Time and temperature dependent magnetic viscosity experiments on Sr/Co nanoferrite particles

Magnetic viscosity experiments have been performed in order to investigate the magnetization reversal in Sr nanoferrite particles (nanoscale SrFe 12 O 19 ) and interacting Sr/Co nanoferrite particles (SrFe 12 O 19 – CoFe 2 O 4 nanocomposites). The magnetic viscosity S ¼ dM ( t )/ dln ( t ), where M ( t ) is the magnetization as a function of time, has been collected. For Sr nanoferrite S shows a maximum close to the coercive field, reflecting the relation between S and the energy barrier distribution. We evidence that magnetic viscosity experiments on Sr nanoferrite and interacting Sr/Co nanoferrite particles provide reliable qualitative results for the different magnetic field sweep rate and saturating field H sat considered. In addition, the activation volumes extracted from the magnetic viscosity experiments performed at different temperatures on Sr nanoferrite are quantitatively correlated to anisotropy changes.


INTRODUCTION
Time-dependent magnetic phenomena, such as the magnetic viscosity effect and the frequency dependence of the coercivity, are characteristic of all ferromagnetic materials.Physically, they are the results of thermal fluctuations of magnetic moments over local energy barriers, 1,2 which can arise from a variety of sources, including shape and crystalline magnetic anisotropy in fine particle systems or local defects in materials dominated by domain wall motion.In case of a system with a single energy barrier, thermal activation follows the equation where τ is the relaxation time. 3,4However, for real systems there is a distribution of physical properties (e.g., particle's sizes and magnetic anisotropy constant) resulting in a distribution of energy barriers and relaxation times.Here, we consider a well-known SrFe 12 O 19 (SFO) nanosystem, commonly exploited, for low-cost permanent magnets, whose hysteresis loop at 300 K is shown in Fig. 1(a).The system is composed of interconnected particles with a crystallite size of 118(5) nm, forming porous aggregates. 5The compounded morphology affects the thermal stability, which is a critical parameter for permanent magnets, and leads to a complex variation of magnetization with time.Assuming a constant distribution of activation energies over the investigated time range, time dependence of magnetization follows a quasilogarithmic decay where t 0 is an arbitrary initial reference measurement time (which is often taken to be 1 s), 6 S is the so-called magnetic viscosity, M(0) is a constant M(t ¼ t 0 ), and the ± sign reflects whether M(t) is increasing or decreasing with time.In an experimental run, after the application of a saturating field H sat , at a constant temperature T, reverse fields H rev of increasing value are applied in the opposite direction and M(t) is recorded for each reverse field as a function of time.The viscosity S ¼ dM(t)/dln(t), determined as the slope of the M vs ln(t) curve, is given by 2M S k B Tf (E), where M S is the saturation magnetization, k B is the Boltzmann constant, and f (E) is the absolute value of the energy barrier distribution at the critical point (i.e., at the critical energy barrier that is undergoing a thermal activation while a magnetic field is applied). 7The corresponding distribution of S(H) as a function of the reversed fields shows a maximum around the coercive field H C , as it mirrors the distribution of the energy barrier within the system [Fig.1(b)], 6 and indicates that this simplified approach well describes our system.The maximum of S, deriving from thermal activation in hysteretic materials, involves irreversible changes of magnetization that can be determined experimentally by remanence measurements, e.g., by collecting the direct current demagnetization (DCD) curves; see the inset of Fig. 1(a): 8 by applying and setting back to zero a reverse magnetic field of increasing magnitude to the sample previously saturated by applying 50 kOe in the opposite direction, the remanent magnetization M DCD is recorded and plotted as a function of the reverse field [see the gray curve of Fig. 1(a)].From dM DCD /dH, the irreversible component of the susceptibility χ irr (H) (corresponding to the switching field distribution, SFD) can be obtained.The connection between the viscosity S(H) and the irreversibility is given by the relation where H f is the fluctuation field, first introduced by Néel, to model the effect of thermal energy. 9The fluctuation field is connected to the activation volume V ACT (i.e., the smallest volume of material that reverses coherently in an event), which can be defined for a singledomain particle with uniaxial anisotropy as 6,10 where k B is the Boltzmann constant, T is the temperature, and M S is the saturation magnetization.H f is calculated using Eq. ( 3),

METHODS
Magnetization relaxation experiments were performed at various temperatures by (1) a Physical Property Measurement System (PPMS) with Vibrating Sample Magnetometer (VSM) option from Quantum Design (denoted in the following as PPMS), equipped with a superconducting magnet operating in a ±50 kOe range, and (2) a MicroSense-Model 10 VSM (denoted in the following as VSM) equipped with an electromagnet generating a maximum field of 20 kOe.First, the sample was saturated in a field (H sat = ±50 or 20 kOe), then reverse fields (opposite in sign) were applied and the magnetization decay was measured as a function of time for 60 min.The DCD curves were measured by first saturating the sample and then measuring the remanence M DCD after applying reverse fields up to H sat .

RESULTS AND DISCUSSION
In most systems, there is always a distribution of energy barriers arising from volume and morphology distributions, anisotropy dispersion, and interparticle interactions, which may lead to deviations from the logarithmic decay with time of the magnetization [Eq.( 2)], thus affecting the experimental determination of V ACT values.An example of such deviation can be observed in Figs.1(c Hysteresis loop M(H) and remanent M DCD data (which were acquired with PPMS) are not expected to vary in different setup conditions and, thus, are not discussed here; specifically, the distribution of χ irr is independent of the sweep rate since it was obtained by starting from a demagnetized state.However, for more complex composite systems, we might expect additional time-dependent effects that arise during the cycling of the applied magnetic field (thus directly connected to the magnetic field sweep rate) and affect the height of the energy barriers (i.e., the activation process).As a model system, here we decided to use a SrFe 12 O 19 -CoFe 2 O 4 composite discussed in a previous study, 5 consisting of a small fraction (10 wt.%) of Co-ferrite (CoFe 2 O 4 , CFO) of size ∼20 nm dispersed within a SFO matrix.Figure 2(a) shows the different field sweep rate adopted during a typical experiment for a specific applied reverse field, i.e., by applying a saturating field and then sweeping the external field to a reverse field, H rev = 5 kOe.The curves reported in the figure were collected by using a VSM (red curve) operating with its standard sweep rate, ∼300 Oe/s, while for the PPMS two sweep rates (∼180 Oe/s, fast, and 180 Oe/s, slow) were chosen accordingly.To illustrate the effects on the measured magnetic relaxation, the corresponding viscosity extracted from the different experiments is shown in Fig. 2(b).The position of the maximum viscosity is not significantly affected by changes in H sat and H rev sweep rate.Furthermore, for the PPMS series, with both positive and negative saturating H = ±50 kOe, the absolute value of the viscosity is not affected by the investigated reverse field range.This is, however, not the case for VSM and PPMS experiments conducted with the lower saturating field H sat = ±20 kOe.While the PPMS results match the previous ones, the viscosity data extracted from VSM differ more significantly.The reason was established to be the fitting range of M(ln(t)), whose linear behavior is affected by the much faster sweep rate of the VSM.By comparing the main panels of Figs.2(c) and 2(d) with their respective insets, it results that the linear relaxation is achieved earlier than that for the PPMS case and, therefore, the fitting could be extended to a wider range [from Δln t t0 ¼ 3 to 8, corresponding to a range of ∼50 min].The initial non-linear behavior of the magnetization may be ascribed to complex effects of magnetic coupling/decoupling, which interfere with the sweeping field as also observed for SFO. 12 In addition, it should be considered that time dependence effects during very fast sweep rate of the reverse field (i.e., ∼300 and 180 Oe/s for VSM and PPMS, respectively) become more negligible.4][15] The main difference is eventually the time to reach the applied reverse field (especially when higher reverse fields are applied).In principle, such deviation may be used to estimate the degree of coupling in granular materials as in fine non-interacting systems the linear trend is preserved along the full range. 16,17It should be pointed out that the good agreement (i.e., the same switching field and low discrepancy in viscosity values), considering the intrinsic inhomogeneity of such granular materials, provides a strong case to be explored as a model system for more complex composites (for instance, larger Co-ferrite fractions) and will be investigated in the future.
9][20][21][22][23] A thorough investigation on these classes of materials is necessary to better describe the reversal process of magnetization, thus achieving technological improvements.As shown for the Sr nanoferrite in Fig. 3, the hysteretic behavior investigated over a wide temperature range (10-300 K) reveals that as the temperature increases, the M S decreases (from ∼100 to ∼70 emu/g) while the coercive field increases (from ∼ 4.9 to 5.8 kOe).These changes can be correlated to a variation of energy barrier distribution, around the coercivity values as shown in Fig. 3(b).The corresponding viscosity, and thus fluctuation field, will change accordingly since it follows the switching fields, which is temperature dependent.This demonstrates that magnetic viscosity is intrinsically linked to irreversible magnetization changes induced by temperature, as shown in Fig. 4 (the slightly lower values of coercivities with respect to the switching field are due to the effects of thermal activation).The increasing temperature leads to a larger activation volume, which increases from ∼4 × 10 3 to ∼12 × 10 3 nm 3 at 300 K (Fig. 5), showing that the reversal process of magnetization is assisted by the temperature, since the effective energy barrier is lowered.Furthermore, we note that S max departs from S(H SW ) as T decreases, confirming the active role played by temperature in the activation process.The same growing trend for viscosity (and fluctuation field as well) was observed for similar ferrite systems. 24The fitted distributions in Fig. 3 do not show significant changes in shape and width, thus concluding that the reversal operates in the same way. 25We report a temperaturedependent comparison between Sr-ferrites obtained with different synthetic techniques, to underline the good agreement of calculated activation volume. 11,26,27We note that V ACT is smaller than the physical volume corresponding to a crystallites' size of 118(5) nm, as previously reported for granular materials, which could be ascribed to weak intergranular coupling of demagnetizing nature arising from the microstructure, [28][29][30] and in agreement with the large width of the SFD (i.e., wide distribution of energy barriers).To establish a quantitative relation between the coercive field and the viscosity, the model proposed by Givord and co-workers 31 was adopted (the inset of Fig. 5): the experimental coercivity (not corrected by demagnetizing factors that would affect the critical volume) 32 was found to be proportional to V À2/3 ACT according to the model.This good agreement confirms that the activation volume follows the coercive field and, thus, the magnetic anisotropy in first approximation. 33,34It reveals the formation of a critical volume (i.e., activation volume) within a structural defect upon application of a reverse magnetic field, that is at the origin of the nucleation of domain walls and thus found to be proportional to the domain wall length (δ) as δ 3 .Slight variations owing to possible incoherent modes 35 are ascribed to the size distribution of our particles that were anyway found to be below the single-domain size, but could participate within one interaction domain, thus resulting in a complex magnetic configuration. 11,36

CONCLUSIONS
To conclude, the results indicated that (i) the activation volume model constitutes a robust way to characterize switching mechanisms of single-phase hard magnetic materials; (ii) timedependent effects on the magnetic relaxation due to different sweep rates of the cycling magnetic field during the experiments could be considered negligible in a restricted time window; (iii) the temperature dependence of magnetic viscosity of nanostructured Sr-ferrite follows quite well the expected behavior for nucleation-like reversals that are assisted by a decrease in effective energy barrier when temperature increases (and, thus, increase in activation volume).Reversal processes are more complex in nanocomposites of interacting Sr/Co-ferrites and results suggest that while qualitative results associated with the overall S(H) dependence can be obtained, further data are necessary in order to obtain quantitative information in those systems and, for example, distinguish the contributions to the reversal of the different nanophases.

FIG. 1 .
FIG. 1.(a) Magnetization vs magnetic field M(H) hysteresis loop and remanent DCD magnetization vs reverse magnetic field M DCD (H) curve for SFO at 300 K; in the inset, corresponding switching field distributions (SFDs) representing the irreversible component of susceptibility, χ irr (H), normalized by its maximum (∼0.0227 emu/gOe).(b) Distribution of magnetic viscosity vs reverse field S(H) extracted from the linear time-dependent magnetic relaxation M(t) (see the main text for details).(c) and (d) show a typical linear fitting of M(lnt) for a reverse field: ±6 kOe, respectively, obtained from (c) VSM and (d) PPMS magnetometers; the corresponding extracted viscosities (slopes) are reported next to the fits.

FIG. 2 .
FIG. 2. (a) Illustration of the time delays during a typical relaxation experiment with different sweeping field rates, for VSM (the red line is an extrapolation from −20 kOe to illustrate how fast a sweep from −50 kOe to 0 would be (rate of ∼300 Oe/s) and PPMS (two rates considered: 18 and 180 Oe/s).(b) Magnetic viscosity S(H) for a selected SFO-CFO nanocomposite acquired in different experimental conditions: (i) saturating fields H sat = ±50 or 20 kOe, (ii) sweep rates.(c) and (d) show two examples of fittings of M(ln(t)) for PPMS at two different reverse fields (−2 and −4.5 kOe) after saturating in H = +20 kOe and the corresponding calculated slope (i.e., viscosity); the results of the same experiments performed on the VSM are shown in the respective insets.
) and 1(d) for SFO, which show M as a function of ln(t), together with linear fittings of M(ln(t)).Possible effects caused by different magnetic field sweep rates were ruled out by performing the same set of measurements on the same samples on two different magnetometers.The magnetic viscosity values of the SFO nanoferrite were found to be in good agreement with each other [see Fig.1(b)for the extracted S(H) in both experiments].11The comparison of the measured M(ln(t)) data extracted at the same reverse field for both experiments (±6 kOe) is shown in Figs.1(c) and 1(d).

FIG. 3 .
FIG. 3. (a) Temperature-dependent M(H) hysteresis loop; (b) normalized SFDs obtained from differentiating M DCD (H) curves recorded at the same temperatures.Dashed lines correspond to the obtained switching field H SW .

FIG. 4 .
FIG. 4. Magnetic viscosity estimated from the fitting M(ln(t)) for each reverse field around the switching region of SFO, and fitted to a Gaussian-like model, for three different temperatures; dashed lines correspond to H SW .The bottom panel shows the normalized S/S MAX curves.