Innovative study of superparamagnetism in 3 nm CoFe₂O₄ particles

Abstract

Cobalt ferrite particles of 3 nm were prepared through the solvothermal method. The obtained material is superparamagnetic in such a temperature range (161 K-room temperature) allowing a thorough investigation of the superparamagnetic phenomenon in this kind of magnetic system. The dependence of the superparamagnetic moment on temperature and magnetic field has been determined for such composition and particle size, obtaining values up to 5 × 10⁵ μ_B. In addition, three different analyses of the magnetic data reveal that at a temperature of 168 K the superparamagnetic behavior exhibits its highest intensity.

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Introduction

Superparamagnetism is one of the most illustrative and studied finite size effects in materials properties.^1,2 For a certain temperature, this behavior is exhibited by single domain magnetic nanoparticles when they are prepared below a critical size.³ For such conditions of temperature and particle size, the magnetostatic energy is not high enough for the domain-wall formation and particles present only one magnetic domain.³ Superparamagnetism occurs in the temperature range between the blocking and the ordering temperature (T_B and T_O, respectively) and is responsible for the S-shaped M(H) curves with high magnetization values and low saturation fields revealing that the magnetic moments easily rotate and align with an external magnetic field.⁴ Usually, the superparamagnetic moment (μ_SP) of these materials is the result of the contribution of several particles' moments and recently, this parameter has been demonstrated to be fully dependent on the local environment and temperature, as these factors drastically affect the particle interactions.^5,6 Thus, higher values of superparamagnetic moment are found for temperature values close to the T_B and it gradually decreases with the temperature increasing, up to the paramagnetic regime.⁵ It has also been found, that particles interactions are avoided when the particles are contained in porous structures or amorphous silica, illustrating then lower magnetic values.^6,7 Ferrite compounds are one of the best systems to study the superparamagnetic phenomenon since its preparation in such particle size range is facile^8,9 and it constitutes one of the easiest affordable magnetic materials. CoFe₂O₄ compound crystallizes with spinel type structure (S.G. Fd3̄m) in which half of the iron atoms are distributed along the tetrahedral sites and the rest of the cations are alternatively located in octahedral sites.⁸

On the other hand, one of the greatest challenges faced by nanoscience, deals with the preparation of monodispersed materials in the nanometric scale.¹⁰ For such purpose those synthesis routes comprised in the so-called soft chemistry, seem to be appropriate. The typical mild conditions used in these synthesis methods allow obtaining particles with nanometric size as the sintering process occurring at high temperature values is avoided. In this sense, solvothermal method is one of the most employed due to its simplicity and short synthesis times.^11,12

We present in this work a study of the superparamagnetic behavior of monodisperse 3 nm cobalt ferrite particles which have been easily prepared with solvothermal conditions. In addition, the temperature for which the material presents the most intense superparamagnetic behavior for a certain applied magnetic field, has been determined for the first time by three different ways. The superparamagnetic moment dependence with the temperature has been analyzed and it has been included an innovative study concerning the dependence of the superparamagnetism with the applied magnetic field.

Experiment

Cobalt ferrite nanoparticles were synthesized by means of the solvothermal method. Stoichiometric amounts of the nitrate salts were dissolved in ethyleneglycol to form a 0.01 M solution. KOH 2.0 M was then added dropwise and under continuous stirring, until reaching pH 11. The brown mixture was then transferred into a Teflon stainless steel autoclave to be treated at 160 °C for 3 hours. Finally the obtained solid was filtered and washed with ethanol before recovering it.

The ferrite crystalline phase was identified by X-ray diffraction using a Siemens D-5000 powder diffractometer (25 mW, 35 kV) with a CuKα radiation. The data was collected in the 2θ range between 20 and 70° in steps of 0.02°. Microstructural characterization was performed by means of electron transmission microscopy (TEM) using a JEOL-2000FX microscope working at 200 kV. Magnetization measurements were carried out in a quantum Design XL-SQUID magnetometer in the 2–300 K temperature range. Magnetic susceptibility with temperature was measured after cooling the sample at 2 K in zero-field (ZFC) and after applying a 500 Oe magnetic field. In the case of field-cooling (FC) measurements, the sample was cooled in the presence of a 500 Oe field down to 2 K. Measurements of the AC magnetic susceptibility χ_ac = χ′ + iχ′′ were performed under zero dc magnetic field, an excitation field of 1 Oe, and driving frequencies varying from 1 Hz to 1 kHz in the temperature range of 2–300 K.

Results and discussion

TEM image and particle size distribution represented in Fig. 1a reveal an almost monodisperse ferrite sample with an average particle size of 3.0 ± 0.4 nm. The observed X-ray diffraction maxima in the corresponding pattern (Fig. 1b) have been indexed to the spinel type structure (S.G. Fd3̄m) as it is indicated in Fig. 1b, and the so-broad profile of the diffraction peaks is indicative of the nanometric particle size of the sample.^8,13

Fig. 1 Representative TEM image (a) and particle size distribution (inset) corresponding to the prepared CoFe₂O₄ nanoparticles. X-ray diffraction pattern of the sample is also shown (b).

The magnetic behavior of the sample has been fully characterized by means of the different magnetic parameters (collected in the Table 1) that have been obtained after analysis of the measurements. The ZFC and FC susceptibility curves corresponding to the sample measured for an external applied field of 500 Oe are represented in Fig. 2a. A blocking temperature of 161 K can be inferred from the maximum observed in the ZFC curve. Also, broad maxima can be observed in both curves which can be ascribed to two different phenomena, (1) broad particle size distribution and (2) interparticle interactions.¹⁴ Taking into account the information deduced from Fig. 1 that reveals a narrow particle size distribution, the interparticle interactions may be the main factor causing the broad maximum of the ZFC curve. The inverse of susceptibility from the blocking temperature up to 300 K, is shown in the inset of Fig. 2a. The observed no-lineal tendency reveals a superparamagnetic behavior instead of the paramagnetic one which is characterized by a lineal dependence of 1/χ with the temperature.^5,8

Table 1 Magnetic parameters corresponding to 3 nm particles of CoFe₂O₄ obtained by the solvothermal method

TB,500^a (K)	TO^a (K)	HC,5^a (Oe)	MS		K (erg cm^−3)	HK^a (Oe)	HS,250^a (Oe)
			emu g^−1	μB
a TB,500, blocking temperature for an applied magnetic field of 500 Oe; TO, ordering temperature; HC,5, coercive field for a measuring temperature of 5 K; HK, anisotropy field; HS,250, saturation field for a measuring temperature of 250 K.
161	>300	9008	67.4	2.83	6.1 × 10^5	3433	1100

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Fig. 2 Magnetic susceptibility (ZFC and FC) measured at 500 Oe corresponding to the sample (a) together with the inverse of susceptibility (inset) in the T_B-RT temperature range. It is shown the distribution corresponding to the anisotropy barriers (b).

As it is already known, the anisotropy barrier and the blocking temperature are correlated through the expression T_B ∝ (KV/25κ_B), therefore, the anisotropy barriers distribution which can be determined from the derivative of the difference between magnetization of both ZFC and FC susceptibility curves (see Fig. 2b), results a useful tool to evaluate the anisotropy barriers distribution.^15,16 As it can be observed in the graph, at 168 K the distribution curve is close to cero, revealing that the thermal energy for such temperature has overcome all the anisotropy barriers.

Recently, the dependence of the superparamagnetic moment with the temperature, has been described as follows:⁵

where κ_B is the Boltzman constant and M_S is the saturation of magnetization (emu g⁻¹).

The superparamagnetic moment as function of the temperature and for an applied magnetic field of 500 Oe, has been calculated from such equation and represented in Fig. 3a. For such calculation it has been employed a M_S value of 67.4 emu g⁻¹ which has been determined through the approach to saturation law as it is discussed below. The evolution of μ_SP in all the studied temperature range is represented in the inset of Fig. 3a. The superparamagnetic moment takes a value close to cero for low temperature values (dashed line) and it drastically increases once the temperature reaches a value close to the T_B, revealing that the major part of the particles moments forming the sample, contribute to the superparamagnetic moment. The highest value of μ_SP close to 5 × 10⁵ μ_B (see inset of Fig. 3a) corresponds to a temperature value of 168 K. At this temperature, the thermal energy has overcome all the anisotropy barriers as it is observed in Fig. 2b, and therefore, all the particle moments contribute to the μ_SP magnitude giving rise to the highest observed value (Fig. 3b). Above the blocking temperature the thermal energy is large enough to gradually destroy the interparticles interactions leading to a decreasing of the superparamagnetic moment.

Fig. 3 Superparamagnetic moment as function of the temperature in the T_B-RT temperature range (a). This parameter is also represented for all the studied temperature range (inset). Saturation field as function of the temperature is represented in (b) and the derivative is depicted in the inset.

On the other hand, when the potential magnetic energy μH (where μ is the magnetic moment and H is an external magnetic field) is similar to the thermal one κ_BT, the magnetic saturation is reached, and the magnetic field (H_HT) corresponding for such value (μ_HT) can be calculated as follows:^8,17

The temperature ranges for which the material behaves as superparamagnetic and paramagnetic, respectively, can be evaluated then, by means of representing the H_HT parameter vs. the temperature.⁵ Both parameters μ_SP and H_HT can be correlated in Fig. 3. Fig. 3b shows the temperature dependence of the H_HT parameter that has been represented in the T_B-300 K temperature range together with its derivative (shown in the inset). The graph indicates an increasing H_HT with temperature, that is, for reaching the magnetic saturation, higher magnetic field are needed in accordance with eqn (2). Taking into account recent results,⁵ the plateau shown by the derivative (inset of Fig. 3b), reveals a similar H_HT-temperature dependence in all the studied temperature range, revealing that 3 nm particles of CoFe₂O₄ behave as superparamagnetic from the T_B to 300 K, since a dependence with different tendency would be expected for the paramagnetic regime.⁵

M(H) curves corresponding to 3 nm particles of CoFe₂O₄ composition, are depicted in Fig. 4 for measuring temperatures of 5 and 250 K. As it can be observed, at 5 K the sample presents a coercive field of 9008 Oe and the magnetization to saturation (M_S), and the anisotropy constant (K) have been obtained after fitting the data in the high magnetic fields region to the approach to saturation law:^6,18

where a is a constant and c corresponds to the magnetic susceptibility in the high applied magnetic field region. The obtained values for these parameters are collected in the Table 1 and agree well for such composition and particle size.^8,19 The M(H) curve corresponding to a measuring temperature of 250 K, shows the typical S-shape as evidence of the superparamagnetic behavior, that is, absence of coercivity due to a free rotation of the magnetic moment as the thermal energy overcomes the anisotropy barrier and an almost reached saturation that makes a difference also between the superparamagnetism and paramagnetism regimes. The lower values of magnetization in comparison with those observed at 5 K (see Fig. 4), are consequence of the thermal energy action that opposes the magnetic energy which tends to align all the particle moments. As it has been discussed before, 250 K belongs to the temperature range for which 3 nm CoFe₂O₄ particles behave as superparamagnetic, therefore, it can be calculated the evolution of the superparamagnetic moment with the applied magnetic field, μ_SP(H), for such temperature. The following equation can be employed:for which it has been used the M_S value of 67.4 emu g⁻¹ (see Table 1), the M and H values have been extracted from the M(H) curve (Fig. 4a) and T is the measuring temperature of 250 K. The obtained dependence of the superparamagnetic moment with the applied magnetic field has been depicted in Fig. 5. It should be noted that for such representation the absolute values of μ_SP have been placed in the corresponding M–H quartier, positive or negative, on the basis of what is observed in the primitive M(H) curve. Also, it should be noted that due to arithmetical reasons, the μ_SP(H) plot draws a different profile that M(H) curve does. As it can be observed in Fig. 5 for the highest applied magnetic fields, the value corresponding to the superparamagnetic moment is close to 3 × 10⁴ μ_B. From this curve a μ_SP value of 2.08 × 10³ μ_B can be inferred for an applied magnetic field of 500 Oe. This value is quite lower than that of 3.57 × 10³ μ_B corresponding to 250 K, deduced from the graph depicted in Fig. 3a which has been obtained for an applied magnetic field of 500 Oe. This difference in μ_B between the μ_SP deduced from the different μ_SP(T) and μ_SP(H) graphs and for 500 Oe and 250 K conditions, may be due to a different initial magnetic state of the sample. Thus, in the case of the susceptibility measurement (from which it was extracted the μ_SP(T) graph) the sample reaches the magnetic saturation (at T_B < 250 K) and afterwards, the thermal energy gradually destroys the interparticle interactions which are the cause of the magnetic saturation. In the case of the M(H) measurement, the initial state of the sample does not correspond, however, to the magnetic saturation. While in the case of the χ(T) measurement, the evolution with temperature implies a destruction of the saturation, in the second one, M(H), the tendency is to reach the magnetic saturation.

Fig. 4 M(H) curves measured at 5 and 250 K. Inset shows a zoom of the curve measured at 250 K.

Fig. 5 Superparamagnetic moment (μ_SP) dependence with the applied magnetic field for a measuring temperature of 250 K. A zoom of the graph is shown in the inset.

AC susceptibility measurements have been registered in order to investigate the dynamics of the system. The temperature dependence of the in-phase component (χ′) registered with ZFC protocol, is shown in Fig. 6 for measuring frequencies between 1 and 1000 Hz. It can be observed a peak with a corresponding temperature (T_max) that increases as the frequency increases. The Néel model indicates that the magnetic moment of a single domain particle in the absence of an external magnetic field, can fluctuate as consequence of the thermal energy action (κ_BT) with a characteristic relaxing time following an Arrhenius law:^12,20

where τ is the fluctuation time of the magnetic moment, τ₀ is inversely proportional to the frequency of fluctuation of the atomic spin between the opposite directions of the easy axis and takes values of 10⁻¹¹ < τ < 10⁻⁹ s.^21,22 When the temperature is decreased the system slows down (↑τ) and it looks like static, in the so-called blocked state, when the fluctuation time (τ) is higher than the measuring time. The data extracted from the graph of Fig. 6 have been fitted to the Arrhenius law ln(1/ω) vs. 1/T′_max where 1/ω = τ as it is shown in Fig. 7a, an unphysical value of τ₀ = 3.7 × 10⁻³⁷ s is obtained revealing the presence of interparticle interactions or a spin-glass behavior. The FC susceptibility curve shown in Fig. 2a does not illustrate a flattening or dip when decreasing the temperature below the T_B, what leads to believe that interparticle interactions are at the origin of this unphysical τ₀ value. In addition, it has been calculated the relative variation of the T_max peak (ΔT_max/T_max) per frequency decade defined as parameter ϕ
the obtained ϕ = 0.03 value supports the idea of the presence of intermediate interactions^23,24 as small shifts with increasing frequency is due to increased interparticle interactions.²⁵ Thus, the Vogel–Fulcher model can be considered to describe such magnetic system:²⁶
where T_in is the temperature above which the interparticle interactions start to be less intense. When fitting the data to the Vogel–Fulcher law T_max ln(τ/τ₀) = E_a/κ_B + T_in ln(τ/τ₀) (see Fig. 7b) and assuming a pre-exponential factor τ₀ value of 10⁻⁹ s,²³ a T_in value of 168 ± 2 K is found. This value of temperature for which interparticle interactions start to be not important when increasing temperature, is in accordance with that value previously deduced from graphs shown in Fig. 2b and 3a. At 168 K, thermal energy action overcomes all the anisotropy barriers (Fig. 2b) and it is able to align as much as possible particles moments along the magnetic field direction. As a result, the maximum magnetization of the samples is obtained leading to the highest μ_SP observed (Fig. 3a). This temperature value of 168 K is confirmed as well by AC susceptibility measurements.

Fig. 6 Temperature dependence of the in-phase component of the AC magnetic susceptibility (χ′) measured in the frequency range of 1–1000 Hz.

Fig. 7 Arrhenius law fit (a) and Vogel–Fulcher fit (b) corresponding to 3 nm particles of CoFe₂O₄.

Conclusions

CoFe₂O₄ particles with an average size of 3 nm, easily prepared through the solvothermal method, behave as superparamagnetic from 161 K as it has been deduced from the susceptibility curves measured at 500 Oe. Thanks to the representation of H_HT vs. temperature, it was possible to confirm the superparamagnetic behavior of the sample in the T_B-300 K temperature range. The superparamagnetic moment seems to be maximum at 168 K. Thermal energy overcomes all the anisotropy barriers of the nanoparticles for an external magnetic field of 500 Oe and this temperature. Also, AC susceptibility measurements reveal a maximum of interparticle interactions for this temperature value. In addition, it has been studied the increasing dependence of the superparamagnetic moment with an increasing applied magnetic field. Thus, superparamagnetism phenomenon seems to be not only dependent on the composition and particle size, but on the temperature and applied magnetic field, as well.

Acknowledgements

The authors are thankful to Ministerio de Ciencia e Innovación for financial support under project MAT2013-44964-R.

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