Validity of the N\'{e}el-Arrhenius model for highly anisotropic Co_xFe_{3-x}O_4 nanoparticles

We report a systematic study on the structural and magnetic properties of Co_{x}Fe_{3-x}O_{4} magnetic nanoparticles with sizes between $5$ to $25$ nm, prepared by thermal decomposition of Fe(acac)_{3} and Co(acac)_{2}. The large magneto-crystalline anisotropy of the synthesized particles resulted in high blocking temperatures ($42$ K \leqq $T_B$ $\leqq 345$ K for $5 \leqq$ d $\leqq 13$ nm ) and large coercive fields ($H_C \approxeq 1600$ kA/m for $T = 5$ K). The smallest particles ($=5$ nm) revealed the existence of a magnetically hard, spin-disordered surface. The thermal dependence of static and dynamic magnetic properties of the whole series of samples could be explained within the N\'{e}el-Arrhenius relaxation framework without the need of ad-hoc corrections, by including the thermal dependence of the magnetocrystalline anisotropy constant $K_1(T)$ through the empirical Br\"{u}khatov-Kirensky relation. This approach provided $K_1(0)$ values very similar to the bulk material from either static or dynamic magnetic measurements, as well as realistic values for the response times ($\tau_0 \simeq 10^{-10}$ s). Deviations from the bulk anisotropy values found for the smallest particles could be qualitatively explained based on Zener\'{}s relation between $K_1(T)$ and M(T).


I. INTRODUCTION
Ferrites are spinel oxides with formula MFe2O4 (M = 3d transition metal) with cubic crystal structure and a multiplicity of complex magnetic configurations arising from the diverse interactions between the M and Fe magnetic ions. When M = Co 2+ , the resulting cobalt ferrite (CoFe2O4) has distinctive magnetic properties due to its large first order magnetocrystalline anisotropy constant (K1= 2x10 5 J/m 3 ), which is about an order of magnitude greater than any other spinel oxide. 1 Together with its chemical stability, this property make CoFe2O4 magnetic nanoparticles (MNPs) a fundamental material for magnetic recording applications and ferrofluids. 2 Considerable efforts have been made to obtain homogenous and stable water-based nanofluids through different synthesis routes such as hydrothermal, coprecipitation, microemulsion, forced hydrolysis, and reduction-oxidation methods. 2 In particular, the thermal decomposition of organometallic precursors in a boiling solution of organic solvents has been successfully used to produce MNPs with narrow size dispersion, 3,4 and thus they are being increasingly exploited in those applications with critical specifications about size dispersion of the MNPs. 5 The ferrimagnetic order in CoFe2O4 results from the competing super-exchange interactions between the two magnetic sublattices of tetrahedral (A) and octahedral (B) sites in the structure. The Fe +3 ions within the B sublattice are ferromagnetically ordered, as well as the Co +2 ions within the A sublattice. On the other hand, the interactions between A and B spin sublattices are antiferromagnetic, resulting in an uncompensated net magnetic moment. The exchange energy in this material has been reported to be as large as JAF = -24 kB. 6 It is well known that the relation between the anisotropy and exchange energies determines the critical size (Dcr) for the single domain configuration. The existence of a critical diameter Dcr of a (spherical) particle implies that below a certain diameter value d such that d < Dcr, the lowest free energy state is that of uniform magnetization, as proposed by Brown. 7 This critical value has been estimated 8,9 to be = 5.1 √ 0 2 , where is the exchange stiffness 10 and MS is the saturation magnetization of the material. Using = 15 x10 -12 J/m; MS = 425 A/m (bulk CoFe2O4) 11 and 0 = 4 10 −7 H/m, a critical diameter = 40,7 nm is obtained.
Accordingly, reported values of the single domain critical size for CoFe2O4 are between 30 and 70 nm. 12 As a consequence of the large magnetic anisotropy, single domain particles of CoFe2O4 of a few-nanometer size can retain the blocked regime up to room temperature. This particularity allows observing the thermal evolution of some magnetic parameters of MNPs such as saturation magnetization and coercivity of the blocked state in a wide range of temperatures before the superparamagnetic transition wipes out this information.
The energy E of an assembly of uniaxial particles with their easy axes parallel to the z axis under an external applied field is usually described (at T=0) by: where  is the angle between field H and saturation magnetization MS, V the particle volume and Keff is the effective magnetic anisotropy. Assuming the energy of a single particle given by equation (1), the unblocking process occurs through an energy barrier E given by:

(2)
At a fixed temperature T the reversal of the magnetic moment occurs through the energy barrier given by equation (2). This thermally-activated process is described by the Néel-Arrhenius model, which gives a simple expression for the relaxation time = 0 ⁄ . Taking  = 10 2 s for the measuring time window and 0 = 10 -9 s we get KeffV = 25 kBT the coercive field HC (T) can be expressed as: This is the well-known HC vs. T 1/2 relation often used for fitting the temperature evolution of the coercive field in the blocked state, i.e., at low temperatures. It is worth to note here that the thermal dependence of Keff in equation (3) is neglected, although previous studies of bulk spinel oxides have reported large variations of the anisotropy below room temperature. 13 In this work, we report a systematic study on the magnetic properties in a series of Co ferrite magnetic nanoparticles within 5 and 25 nm. An exhaustive study by high resolution electron transmission microscopy (HRTEM) techniques has been performed in order to explore the influence of MNPs size and shape on the observed magnetocrystalline anisotropy, 14 with a precise observation of the crystallographic structure with atomic resolution. The chemical composition at the single-particle level was performed to assess the levels of stoichiometric homogeneity of samples.
Systematic measurements of magnetization, coercive field and magnetic anisotropy were performed for increasing particle size to study the temperature evolution of the magnetic parameters in the blocked regime. The validity of the Neel-Arrhenius law for explaining the temperature dependence of the relaxation time has been re-gained by taking into account the variation of the anisotropy constant with the temperature.  The ratio between iron and cobalt content was determined through Energy-Dispersive X-ray spectroscopy (EDX) performed on a macroscopic zone of a powder sample (about 10000 m 2 ) in SEM analyses, and on a small area (about 1000 nm 2 ) containing many particles as well as on single particles using the TEM.

II. EXPERIMENTAL
The total iron concentration was determined from UV/Vis spectroscopy in a Varian Cary 50 Spectrophotometer operating at a fix wavelength of 478 nm. For the absorbance measurements, Potassium thiocyanate (KSCN) was used following the standard protocol described elsewhere. 15,16 Magnetization measurements M (T, H) and ac magnetic susceptibility measurements were performed on a MPMS-XL SQUID Quantum Design magnetometer. All measurements were performed on dried samples, after conditioning the dry powder inside plastic capsules. The temperature dependence of the magnetization was measured following zero-field-cooling (ZFC) and field cooling (FC) protocols, applying 7.9 kA/m, and the data were collected increasing the temperature from 5 to 400 K. The magnetization isotherms were measured between 5 and 400 K up to a maximum magnetic field of 3.96 MA/m. The susceptibility versus temperature was measured applying an excitation ac field of 0.24 kA/m, at frequencies from 0.1 to 10 3 Hz, under zero external dc magnetic field.

A. Particle morphology and composition analyses
The analysis of the TEM images ( Fig. 1) showed that for each particular synthesis, the MNPs obtained can be considered as uniform in size. The statistical analysis of the MNPs size distribution done by fitting the respective size-histograms to a Gaussian distribution yielded mean diameters ranging from d = 5 to 25 nm and standard deviations size distribution widths  between 0.7 and 3 nm (see Table I). As previously reported for this synthesis route, the final average particle size reflected the influence of both the boiling point of the solvent and boiling time. 3,17 Specifically, a systematic increase of the average particle size <d> for increasing boiling temperature of the solvent was observed. In the case of sample AV11 the final size is a combination of the higher boiling temperature and a shorter time at TFst (10 minutes, see Table SI of supplemental material). 18 Regarding the MNPs morphology, the analysis of HRTEM images showed that for <d>  13 nm a noticeable population of rounded-shaped particles were present, whereas the largest ones showed a more faceted structure (see  The CS-corrected STEM-HAADF analysis at atomic resolution revealed that all the synthesized nanoparticles crystallized in the spinel structure with Fd-3m space group and unit cell parameter a = 8.394 Å. The data showed no evidence of distortions, crystal defects or any preferential orientation of the nanoparticles. As an example, Fig. 3(a) shows a high resolution CS-STEM-HAADF image of a particle of sample AV13.  [111] resulting composition of the CoxFe3-xO4 MNPs extracted for these analysis yielded x values ranging from 0.90 (sample AV05) to 0.54 (for sample AV08).  Fig. 4. For all analyzed samples, the Fe:Co ratios derived from individual particles and from particle clusters coincide, as in the EDX-SEM analysis. The close values of both TEM and SEM analysis in each case (see Table   I) indicate that the chemical composition of the MNPs is homogeneous throughout the samples and, more importantly, within individual particles. Clearly, this analysis of the homogeneous internal structure of single MNPs is performed only for a few selected MNPs. However, the consistency of these data from several particles has been verified in all synthesized samples and therefore, gives support to the statistical confidence of these results.

C. Magnetic field dependence of the magnetization.
The magnetization of all samples was studied at temperatures from 5 K to 400 K, in applied field H up to 11.2 MA/m (14 T). For the M(H) performed at T = 400 K, the obtained coercive field values HC decreased with decreasing particle size (Table II) attaining  4 . 28 A second explanation for the observed reduction in MS could be the existence of spin canting at the particle surface 11,29 originated from competing interactions between A and B sublattices when a symmetry break and oxygen vacancies are produced at the particle surface. Monte Carlo simulations using different models 30,31 and approximations have shown that the reduction of MS is size dependent, and is related to the canted configuration of the spins at the surface. For all but AV05 and AV08 samples (i.e., the two smallest particle sizes), the magnetization was nearly saturated at H = 2x10 3 kA/m. Samples AV05 showed a marked decrease in the magnitude of M, and no signs of saturation up to the highest field. We further investigate this behavior of sample AV05 through measuring the M(H) curves up to H = 11.2 MA/m at 400 K and at 5 K (see Figure 7). As expected for a minor loop, saturation was not reached even at this high field and the cycle remained  48 44 open showing that the irreversibility field Hirr, defined as the field where the two branches of the hysteresis loop merge, was larger than our attainable maximum field. The hypothesis of the surface spin canting that could explain the reduction of magnetization, also would originate the non-saturating behavior of the M(H) curves even at large applied fields, similarly to previous reports on small-sized ferrite nanoparticles. 21,28,32 This is likely to be the case in our samples AV05 and AV08, with a less pronounced effect in AV08 since surface effects are attenuated in particles with increasing volume.
For the rest of the samples, however, the decreasing surface/volume ratio would imply that surface spin canting cannot be a major cause for magnetization reduction.
Additionally, for these samples the observed reduction in MS is not accompanied by the linear increase of the magnetization at high fields. On the contrary, the M (H) curves showed that the magnetic saturation is attained at moderate fields (H  2 MA/m), consistent with previous findings using polarization-analyzed small-angle neutron scattering experiments on Co-ferrite nanoparticles of 11 nm. 33,34 These results are in agreement with our observation of the concurrent low value of the saturation magnetization and the small fields required to reach MS. 33,34 There is experimental evidence that the above mentioned spin canted structure extends over the whole particle volume, instead of forming a shell. 28 In moderate/high magnetic fields the measured magnetization is due to the net sum of spin components parallel to the applied field, and the reduction with respect to the bulk magnetization is due to the cancellation of the components perpendicular to the field, as the result of the competition between Zeeman and anisotropy energies. This might be the case of our nanoparticles with <d>  11 nm, being the particles with <d> = 13 nm, those in which the canting angle is lower (and therefore the magnetization is higher). However, local probe and/or neutron scattering experiments would be necessary to confirm this hypothesis. The values of HC measured at T = 5 K and 400 K (see Table II

D. Temperature dependence of the coercive field.
We have studied the evolution of the coercive field, HC, with the temperature by plotting the experimental HC(T,V) data for 5 ≤ T ≤ 400 K. The expected decrease of HC(T) for increasing temperature was observed in all samples, reaching the HC=0 value at the corresponding superparamagnetic transition temperatures. The exact functional dependence of HC with temperature for single/domain magnetic nanoparticles in the blocked state has been discussed since decades ago. Within the simple Neel-Arrhenius model already presented in the introduction section, a ∝ 1/2 is expected. However, equation (3) neglects the particle size dispersion existing in any real sample, which is an oversimplification in most cases. 42 Recent works have pointed out the difficulties of including the size distribution into a realistic model 43 because the measured HC is not a simple superposition of individual particle coercivities. An analytical expression for the dependence of HC(T) with T and particle size has been proposed, 44 obtaining a T 3/4 for the thermal dependence in a randomly-oriented ensemble of particles. The fact that this approximation was unable to fit our experimental data for any sample, together with the quite narrow size distributions observed in our samples (see Figure 1) suggest that deviations from the T 1/2 law for HC were not due to size distributions.
The departures observed from the HC(T) vs. T 1/2 graphs of our samples (see Figure S3 in the supplementary material) were increasingly marked for the larger particles, strongly suggesting that this feature was related with some neglected Tdependence of the magnetic parameters involved. As equation (3) assumes that the magnetocrystalline anisotropy is a temperature-independent parameter, the corresponding HC expression should be a valid approximation only for a narrow Trange where K1 is not expected to vary substantially. 45,46 This is the case for particles with low blocking temperatures, since only in the blocked state HC>0 can be effectively measured. Indeed, a good T 1/2 fits have been reported for small and/or low-anisotropy MNPs (e.g., T< 50 K). 47,48,49 However, this approximation fails completely for particles with large size and/or anisotropies like CoFe2O4, for which the blocked state may span a temperature range from 5 to 400 K. In such a wide temperature interval K1(T) can change markedly 50 with n = 10 for full correlation between adjacent spins and n=6 for incomplete correlation. 59 In cubic ferromagnetic crystals like spinel oxides, this relation is expected to hold for temperatures below 0.9TC, being TC the Curie temperature of the material.
Based on these relationships Shenker 60 has demonstrated that for bulk cobalt ferrite K1(T) can be expressed by the empirical Brukhatov-Kirensky relation 60 valid for the 20 K< T < 350 K temperature range, with 1 (0) = 1.96 10 6 / 3 and = 1.9 10 −5 −2 . Incorporating this dependence into the HC(T) expression given by eq. (3) and considering that Keff as the first magnetocrystalline anisotropy constant K1 we obtain: As seen in Figure 8, this expression provides an excellent fit of the experimental data for a wide range of particle sizes and temperature, and makes clear that any attempt of describing the thermal evolution of any magnetic parameter depending on Keff over more than a few-degrees temperature range should consider the impact of K1(T). parameter. Similar arguments could be applied to qualitatively explain the additional contribution to the anisotropy observed for K1(0) in AV05 and AV08 samples. Table III. Parameters K1(0) and B obtained from a) fitting the HC(T) data using eq.(6); and b) Néel-Arrhenius model using the eq. (8). For the latter, the values of 0 are also listed.

E. Temperature and frequency dependence of the AC magnetic susceptibility
In order to get a deeper insight into the effective magnetocrystalline anisotropy obtained from dc data, the magnetic dynamics of these nanoparticles was studied through the temperature dependence of and at fixed field amplitude and increasing frequency from 100 mHz to 1 kHz. Typically, both   (T) and   (T) components for all samples exhibited the peak at a temperature TP expected for a single-domain magnetic particle, which shifted towards higher T values with increasing frequency. Typical curves are shown in Figure 9 as examples for <d> = 8.8 and 11 nm (samples AV08 and AV11, respectively). The dynamic response of an ensemble of single-domain magnetic nanoparticles can be described by the thermally-assisted magnetic relaxation of a singledomain magnetic moment over the anisotropy energy barrier Ea. 47 The relaxation time  associated to this process is given by a Neel-Arrhenius law where 0 is in the 10 -9 -10 -11 s range for SPM systems.
In the absence of an external magnetic field, the energy barrier Ea can be assumed to depend on the particle volume V and the effective magnetic anisotropy Keff through the expression = 2 , where  represents the angle between the magnetic moment of the particle and its easy magnetization axis. A linear dependence of ln . −1 is expected from eq. (7) if Keff is assumed to be temperature-independent. However, the extrapolation of the linear fit of the experimental data to T -1 = 0 usually gives too small, unphysical values of 0, from 10 -12 to less than 10 -32 s. 51 Several attempts to fit the frequency dependence of the AC susceptibility maxima included the Vogel-Fulcher law 67   The K1(0) and B parameters obtained from dynamic data were found to be in agreement with the previously discussed values obtained from the fit of HC(T) curves, and consistent to those reported for bulk CoFe2O4 (see Table III). These values should be considered as the actual effective magnetic anisotropy (Keff), since additional shape/stress contributions to the energy barrier could not be discarded. However, the close values obtained from both methods to the bulk counterpart indicate that these effects, if present, have no major influence over the overall magnetic anisotropy. Also consistent with the results from HC of the previous section, the two smallest particles AV08 and AV05 showed deviations of both K1(0) and B. Nonetheless, as our measurements of dynamic data was limited those four samples with TB < 400 K, further measurements at T < 400 K would be needed to draw conclusions for the actual behavior of these parameters.
The effective magnetic anisotropy reported for many small and ultrasmall MNPs has been found to be largely enhanced with respect to the corresponding bulk materials.
Furthermore, theoretical calculations have also led to expect an increase in Keff as the particle size decreases. [70][71][72][73] Models for this increased value have been attempted through an additional surface contribution to the total anisotropy, 74 of the form = + 6 with KV and KS being volume and surface anisotropies for a particle of diameter , although it is not clear how this approach could be applied to spherical particles, for which symmetry arguments yield a zero net contribution from the surface term. In any case, the Néel-Arrhenius or any other simple model would be expected to fail for ultrasmall particles, composed by a few number of atomic layers, and a more complete approach such as the Landau-Lifschitz-Gilbert equation should be employed. 75

V. Conclusions.
Our systematic exploration of these high-anisotropy particles having d between 5-25 nm showed a consistent magnetic behavior over a wide range of temperatures.
Interestingly, some deviations in the stoichiometry of the samples measured in macroscopic sample volumes were found to extend to the single-particle level, opening questions about the actual magnetic structure in cobalt-ferrite nanoparticles.