Critical behavior and magnetocaloric study in La_0.6Sr_0.4CoO₃ cobaltite prepared by a sol–gel process

Abstract

The structure, magnetic properties, critical exponents and magnetocaloric effect (MCE) of La_0.6Sr_0.4CoO₃ cobaltite are studied. A sol–gel method is used in the preparation phase. Phase purity, structure, size, and crystallinity are investigated using XRD and SEM. The temperature dependent magnetization exhibits a sharp paramagnetic–ferromagnetic transition (FM–PM) at the Curie temperature T_C = 230 K. By means of several methods such as modified Arrott plots, the Kouvel–Fisher method, and Widom scaling relation, the values of β (corresponding to the spontaneous magnetisation), γ (corresponding to the initial susceptibility), δ (corresponding to the critical magnetisation isotherm) and T_C are estimated for La_0.6Sr_0.4CoO₃ around the FM–PM phase transition. Compared to standard models, the critical exponents obtained in our work are located near to those expected for the 3D Heisenberg model. Such results demonstrate the existence of ferromagnetic short-range order in La_0.6Sr_0.4CoO₃. With the obtained values, one can scale the magnetic-field dependences of magnetization below and above T_C following a single equation of state. In the vicinity of T_C, the magnetic entropy change ΔS reached maximum values of 1.1 and 2.1 J kg⁻¹ K⁻¹ under magnetic field variations of 2 and 5 T, respectively. The field dependence of the magnetic entropy changes are analyzed, which show power law dependence ΔS_max ≈ a(μ₀H)ⁿ at the transition temperature. Moreover, the temperature dependence of the exponent n for a different magnetic field is also studied. The values of n obey the Curie Weiss law above the transition temperature. It is shown that for La_0.6Sr_0.4CoO₃ cobaltite, the magnetic entropy change and n(T) follow a master curve behavior.

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1. Introduction

Research on the nature of the phase transitions in perovskite-type cobaltites has been and still remains one of the actual directions in condensed state physics.¹ Cobaltites with the chemical formula R_1−xA_xCoO₃ (R and A are rare-earth and divalent-alkaline elements, respectively) are still of intense interest because they exhibit very rich and complex magnetic phase diagrams due to the occurrence of thermal, compositional, and pressure induced spin-state changes.¹ They are categorized as perovskites with unusual electromagnetic properties such as the large negative magnetoresistance (MR),^2–4 spin- (or cluster-) glass magnetism,^5,6 spin-state transition,^7,8 and insulator-metal (IM) transition induced by temperature, hydrostatic pressure or doping level.⁹ The Hund coupling and the crystal-field splitting between t_2g and e_g states lead to the temperature dependence of the spin-state transitions between the low-, intermediate- and high-spin states in the Co ions.¹⁰ The nonhomogeneity of the spin state of these ions governs the magnetism and the electrical conduction and might affect the cooperative behavior of the Co sublattice, and thereby, the nature of ferromagnetic (FM) to paramagnetic (PM) phases transition. Thus, from the phase transition point of view one may ask whether a true long-range magnetic ordering exists in this system and, if yes, what is the universality class of this magnet? Experimental observations indicate that the replacement of La³⁺ by a Sr²⁺ introduces more holes to the valence band, and thus changes the Co valence from Co³⁺ to Co⁴⁺. The choice of La–Sr-based cobaltite as magnetic potential materials is due to their variety of phenomena like ferromagnetism and magnetocaloric effect, jointly with multiple possible magnetic states, such as paramagnetic insulator, ferromagnetic insulator and ferromagnetic metallic.^11,12 Earlier studies suggest that La–Sr-based cobaltite leads to a gradual increase of the magnetic moment and orders such that for of strontium exceeding 20% the system becomes ferromagnetic.⁵ Ferromagnetic cobaltites are intrinsically inhomogeneous both above and below PM–FM transition. However, the existence of the mixed-valence in the system with complex spin states affects the observed properties. Particularly, it is unclear how the magnetic interactions and behavior are renormalized near the PM–FM transition range and what universality class governs the PM–FM transitions in these materials.¹³ Furthermore, the interpretations of a few relevant experimental results pertaining to the estimated exponents and the universality class of different inhomogeneous perovskite systems are still controversial.^1,14 Therefore, to understand better the FM–PM transition nature, it is important to investigate in detail the critical exponents associated with the transition. This investigation can provide us the order, the universality class, and the effective dimensionality of the phase transition around the Curie temperature T_C. Experimental observations indicated that the critical behaviors is an effective way playing important roles in revealing interaction mechanisms and properties near T_C.¹⁵ The study of critical exponents around T_C would introduce the exponents (β, γ, and δ) far from those obtained by conventional theoretical models of the mean-field theory, 3D Ising model, and 3D Heisenberg model.^16–18 Currently, the nature of magnetic phase transitions was the subject of several investigations because of the demand for environmentally aware science. In fact, study into materials showing large magnetocaloric effects (MCE) close to room temperature is one of the areas at this time being explored to develop new “green refrigerators”. The knowledge of field dependence of magnetic entropy change ΔS of a magnetic refrigerant compound is important.^19–21 The field's dependence of the magnetic entropy change curve helps us to predict the response of a particular material under different extreme conditions, which can be useful for designing new materials for magnetic refrigeration. Therefore, a study of the MCE for a particular material is not only important from a practical application point of view but also it provides a tool to understand the properties of the material. In particular, the details of the magnetic phase transition and critical behavior can be obtained by studying the MCE of the material.^22,23

Since, ferromagnetic behavior has been considered for the La_1−xSr_xCoO₃ with the range 0.30 ≤ x ≤ 0.50.⁵ However, the critical exponents study in the upper limit x = 0.5 has shown that the exponent γ value is close to 3D Ising value whereas the exponent δ approaches to mean-field one.²⁴ The critical behavior of in polycrystalline cobaltite La_0.7Sr_0.3CoO₃ (lower limit x = 0.3) has been investigated and the exponents are found to approach to 3D-Ising values.²⁵ This implies that the La_1−xSr_xCoO₃ with 0.30 ≤ x ≤ 0.50 systems does not belong to a single universality class. Motivate by this, we considered it important to study the critical behavior in the intermediate value x = 0.4. The intention is to complete the image of the magnetic behavior of La_0.6Sr_0.4CoO₃ with the magnetocaloric properties, in the view of some potential technical applications. We have found quite large values of magnetic entropy change in this compound making this interesting for magnetic refrigeration over a wide temperature range. To get more insight into this compound system, we prepared La_0.6Sr_0.4CoO₃ and then studied the critical phenomena around its ferromagnetic–paramagnetic phase transition temperature. We present an estimate of the critical exponents for this system near Curie temperature by analyzing magnetization isotherms through different methods such as modified Arrott plot and Kouvel–Fisher and show that the critical exponents are short range 3D-Heisenberg model like in the present compound. We used the scaling hypotheses of the thermodynamic potentials to scale the magnetic entropy change to a single universal curve for La_0.6Sr_0.4CoO₃.

2. Experimental details

The La_0.6Sr_0.4CoO₃ sample was synthesized using the sol–gel method. Stoichiometric amounts of La₂O₃ (dried at 1000 °C), SrCO₃ and Co₃O₄ were dissolved in nitric acid to obtain a clear pink solution. Citric acid (CA) and ethylene glycol (EG) were added to the obtained solution with the molar ratio total metals: CA : EG = 1.5 : 5. The resulting mixture was heated at 80 °C under constant stirring to accelerate the polyesterification reaction between CA and EG. Then, this solution was heated at 130 °C and the excess of nitric acid and water were boiled off giving a homogeneous red resin, which was then dried at 150 °C overnight to evaporate a nitrogen element and obtain a black fine powder. This powder was crushed and heated in air at 300 °C before sintering at various temperatures (400–800 °C) for 6 h. After cooling, the sample was pressed into a pellets (of about 1 mm thickness and 13 mm diameter), and then sintered at 900 and 1000 °C for 12 h in air with intermediate regrinding and repelling. Phase purity, homogeneity and cell dimensions were determined by powder X-ray diffraction (with a diffractometer using Cu Kα radiation) at room temperature. Structural analysis was carried out using the standard Rietveld technique.^26,27 The microstructure of the prepared samples was observed in a scanning electron microscope that operated at an acceleration voltage of 10 kV. Magnetization measurements versus temperature in the range 20–300 K and versus magnetic applied field up to 5 T were carried out using a vibrating sample magnetometer. For critical exponent investigation, the isothermal magnetization measured in the range of 0–5 T and with a temperature interval of 2 K in the vicinity of Curie temperature (T_C). These isothermals are corrected by a demagnetization factor D that has been determined by a standard procedure from low-field dc magnetization measurement at low temperatures (H = H_app − DM). MCE were deduced from the magnetization measurements versus magnetic applied field up to 5 T at several temperatures.

3. Scaling analysis

As it is well known, critical phenomena of a magnetic system is not defined around his first-order transition as the magnetic field can shift the transition, leading to a field-dependent phase boundary T_C (H). According to the scaling hypothesis, the second order magnetic transition near T_C is characterized by a critical region defined by the interrelated exponents β, γ and δ which are associated with the spontaneous magnetization M_SP below T_C, the magnetic susceptibility χ⁻¹₀ above T_C and the field dependence of the magnetization at the Curie temperature, respectively. Those exponents are given by the following relations:^28,29

M = D(μ₀H)^1/δ, ε = 0 T = T_C (3)

where M₀, h₀ and D are the critical amplitudes β, γ and δ are the critical parameters, and ε = (T − T_C)/T_C is the reduced temperature. The magnetic equation of state M(μ₀H, ε) in the critical region obeys a scaling relation, which can be expressed according to the scaling hypothesis as:

M(μ₀H, ε) = ε^βf_±(μ₀H/ε^β+γ) (4)

where f_± are regular functions with f₋ for T < T_C and f₊ for T > T_C. Eqn (4) indicates that for true scaling relations and right choice of β, γ and δ values, M(μ₀H, ε)ε^−β versus μ₀Hε^−(β+γ) forms two universal curves of temperature T > T_C and T < T_C.

4. Results and discussion

The XRD pattern of La_0.6Sr_0.4CoO₃ sample registered at 300 K including the observed and calculated profiles as well as the difference profile is depicted in Fig. 1. The structural refinement was performed using the FullProf program²⁶ and.²⁷ The compound is single phase without any detectable impurity and crystallizes in the rhombohedric system with the R3̄c space group (Z = 6) in which La/Sr atoms are located at 6a (0, 0, 1/4) position, Co at 6b (0, 0, 0), and O at 18e (x, 0, 1/4) position. The quality of the refinement is evaluated through the goodness of the fit indicator χ². The cell parameters deduced from the refinement were found to be a = b = 5.434(7) Å, c = 13.217 (5) Å and the unit cell volume V = 238 (4) ų. The average crystallite size D_XRD was calculated from the XRD peaks using the Debye-Scherrer formula: where λ is the X-ray wavelength employed, θ is the diffraction angle for the most intense peak (1 0 4), and β is defined as . Here, β_m is the experimental full width at half maximum (FWHM) and β_s is the FWHM of a standard silicon sample. The obtained D_XRD is estimated to be 25.71 nm. The morphology and the average grain size D_SEM determined by SEM micrographs (inset of Fig. 1) shows that the sample is constituted of particles with much porosity and D_SEM is about ∼783 nm. Obviously, the particle sizes observed by SEM are larger than those calculated by XRD, which indicates that each particle observed by SEM consists of several crystallized grains. Similar results were also observed in another perovskite family.¹⁹

Fig. 1 (a) The powder X-ray diffraction pattern at room temperature and the result of the Rietveld analysis for La_0.6Sr_0.4CoO₃ compound. (b) represents a SEM micrograph for the sample.

We plot in Fig. 2(a) the temperature dependence of the magnetization M (T) at low magnetic field (μ₀H = 0.05 T) for the La–Sr based cobaltite. The measurements were carried out after field cooling (FC) process. The sample exhibits a clear transition from paramagnetic to ferromagnetic state with decreasing temperature. The magnetic transition temperature T_C (∼230 K) is defined as the inflection point of dM/dT (see in the inset of Fig. 2(a)). From the slope of the linear 1/M vs. T curve, the effective paramagnetic moment (μ^exp_eff) can be determined and the temperature at which 1/M intercepts the temperature axis is the Curie Weiss temperature θ_p. To get a clear knowledge about the magnetic interaction for the La_0.6Sr_0.4CoO₃, the magnetic susceptibility (χ) could be fitted to the Curie–Weiss law:

χ = C/(T − θ_P) (5)

where C is the Curie constant and θ_P is the Weiss temperature. Inverse susceptibility 1/χ(T) ≈ 1/M(T) deduced from FC–M(T) is plotted in Fig. 2(a). A linear fit to paramagnetic region (high temperature) yields positive Curie–Weiss temperature θ_P = 231.12 K. The positive sign of θ_p value implies the ferromagnetic nature of the magnetic interactions between Co ions in this material. To additional understand the nature of magnetic properties and the kind of the phase-transition in La_0.6Sr_0.4CoO₃ cobaltite, we studied their magnetic hysteresis loops at T = 10K from −5 to 5 T applied field. Fig. 2(b) shows the hysteresis loops for our material and reveals that La_0.6Sr_0.4CoO₃ exhibiting a typical of soft FM behavior as the saturation magnetization (M_s), the remanent magnetization (M_r) and the coercitive field (μ₀H_C) are 32.44 emu g⁻¹, 22.33 emu g⁻¹ and 0.238 T respectively. The obtained value for coercive field confirms that the magnetic domains can rotate easily to the direction of the applied magnetic field.³⁰ The isothermal magnetizations M(μ₀H) curves for the La_0.6Sr_0.4CoO₃ sample in the temperature range 200–260 K are shown in the inset of Fig. 2(b). In the paramagnetic state, the curves M(μ₀H) begin to be linear only at elevated temperatures, well above the T_C. Below T_C, the M(H) increases rapidly at low fields, signature of a ferromagnetic behavior but does not saturate even in a field of 5 T. Such results confirm the presence of ferromagnetic clusters as is known in cobaltites.³¹ In order to discard the possibility of a first order phase transition we have used the criterion given by Banerjee.³² The criterion consists in inspecting the slope of isotherm plots of M² versus μ₀H/M curves, known as the standard Arrott plot assumes the critical exponents following mean-field theory β = 0.5 and γ = 1.³³ The positive or negative slope of μ₀H/M vs. M² indicates whether the magnetic phase transition is second-order or first-order.³² Fig. 3(a) shows the Arrott plot for La_0.6Sr_0.4CoO₃ sample and reveals the positive slope, so following Mira et al.,¹ we assume that the FM–PM phase transition in these La–Sr based cobaltites is basically of second order which is in agreement with earlier reports.^7,24 The same figure demonstrate a nonlinear M² versus μ₀H/M parts in the low-field region at temperatures below and above T_C are driven towards two opposite directions, revealing the FM–PM separation. Also, according to the mean-field theory near Curie temperature, M² vs. μ₀H/M at several temperatures should form a progression of parallel lines, and the line at T = T_C should pass through the origin. In the present case, the curves in the Arrott plot show a considerable downward curvature and the criterion is not met in the La_0.6Sr_0.4CoO₃ material. The result is similar to that observed in other FM cobaltites, such as La_0.67Sr_0.33CoO₃.¹⁵ This fact itself implies that the FM short-range order exists in our sample and proves that the framework of Landau mean-field model is unsatisfied. Therefore, a modified Arrott plot should be employed.

Fig. 2 (a) Temperature dependence of magnetization and 1/M(T) curve measured at μ₀H = 0.05 T in the field cooled mode for La_0.6Sr_0.4CoO₃. The inset shows the dM/dT as a function of temperature. (b) The M(H) curves recorded at 10 K from –5 T to 5 T. The inset represents the isothermal magnetizations around T_C for La_0.6Sr_0.4CoO₃.

Fig. 3 Modified Arrott plots: M^1/β vs. (H/M)^1/γ with (a) mean-field model (β = 0.5, γ = 1) and (b) 3D-Heisenberg model (β = 0.365, γ = 1.336). (c) Relative slope (RS) as a function of temperature for La_0.6Sr_0.4CoO₃.

Hence, we tried to analyze our data according to the modified Arrott plot, based on the so-called Arrott–Noakes equation of state given by:³⁴

where a and b are considered to be constants.

Four kinds of trial exponents belonging to the mean-field model (β = 0.5 and γ = 1), Tricritical mean-field model (β = 0.25 and γ = 1); 3D-Heisenberg model (β = 0.365 and γ = 1.336) and 3D-Ising model (β = 0.325 and γ = 1.24) are used to construct the modified Arrott plot of M^1/β vs. (μ₀H/M)^1/γ at several temperature for our cobaltite system. These models exhibit quasi-straight lines in the high field region, so it seems difficult to distinguish which one of them is the best for the determination of critical exponents. The mean-field, the 3D Heisenberg and the 3D-Ising model are quite similar. By using the conventional 3D Heisenberg model of critical exponents we plot in Fig. 3(b), the modified Arrott plot for La_0.6Sr_0.4CoO₃ sample. Thus, we determined the relative slopes (RS) defined at the critical point as RS = S(T)/S(T_C) for La_0.6Sr_0.4CoO₃ which enables us to identify the most suitable model by comparing the RS with the ideal value of ‘1’.³⁵ The RS vs. T plots for the four different models are presented in Fig. 3(c). We can see that the RS for La_0.6Sr_0.4CoO₃ using mean-field model, tricritical mean-field model and 3D-Ising deviate from RS = 1, as against, the RS of 3D Heisenberg model is close to it. So, it is the appropriate one to determine the critical exponents and describe our sample. This result indicates that the critical behavior of La_0.6Sr_0.4CoO₃ may belong to a single universality.

The linear extrapolation from the high field region to the intercepts with the axes M^1/β and (μ₀H/M)^1/γ yields reliable values of the temperature variation of the spontaneous magnetization M_SP(T, 0) and the magnetic susceptibility χ₀⁻¹(T, 0), which are plotted in Fig. 4(a). The values of β, γ and T_C correspond to that of optimum fitting obtained from eqn (1) and (2) using a self-consistent method.¹ The best fits give the values (β = 0.396(4), T_C = 230.25(2) K) and (γ = 1.320(1), T_C = 229.85(1) K) for La_0.6Sr_0.4CoO₃. These values are in good agreement with those predicted by the 3D Heisenberg model. The critical temperature T_C from the modified Arrott plot is in good agreement with that obtained from the inflection point of dM/dT plot, indicating strong critical fluctuation before the formation of the long-range ordering in La_0.6Sr_0.4CoO₃.

Fig. 4 (a) Spontaneous magnetization M_S (T, 0) (left axis) and the inverse initial susceptibility χ₀⁻¹(T, 0) (Right axis) versus temperature of La_0.6Sr_0.4CoO₃. (b) shows the K–F construction for determining the critical exponents and the Curie temperature; the solid lines are fits to eqn (8).

Further processing of M_SP and χ₀⁻¹ was performed using the Kouvel-Fisher (KF) method by constructing the functions defined by the expressions:³⁶

This method is known to allow a more accurate determination of the critical exponents β, γ and T_C. To further support the correctness of the obtained exponents and T_C, eqn (7) is applied to deduce the critical exponents β and γ along with T_C for the La_0.6Sr_0.4CoO₃. The temperature dependence of the M_S (dM_S/dT)⁻¹ and χ₀⁻¹(dχ₀⁻¹/dT)⁻¹ should yield straight lines with slopes 1/β and 1/γ, respectively and the intercepts on T axes are equal to Curie temperature (T_C). The obtained plots are depicted in Fig. 4(b). The critical exponents obtained from KF method are β = 0.391(2) with T_C = 229.97(4) K and γ = 1.314(3) with T_C = 229.77(8) K for La_0.6Sr_0.4CoO₃. From the obtained data, it is evident that the Heisenberg exponents describe well the temperature dependencies of the spontaneous magnetization M_SP and of the initial susceptibility χ₀⁻¹ only in a restricted temperature range close to T_C. Clearly, these values are also in a good agreement with those obtained from modified Arrot plots.

Fig. 5(a) shows the critical isotherm M(μ₀H) curves a T_C=230 K for our sample, with the inset plotted on a log–log scale. Using eqn (3), the best fits give the value of the third exponent δ and the obtained value for the currently investigated sample is 4.761(1). According to the statistical theory, these critical exponents should fulfill the Widom scaling law which critical exponents β, γ, and δ are related in following way:³⁷

δ = 1 + γ/β (8)

Fig. 5 (a) Critical isotherms of M vs. μ₀H for La_0.6Sr_0.4CoO₃. (b) Scaling plot of M|ε|^−β versus μ₀H|ε|^−(β+γ) for La_0.6Sr_0.4CoO₃ compound at temperature T < T_C and T > T_C.

As a result, δ is obtained from the modified Arrott plot (δ = 4.330(2)) and from the KF method (δ = 4.359(6)). The value obtained from critical isotherms M (T_C, μ₀H) is similar than determined from the Widom scaling. The small deviation can be explained probably by the experiments errors. The self-consistency of the critical exponents demonstrates that they are reliable and unambiguous. To learn about the exponent δ obtained for the La_0.6Sr_0.4CoO₃ system we have compared them with the theoretical models. Here, we noted that these values of δ are in agreement those given by the 3D-Heisenberg model. It is known that the 3D-Heisenberg model with short range magnetic exchange interaction entirely describes the magnetic phase transition.

As confirmation, we use more rigorous method to check the accuracy of the deduced exponents with the prediction of the scaling hypothesis. Thus, the isothermal magnetizations around T_C are shown as this prediction in the inset of Fig. 5(b) for our compound, with the log–log scale in Fig. 5(b). All experiment data collapses into two independent branches, one for temperatures below T_C and the other for temperatures above T_C. This finding proves that eqn (5) is obeyed over the entire range of the normalized variables, which indicates the reliability of these obtained critical exponents and this shows that the critical exponent values obtained from the present investigation are reasonably accurate.

In order to evaluate the magnetocaloric effect in the investigated material, the isothermal magnetization curves of the sample are also used to determine the magnetic entropy change ΔS under a magnetic applied field change Δμ₀H in a range of temperatures around the Curie temperature T_C. From these M(μ₀H) isotherms the magnetic entropy change ΔS of the La_0.6Sr_0.4CoO₃ has been calculated using Maxwell's relations:

In this equation, M_i and M_i+1 are the experimental values of magnetization measured at temperatures T_i and T_i+1, respectively, under magnetic applied field μ₀H_i. Fig. 6(a) shows the ΔS(T) for different magnetic field changes up to 5 T for La_0.6Sr_0.4CoO₃. The ΔS shape shows a symmetrical broadening with a field applied, indicating that the large ΔS originates from a reversible second-order magnetic. It is interesting to note that the maximum of the magnetic entropy change is obtained around T_C = 230 K, which is a property of simple ferromagnets due to the efficient ordering of magnetic moments induced by the magnetic field at the ordering temperature. The maximum values of −ΔS are found to be 1.1 and 2.1 J kg⁻¹K⁻¹ for the field changes of 0–2 T and 0–5 T, respectively. It is found that ΔS values of La_0.6Sr_0.4CoO₃ are comparable to or larger than those of some magnetic cobaltites materials with a similar transition temperature. For example, the maximum values of entropy changes are 1.9 J kg⁻¹K⁻¹ for Pr_0.6Sr_0.4CoO₃,³⁸ 2.2 Jkg⁻¹K⁻¹ for Pr_0.5Sr_0.5CoO₃,³⁸ 1.35 J kg⁻¹K⁻¹ for La_2/3Sr_1/3CoO₃,³⁹ and 1.69 J kg⁻¹K⁻¹ for La_0.55Bi_0.05Sr_0.4CoO₃ (ref. 31) for a field change of 0–5 T. The field dependence of the isothermal magnetic entropy change of La_0.6Sr_0.4CoO₃ sample, is consistent with a simple ferromagnetic order (i.e., a monotonic, almost linear increase is seen) over the total investigated field and temperature range. The inset of Fig. 6(a), for temperatures below and above the Curie temperature, respectively, shows −ΔS changes to a positive value with the increase of the magnetic field at the individual temperatures, corresponding to the magnetic transition from ferromagnetic to paramagnetic states.

Fig. 6 (a) Magnetic field dependence of magnetic entropy change for La_0.6Sr_0.4CoO₃. (b) The magnetic field dependence of ΔS_max, δT_FWHM and RCP for La_0.6Sr_0.4CoO₃.

Concerning the magnetic-field dependence of the maximum magnetic entropy changes it has been found that this dependence can be expressed by a power law ΔS^max ≈ a(μ₀H)ⁿ where n is related to the magnetic state of a sample. Using the obtained data for different magnetic-field changes, the experimental data of field dependence of ΔS at T_C for La_0.6Sr_0.4CoO₃ sample is depicted in Fig. 6(b). The fitting value of n is about 0.805(1) which is significantly higher than 2/3, as predicted by the mean field model. Recently, for materials that do not obey the mean-field the relationship between the exponent n at the Curie temperature and the critical exponents of the ferromagnetic sample has been established:

the value of n at T_C obtained from the (−ΔS) data is deviate with this calculated from eqn (10) using the values of β and δ determined from the K–F method (n(T_C) = 0.709 (6)). The disagreement between the exponent n obtained from the K–F method, the MCE analysis and the predicted value of 2/3 in the mean field approach is due to the local inhomogeneities or superparamagnetic clusters in the vicinity of Curie temperature existed in the La_0.6Sr_0.4CoO₃ specimen.⁴⁰ A similar behavior has been observed in other perovskite materials.^21,41,42

The RCP is evaluated as RCP = |ΔS^max(T,μ₀H)| × δT_FWHM where δT_FWHM is the full-width at half-maximum of −ΔS(T). For our compound, the field dependence of the RCP and δT_FWHM for the present sample are also calculated and depicted in Fig. 6(b). By fitting the RCP data to a power law of where N being the critical exponent of the magnetic transition, the N value obtained (from experimental data is N = 1.154(1)) is comparable with values previously reported for (Na, K) doped Pr_0.6Sr_0.4MnO₃,^43,44 Pr_0.5Eu_0.1Sr_0.4MnO₃,¹⁹ and those observed for Gd alloys (N = 1.16) by Law et al.⁴⁵ Under a magnetic field change of 5T, δT_FWHM value is about 64.2 K and RCP value is about 135.1 J kg⁻¹ for La_0.6Sr_0.4CoO₃, respectively. Our results indicate that RCP value is comparable with those determined from some perovskite cobaltites,^38,46 revealing the applicability of La_0.6Sr_0.4CoO₃ in cooling devices.

Franco et al.⁴⁷ proposed a useful fundamental method to construct the universal curve based on the magnetic entropy change that can be used to access the nature of phase transitions for a wide range of ferromagnetic compounds. This process is carried out by normalizing all the ΔS(T) plots using their respective maximum value:

The temperature axis was rescaled in a different way below and above T_C, just by imposing that the position of the two additional reference points of each curve correspond to in the curve corresponds to θ = ±1:^48,49

where T_r1 and T_r2 are the temperatures of the two reference points of each curve satisfy T_r1 < T_C < T_r2. For the present work, these temperatures were selected corresponding to the value ΔS/ΔS^peak = 1/2. This procedure has been successfully applied to different families of magnetic material.^50–52 For La_0.6Sr_0.4CoO₃, we attempt to confirm whether the magnetocaloric data obey the phenomenological curves. Within the experiment error, it is found that all the data collapse into a single curve (see figure Fig. 7(a)), which further confirms the ΔS value of La_0.6Sr_0.4CoO₃ originates from a reversible second-order magnetic transition. In the inset of Fig. 7(a) we show the field dependences of the two reference temperatures (T_r1 and T_r2) of the compound.

Fig. 7 (a) Universal behavior of the scaled entropy change curves at different fields. The inset shows the dependences of the two reference temperatures (T_r1 and T_r2) at various magnetic fields (b) the local exponent n at several magnetic fields for La_0.6Sr_0.4CoO₃ compound. The inset shows θ dependence of the local exponent n for the studied sample for typical field changes.

The value of n depends on the values of applied field and temperature. This exponent, in general, can be locally calculated as:

The temperature dependence of n is depicted in Fig. 7(b). For our compound, it can be noted from the obtained curves that the value of n reaches 2 above the T_C and 1 far below the T_C. The n exponent exhibits a moderate decrease with increasing temperature, with a minimum value in the vicinity of the transition temperature, sharply increasing above T_C. The n(T,μ₀H) curves also collapse when plotted against the same rescaled temperature axis for which the normalized values of −ΔS(T) collapse onto the universal curve. The procedure consists of identifying the reference temperatures as those which have a certain value of n. This value has been arbitrarily selected as n(Tr) = 1.5. For several magnetic fields, all the data points collapse into a single master curve revealing universal behavior in La_0.6Sr_0.4CoO₃. The method consists of identifying the reference temperatures as those which have a certain value of n. This value has been arbitrarily selected as n(Tr) = 1.5. The master curve is depicts in the inset of Fig. 7(b).

5. Conclusion

We have studied the structural, magnetic, critical parameters and magnetocaloric properties in La_0.6Sr_0.4CoO₃ cobalitite prepared by sol–gel method. Rietveld refinement of XRD pattern shows that La_0.6Sr_0.4CoO₃ possesses rhombohedral structure with R3̄c space group. We have analyzed the critical behavior of the sample from isothermal magnetization in the critical region based on various methods including modified Arrott plot, Kouvel–Fisher method, and critical isotherm analysis. The experimental results have revealed that the samples exhibited the second-order magnetic phase transition and the critical exponent values of β, γ, and δ obtained for the La_0.6Sr_0.4CoO₃ are close to those found out by the 3D-Heisenberg model. The reliability of these critical exponents was confirmed by the scaling equation, which shows the magnetization-field-temperature data below and above T_C collapse into two independent branches. The magnetic entropy change and the exponent n can be calculated and characterized by one universal curve respectively for the studied material. A study of the local exponent n controlling the field dependence of ΔS was carried out and reached the value of n = 0.801(5). The deviation of the exponent n values from the mean field model confirms the insufficient description of the mean field model at the peak temperature in the present material.

Acknowledgements

This study is supported by the Tunisian Ministry of Higher Education and Scientific Research and the Neel Institute.

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