temperature dependence of the coercive eld of MgFe2 O4 samples successfully. The contribution of superparamagnetic particles mono and multi-domain and the use of a temperature-dependent average blocking temperature was shown to be important to describe the coercive eld throughout the measured temperature range.

Introduction The dynamics of magnetic nanoparticles systems with dierent interaction strengths has been widely studied in recent years. Understanding and controlling the behavior of magnetic nanoparticles is indeed extremely relevant for data-storage protocols, spin-electronic devices, and biomedical applications 114 . The model describing the magnetic behavior of a system of monodispersed and noninteracting single-domain nanoparticles with uniaxial anisotropy 2


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proposed by Néel 15 has been successfully tested by several experiments 1619 . In the Néel model the relaxation time τN is described by an Arrhenius law

τN = τ0 exp

Ean kB T

,

(1)

where τ0 is attempt time typically ranging from 10−10 to 10−12 seconds, 20,21 Ean is anisotropy energy barrier, kB is the Boltzmann constant, and T is the environment temperature 22 . The quantities τ0 and Ean are two main parameters for understanding magnetic processes related to nanoparticles in devices that use magnetic ordering. The frequency dependence of ac magnetic susceptibility, χAC , allows the evaluation of

Ean and τ0 parameters, which are related to the reorientation of nanoparticle magnetic moments. This technique is adequate for systems governed by thermally-activated processes with anisotropy energy barrier since for small applied ac magnetic elds the dynamics of attempt time can be obtained from the dependency on the temperature. 2325 In the case of mono-domain particles and uniaxial anisotropy, Ean = Kef f V , where Kef f is a constant addressing the eective magnetic anisotropy and V is the average particle volume. The temperature where the transition from slow magnetic moment relaxation to rapid relaxation occurs, within a given measurement time window, is directly proportional to the volume of the particles and is called blocking temperature, TB . However, it is known that when taking a particle-size distribution into account a distribution for the blocking temperature, f (TB ), will be observed and consequently also for the anisotropy energy barrier 2629 . In added, the average anisotropy energy barrier can be extracted from the imaginary component of χAC , 00 by plotting ln(τN ) versus 1/T 00 m and carrying out ttings using Eq. (1), where Tm is the

temperature whose imaginary component of the ac magnetic susceptibility is most intense. With this procedure, we can also determine the average attempt time τ0 . Another way of characterizing the dynamics of magnetic nanoparticles is through analysis of temperature dependent magnetic coercivity, HC . A lot of eort has been put for-

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ward in modeling HC versus T , thus making possible to determine an eective magnetic anisotropy for noninteracting mono-domain particles with uniaxial anisotropy 3035 . This approach alongside with the one discussed above provides us a powerful tool for describing the behavior of magnetic nanoparticles. However, the related models are no longer valid when the nanoparticles system has the behavior of magnetic multi-domains. In this paper, we propose a model for the magnetic behavior of systems with the presence of magnetic multi-domains. We characterize the magnetic dynamics of a mixed spinel structure (AB2 O4 ), i. e., of a system consisting of magnesium ferrite (MgFe2 O4 ) nanoparticles grown in a single-crystalline matrix of iron magnesium oxide, [Mg;Fe]O (magnesiowüstite). Such nanoparticles grow coherently with the matrix 36 and take the form of an octahedron with the diagonals parallel to the h100i directions of the [Mg;Fe]O host lattice. 37,38 This system has a cubic magnetocrystalline anisotropy with the easy-magnetization axis parallel to h100i directions. 37,39

Experimental procedure Magnesium ferrite samples were prepared packing a single crystal of MgO in a MgO+Fe2 O3 mixture containing 2.2 mol% Fe, heating at 1673 K for two weeks, and quenching it in cold water. Then, the single crystal was cleaned and broken up into pieces. The annealing temperatures were 773, 873, 973, 1073 and 1173 K for 8 hours to precipitate the MgFe2 O4 phase. After annealing processes, a piece of the single crystal was cleaved and ground into powder in a ponder. This powder was used for measurements of X-ray diraction (XRD) and transmission electron microscopy (TEM). X-ray powder diraction patterns were obtained at room temperature using a Rigaku diractometer (DMAX100) with Bragg-Brentano geometry in continuous mode at a scan speed of 1/4o /min in the 2θ range from 25o to 85o with a Cu-Kα radiation source at 40 kV and 40 mA. The crystalline phases and mean crystallite size were addressed by Rietveld

4


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renements. We TEM was employed to characterize the crystalline structure, morphology, and size of the nanoparticles. TEM images and electron diraction patterns (selected area electron diraction technique, SAED) were obtained using a JEOL JEM−2010 microscopy (200 kV). The powder with nanoparticles was dispersed in isopropyl alcohol, sonicated for 30 minutes, and dripped in an ultra-thin carbon lm supported by a copper grid (300 meshes). The ac magnetic susceptibility χAC = χ0 + iχ00 was obtained with a Quantum Design PPMS Dynacool

TM

magnetometer (Physical Property Measurement System) for several

temperatures, applied elds, and excitation frequency values. Magnetization−Field (M −H ) hysteresis loop measurements were performed from 2 to 70 K and M − T measurements were done in zero-eld-cooled (ZFC) and eld-cooled-warming (FCW) modes with an applied magnetic eld (HDC ) of 100 Oe. The whole set of measurements discussed above was performed setting the eld orientation along the easy h100i directions.

Results and discussion

Structural analysis In Fig. 1 we show the XRD patterns taken at room temperature. The cubic phases of MgO (space group f m-3m) feature the very same structure as of [Mg;Fe]O for every sample. The phases of MgFe2 O4 (with space group f d-3m and cell parameters a = b = c) can be easily identied in samples annealed above 873 K. Our XRD analysis was done using the PDF-01-077-2179 for [Mg;Fe]O and PDF-01-073-2410 for MgFe2 O4 patterns which we set as initial estimates for the diractogram numerical analysis carried out via Rietveld method. The (220), (311), (511), and (440) MgFe2 O4 peaks at 2θ ≈ 30.1◦ , 35.5◦ , 57.1◦ , and 62.7◦ , respectively, was used to evaluate the mean particle size and cell parameters. For annealing temperatures of 773 and 873 K, we could not identify MgFe2 O4 phases and we found cell parameters of 4.2100(1) and 4.2111(1) , respectively, corresponding to 5



Figure 1: XRD patterns for the samples annealed at 773 (a), 873 (b), 973 (c), 1073 (d) and 1173 K (e) for 8 hours. Solid lines denote the corresponding ttings obtained from the Rietveld protocol. the [Mg;Fe]O matrix. For samples annealed at 973, 1073, and 1173 K the cell parameters were found to be 4.2124(2) (8.4410(2)), 4.2100(1) (8.4521(2)), and 4.2091(2) (8.4815(1) ) for [Mg;Fe]O (MgFe2 O4 ), respectively. For samples annealed at 973, 1073, and 1173 K, the mean particle sizes obtained from Rietveld analysis were 9.1(5), 11.8(6), and 12.4(6) nm, respectively. The XRD results showed above clearly indicate that the nucleation and growth of the MgFe2 O4 nanoparticles worked out successfully for samples annealed above 873 K. Even though the nanoparticle growing process of MgFe2 O4 phase has a dependence on annealing temperature, a signicant increase in the size of the crystallite is not observed with temperature. Besides, the measured cell parameters for MgFe2 O4 were roughly as twice as high as

6


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the values found for [Mg;Fe]O conrming the growing of nanoparticles. Fig. 2 shows TEM results in nanometric akes from samples heated at 973 and 1173 K, for 8 hours. Bright-eld (BF) images were obtained of akes with approximately 200 nm for both samples, i. e. 973 K (Fig. 2 (a)) and 1173 K (Fig. 2 (f)). The BF images show regions with dierences, allowing to identify nanoparticles with sizes varying between 5 to 30 nm. Selected area electron diraction patterns were acquired and MgO, and MgFe2 O4 crys-

(a)

(b) (III) (111) MgO

(II) (400) MgFe2O4

(I) (400) MgFe2O4

(c)

(e)

(d)

(f)

(g)

(VI) (420) MgO

27.36 nm 24.59 nm 31.53 nm

(V) (400) MgFe2O4 (IV) (511) MgFe2O4

(h)

(i)

( j)

Figure 2: (a) [(f)] Bright-eld image and (b) [(g)] SAED patterns acquired for heat-treated sample at 973 K [and 1173 K]. The MgO and MgFe2 O4 phases were identied in the SAED patterns. DF images were acquired and the MgFe2 O4 phase was identied by the (400) plane, (c), (d) and (i). The (511) plane was observed only in the heat-treated sample at 1173 K (h). The MgO phase was identied from (111) and (420) planes, (e) and (j), respectively.

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talline phases were identied, corresponding to fcc system. For the sample heat-treated at 973 K, the pattern (Fig. 2 (b)) shows spots with interplanar spacing (dhkl ) corresponding to diraction rings identied as the (400) plane of MgFe2 O4 and the (111) plane of MgO. For the sample heat-treated at 1193 K (Fig. 2 (g)) were identied spots with dhkl -spacing corresponding to (400) and (511) planes of MgFe2 O4 and plane (420) of MgO. The identication of phases in both patterns concurs with the XRD results. The values of dhkl -spacing obtained from SAED patterns are in agreement with those indicated in the standard data (CIF:1000053 and 9001484). The spot referent to (400) plane was selected to obtain dark-eld (DF) images for both samples (Fig. 2 (c), (d) and (i)) and it is possible to observe structures that diract with size less than 10 nm, which is in agreement with the average size of crystallites observed by XRD. Fig. 2 (h) shows a DF image obtained by selecting the spot associated with the (511) plane. In this case, it was possible to observe the presence of structures with size around 30 nm in the ake from sample heat-treated at 1173 K. The MgO phase was identied from (111) and (420) planes, see Figs. 2 (e) and 2 (j), respectively. From these results, we could conclude, that the MgFe2 O4 phase occurs with two distributions of crystallite size in the observed samples.

Magnetic properties ac

magnetic susceptibility

Now, in Fig. 3 we address the temperature dependence for both real χ0 and imaginary χ00 components of the ac magnetic susceptibility. There are a few interesting features we would like to point out for the samples annealed at 973, 1073, and 1173 K. First, χ0 and χ00 have 0 00 maxima at temperatures Tm and Tm , respectively. Such maxima are not well dened in the

range of measured temperatures for samples annealed at 773 and 873 K. We note in Fig. 0 00 3 that both Tm and Tm are signicantly shifted towards higher measurement temperatures

as the measurement frequency increases. Furthermore, the frequency-dependent behavior of 8


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χ0 for T ≤ Tm0 stands out as a signature of the blocking process of MgFe2 O4 nanoparticles, thus witnessing a superparamagnetic regime. This behavior is expected for a wide variety of superparamagnetic materials 40 and it was reported for this same experimental system in previous publications 37,40 . However, in those cases the samples were prepared at the same annealing temperature (973 K) for dierent times - a procedure inverse to the employed in the present work, in which the samples were annealed for the same time (8 hours) at dierent temperatures. The use of dierent annealing temperatures has at least two kinds of implications on the nal magnetic properties of the resulting magnesioferrite nanoparticles. The rst one arises from the fact that the inversion parameter of its spinel structure varies with the annealing temperature and, consequently, the molecular magnetic moment also changes 41,42 . The second one arises from the inuence of the annealing temperature on the size distribution of the particles. For samples annealed at 973 K, the frequency-dependent behavior most likely stems from the superparamagnetic blocking/unblocking process arising mostly from noninteracting small particles. On the other hand, higher annealing temperatures (1073 and 1173 K) supposedly favor the particle clustering and the growth controlled by coalescence 36,43 , which may increase the occurrence of large multi-domain particles as well as the probability of dipolar interactions between them. 00 The temperature Tm increases with the annealing temperature for a given frequency f 0 00 . Considering pairs (Tm , f ) we have computed and a similar behavior was observed for Tm 00 for the samples annealed at 973, 1073 and 1173 K, and the results were ln(τ ) versus 1/Tm

plotted in the Fig. 3(d). The attempt time τ0 and the average anisotropy barrier h∆Ean /kB i obtained from linear ttings with the Néel-Arrhenius law [Eq. (1)] are shown in the inset of Fig. 3(d). Overall, we that observe for the samples annealed at higher temperatures the τ0 decreases while h∆Ean /kB i grows, which indicate a progressive emergence of dipolar interactions between particles. For samples annealed at lower temperatures, we have fewer particles and they are very small and dispersed in the matrix. In such a condition, we can 9



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(d)

Figure 3: Temperature dependence of the real χ0 and imaginary χ00 component of the ac magnetic susceptibility for samples annealed at 973, 1073 and 1173 K for 8 hours on panels a0 , a00 ; b0 , b00 and c0 , c00 , respectively, at the frequencies showed on the inset. (d) Classical plots of ln(τ ) versus 1/T 00 m for samples annealed at 973, 1073, and 1173 K. Solid lines represent the optimal ttings evaluated via Néel-Arrhenius law [see Eq. (1)]. The inset displays the corresponding parameters ∆Ean and τ0 . assume that the dipolar interactions between particles are negligible. On the other hand, for samples annealed at higher temperatures, we have a larger population of particles, many of them quite large and close to each other. This occurs due to the availability of iron ions in the matrix and the consequent change in the particle growth regime from growth controlled by diusion to growth by coalescence 36 . This conguration favor the dipolar interaction 10


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between nearby particles, mainly when they form clusters. This behavior is remarkably dierent from those reported for this same material in previous publications 37,40 . In those studies, the samples were annealed at a xed temperature for dierent times, and the authors do not observe changes in τ0 with the annealing time.

Magnetization and distributions of energy barriers In Fig. 4 we show the normalized magnetization as a function of temperature (ZFC and FCW curves at HDC = 100 Oe) of samples annealed at 973, 1073 and 1173 K. The ZFC curves exhibit a maximum that is dependent on the temperature of annealing. In general, the FCW curves show that magnetization decreases with increasing annealing temperature. However, this behavior is signicantly suppressed at higher temperatures. Furthermore, the ZFC curves exhibit a monotonic increase of the magnetization with temperature that is dependent on the annealing temperature, i. e., higher annealing temperatures results in samples with weaker dependence of the magnetization on the temperature. In other words, higher temperatures are needed to attain the maximum values of magnetization in samples submitted to higher annealing temperatures. For samples annealed below 973 K, we did not observe any dierence whatsoever among the ZFC-FCW curves, and their behavior was typically paramagnetic (results not shown). The results above allow us to distinguish two main characteristic temperatures: the blocking temperature TB which is associated with the maximum of the ZFC curve; and the temperature that stands for the maximum of f (TB ) ∝ d(MF CW − MZF C )/dT versus T , that provides information about the energy-barrier distribution. When the anisotropy is proportional to the volume, as in our case, it also becomes proportional to the size-distribution prole. Therefore, from the FCW-ZFC measurements the coexistence of blocked and unblocked particles can be represented by a distribution of blocking temperatures,f (TB ), holding not a unique value for TB due to particle-size distribution eects and interparticle interactions. By looking at f (TB ) carefully, see the experimental data in Fig. 5(b) and Fig. 5(c), we note 11



Figure 4: Zero-eld cooled (square symbols) and eld cooled warming (circles) curves measured for samples annealed at 973 (a), 1073 (b), and 1173 K (c) for 8 hours at HDC = 100 Oe. that the anisotropy-energy barrier is wider for higher annealing temperatures. This can be accounted for the increase in the average particle size. The existence of large particles with higher blocking temperatures is also seen in the low-temperature part of FCW curves (see inset Fig. 4(b) and 4(c)). In our case the results present two inexion points suggesting two mean blocking temperatures, hTB1 i and hTB2 i. This conclusion is corroborated by the TEM results, in which we notice the formation of the phase MgFe2 O4 in two distributions of crystallite size that can be directly related to the two assumed blocking temperatures. This

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Figure 5: f (TB ) distribution curves for samples annealed at 973 (a), 1073 (b), and 1173 K (c) for 8 hours: experimental and best adjustment with the generalized model described by Eq. 2. leads us to assume f (TB )12 as being the bimodal log-normal distributions can be written as

f (TB )12 = f (TB )1 + f (TB )2 1 TB A 2 √ exp − = ln 2σ1 2 hT i TB σ1 2π B1 1−A 1 TB √ exp − ln2 , + 2 2σ2 hTB2 i TB σ2 2π

(2)

where A is weighting factor, σ1 (σ2 ) is the standard deviation and hTB1 i (hTB2 i) is the median blocking temperature. The lines in Fig. 5 were ttings of the experimental data by Eq. 2. The hTB1 i and 13



hTB2 i where initially estimated from the experimental distribution empirically and the free parameters used in the ts were σ1 , σ2 and A, using the algorithm presented in Ref. 32 . The parameter A varies between 0−1 because it will represent the percentage of each distribution function, i. e., for A = 1 only one blocking temperature distribution is present, values smaller than 1 indicates the presence of a second blocking temperature distribution. The functions

A and 1 − A are thus added to obtain the percentage of each distribution function. However, the σ parameters are the polydispersion parameters. Experimentally the data do not reveal the negative slope for the annealed sample at 973 K because the equipment used does not measure at temperatures below ∼2 K. However, we assume this slope due to the transition of magnetization at this temperature. When we performed the ttings, we assume an additional in the origin of coordinate system (0,0). The optimized parameters of the tting are shown in Table 1. Although in the present case this adjustment is not appropriate at the lower temperature range due to the lack of experimental data at temperatures below 2 K, we can see that the agreement between the theoretical and experimental curves is good for the three samples since R2 is high. The characteristics of the nanoparticles can be deduced from f (TB ) that represents the distribution of energy barriers since KV ∝ TB . The smaller particles are responsible for the low-temperature peak in all samples and the larger ones for the second peak at higher temperatures observed in samples annealed at 1073 K or above. The results shown in Table 1 are meaningful and can be interpreted as a consequence of the size distributions of small individual particles (parameters σ1 and hTB1 i) as well as the size distribution of particle clusters and larger particles (parameters σ2 and hTB2 i). Moreover, a visual inspection of Table 1 reveals a progressive increment of the blocking temperatures as the ones attributed to smaller particles hTB1 i as well as to particle clusters and large particles hTB2 i with the annealing temperature. In addition, we observe a progressive narrowing of the blocking temperature distribution associated with small particles (decrease of σ1 with the annealing temperature) and a progressive broadening of such distribution associated to large particles 14


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Table 1: Optimal tting parameters of the blocking temperature distribution. Parameters σ1 hTB1 i (K) σ2 hTB2 i (K) A (%) R2

973 K 0.84 4.88 − − 100 0.99

1073 K 0.81 5.55 0.37 8.94 71 0.97

1173 K 0.65 6.8 0.56 24.4 27 0.96

and clusters (increase of σ2 with the annealing temperature). Magnetic hysteresis loops of the various temperatures are displayed in Fig. 6. This behavior is expected from the ZFC-FCW curve (Fig. 4) that have a superparamagnetic behavior when both curves converge. For samples annealed at 873 K, the coercivity (HC ) and remanent magnetization (MR ) was not observed even at very-low temperatures thus following a typical Brillouin function, and samples annealed at 773 K exhibited only a linear region, thereby featuring a paramagnetic behavior (not shown here). The M − H taken for

T > TBmax , where TBmax is a temperature above which the magnetization of all particles are unblocked, exhibit a Langevin-like superparamagnetic behavior with negligible hysteresis. By looking at the magnetic hysteresis loops, we observe that raising the annealing temperature leads to the increase of the magnetic unblocking temperature. Also, the MR .HC product decreases abruptly with the rise of temperature for the sample annealed at 973 K. Conversely; it increases smoothly with the increase of annealing temperature. Therefore, the energy required to magnetize and demagnetize the material grows up considerably, indicating a progressive strengthening of the magnetic inertia and possible interaction between granules with the annealing temperature. The MgFe2 O4 undergoes a T -activated cation order-disorder transition, which involves an atomic exchange between tetrahedral and octahedral sites, thus resembling the behavior of spinels. Ideally, the MgFe2 O4 has an inverse structure, meaning that Mg and half of the Fe atoms occupy the octahedral sites, whereas the remaining Fe atoms distribute over the tetrahedral sites. 38,42 However, this system has partially inverse spinel with dependency on 15



Figure 6: Magnetic hysteresis loops measurements performed for various temperatures at the blocked regime for samples annealed at 973 (a), 1073 (b), and 1173 K (c) for 8 hours. 2+ 3+ 2− the inversion parameter (1 − ξ ) of the form (M gξ2+ , F e3+ 1−ξ )[M g1−ξ , F e1+ξ ]O . The inversion

parameter takes the range 0 ≤ (1 − ξ) ≤ 1, with ξ = 1 being direct spinel and ξ = 0 denoting an inverse spinel, that is Fe cation coupled ferromagnetic and antiferromagnetic, respectively. Note that MR increases with the annealing temperature, thus highlighting the decrease of the inversion parameter (higher magnetic ordering). In other words, it results in a higher concentration of Fe cations in octahedral sites corroborating with the increase of MR .

Coercive eld The temperature dependence of the coercivity for an idealized particle system featuring noninteracting mono-domain magnetic nanoparticles with uniaxial anisotropy can be tted

16


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using HC = α(2Kan /MS )[1 − (T /hTB i)1/2 ] in the temperature range 0 − TB when all particles remain blocked, where MS is saturation magnetization, Kan is anisotropy constant, and

α = 1 if the particles easy axes are aligned and α = 0.48 if they are randomly oriented. 44,45 However, when taking the particle-size distribution into account, we can evaluate the average blocking temperature only for blocked particles at specic temperatures TB > T by

hTB iT1(2)

R∞ TB f (γ1(2) TB )1(2) dTB . = TR ∞ f (γ T ) dT B B 1(2) 1(2) T

(3)

where γ1(2) is a parameter that takes into account the eect of interparticle interactions similar to that proposed by other authors 30,46 . As described in previous sections, the existence of two well-dened distributions of blocking temperature, due to the two distribution of particle size, implies two behaviors of HC . Thus, these two components may be superposed in order to obtain HCB12 = HCB1 + HCB2 , where HCB is coercive eld taking into account only the blocked particles at a given temperature (T ), i. e., only particles with TB > T , can be written as

" 1/2 # 2hKan i1(2) T 1− . HCB1(2) (T ) = α MS1(2) hTB iT

(4)

The magnetic behavior of superparamagnetic particles at H < HCB it is linear and corresponds to M1(2) = MR1(2) + (MR1(2) /HCB1(2) )H . Hence, the total magnetization can be obtained by sum MT (T ) = M1 + M2 + MχT where MχT = χT H and χT is the total superparamagnetic susceptibility (which will be detailed below). In such a way, considering

M (hHC iT ) = 0, the average coercive eld can be written as,

hHC iT =

MR1 (T ) + MR2 (T ) . MR1 (T ) MR2 (T ) + + χT (T ) HCB1 (T ) HCB2 (T )

(5)

The superparamagnetic susceptibility has three contributions: free spins of few Fe atoms,

χS , smalls particles superparamagnetic in the unblocked regime, χSP1 and large particles 17



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superparamagnetic in the unblocked regime, χSP2 . Then, the total susceptibility χT is the sum of these three contributions, χT = χSP1 +χSP2 +χS . For calculation of superparamagnetic susceptibility in the low eld limit for systems with a particle size distribution, it can be considered the relation between V and TB described by Néel model, 15 which can be written as,

25MS1(2) 2 χSP1(2) (T ) = 3hKan i1(2) T

Z

T

TB f (γ1(2) TB )1(2) dTB ,

(6)

0

the total susceptibility will be given by,

25 MS1 2 MS2 2 χT (T ) = ··· + 3T hKan i1 hKan i2 Z T C ··· TB f [(γ1 TB )1 + (γ2 TB )2 ] dTB + , T 0

(7)

and the remanence magnetization, presented in Eq. 5, is related to f (TB ) according to,

Z MR1(2) (T ) = α1(2) MS1(2)

∞

TB f (γ1(2) TB )1(2) dTB .

(8)

T

Fig. 7 shows the experimental and optimal adjustments of the hHC iT − T curves performed through the model proposed in this paper through Eqs. 5, 7 and 8 and also using the adapted sampling method described in Ref. 32 which considers only one blocking temperature distribution. The experimental values of HC were obtained by (|HC+ | + |HC− |)/2 because

Table 2: Optimal tting parameters of the coercive eld. Parameters 973 K 1073 K 1173 K hKan i1 (erg/cm3 ) 845 830 460 γ1 (dimensionless) 1 1 1 MS1 (emu/g) 2.1 1.73 0.92 hKan i2 (erg/cm3 ) − 710 1950 γ2 (dimensionless) − 1.1 1.145 MS2 (emu/g) − 0.46 0.85 3 C (emu K/Oe cm ) 0.012 0.02 0.02 2 R 0.98 0.97 0.96

18


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Figure 7: Theoretical (HC−CAL ) and experimental (HC−EXP ) dataset of the coercive eld dependence on the temperature for samples annealed at 973 (a), 1073 (b), and 1173 K (c) for 8 hours. The dotted lines are ttings using the mono-domain model 32 and the solid line is tted using mono and multi-domain proposed model. The inset shows the very same properties for samples annealed at 973 K for 6, 8, and 10 hours. the remanence in the superconducting coil smoothly displaced the M − H curves to positive elds. These results indicate that the contribution of superparamagnetic particles and the dependence on the average blocking temperature are rather crucial to describe the coercive eld over a wide temperature range. On the other hand, experimental and theoretical data from the model that considers only one blocking temperature do not match in the low-temperature region for the samples annealed at higher temperatures. For sample annealed at 973 K, a single blocking temperature distribution describes the experimental data, and we can observe that the model proposed here did not make it necessary to add a second distribution as shown in Table 2. Therefore, it is noted that for 973 K the behavior is characteristic of a magnetic mono-domain, in agreement with TEM results. Moreover, in the insertion in Fig. 7, we show that this behavior remains even for dierent annealing times. For annealing temperatures higher than 973 K the model with only one blocking tem-

19



perature distribution from Ref. 32 tends to disagree from the experimental data at low temperature, in particular for the sample annealed at the highest temperature. The model we are proposing considers two distributions of blocking temperature due to the existence of large particles with multi-domain and small particles with mono-domain magnetic behavior and accurately describes the experimental data, as displayed in Fig. 7. The obtained results are following the sudden increase of the magnetic anisotropy energy barrier obtained from ln(τ ) versus 1/T 00 m .

Conclusion The thermal annealing of magnesium ferrite (MgFe2 O4 ) samples provides a strong magnetic contribution of the direct spinel increasing the magnetic remanence. The time dependence on thermal energy did not signicantly change the structure of the material as well as the magnetic regime. The Rietveld renement of the XRD diractograms showed an average particle size with small growth with the increase in annealing temperature. Also, the TEM images showed that the crystal patterns suered signicant changes with annealing temperatures. Instead of increasing the average size, it was observed the formation of crystallites small agglomerates and not indicating the return of the phase magnesiowüstite in the range of temperatures studied (MgFe2 O4 → [Mg;Fe]O). The magnetization curves showed at the beginning a magnesium ferrite phase with a single blocking temperature distribution. With the increase of the annealing temperature, two blocking temperature distributions were observed. It was also found that the blocking temperature distribution is related to the particle size distribution observed in TEM images. We associate one of the blocking temperature distributions to noninteracting particles and magnetic mono-domain and the other distribution to magnetic multi-domain particles. Then, we presented a generalized model for the description of magnetic coercivity as a function of temperature of systems composed by magnetic mono and multi-domains. With

20


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this model, we described the temperature dependence of the coercive eld of MgFe2 O4 samples successfully. The contribution of superparamagnetic particles mono and multi-domain and the use of a temperature-dependent average blocking temperature have shown to be essential for describing the coercive eld in the measured temperature range. The presented model will serve for other authors to identify and characterize nanostructured magnetic systems with magnetic multi-domains.

Acknowledgments The authors acknowledge support from the Brazilian agencies FINEP, CAPES (Finance Code 001), FAPESC and CNPQ (Grant No. 152026/2016-9) and LCME/UFSC (LCMEMAT-2019) for the access to electron microscopy facilities.

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Graphical TOC Entry Increase of annealing temperature

+

Mono and multi-domain

Mono-domain

Coervive Field Model

Experimental

Coercive Field

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Mono-domain Mono and multi-domain

Temperature

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Coercive Field Model for Description of Mono- and Multidomain

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