Structural, magnetic and theoretical investigations on the magnetocaloric properties of La_0.7Sr_0.25K_0.05MnO₃ perovskite

Abstract

The La_0.7Sr_0.25K_0.05MnO₃ (LSKMO_0.05) manganite compound was synthesized by the solid state reaction method. X-ray diffraction analysis revealed that this sample crystallizes in the distorted rhombohedral system with the R3̄c space group. The magnetic study showed a second order paramagnetic (PM)-ferromagnetic (FM) transition at the Curie temperature T_C = 332 K. In addition, isothermal measurements of magnetization allowed us, through thermodynamic Maxwell relations, to determine the magnetic entropy change (ΔS_M). The maximum magnetic entropy change (ΔS_max) and the relative cooling power (RCP) were found to be, respectively, 2.37 J kg⁻¹ K⁻¹ and 102 J kg⁻¹ for a 2 T magnetic field change along with a negligible hysteresis loss, making this material a promising candidate for magnetic refrigeration. In order to eliminate the drawbacks due to the use of multistep nonlinear fitting in a conventional manner, the field dependence of magnetic entropy change was applied to study the critical behavior. As expected, our results are consistent with the values derived for the 3D-Heisenberg model.

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1. Introduction

Magnetic refrigeration based on the magnetocaloric effect (MCE) is a promising alternative to conventional gas compression refrigeration. This is thanks to its high efficiency, small volume, energy saving and ecological cleanliness; in fact, this technology does not use toxic gases that can negatively influence the ozone layer.^1,2 Therefore, new and improved promising magnetic materials are one of the cornerstones in the development of room temperature magnetic refrigeration.

In the context of MR applications, several materials with giant magnetic entropy changes have been proposed. Among these materials we can cite Gd₅Si_4−xGe_x,³ MnAs_1−xSb_x⁴ and La(Fe_1−xSi_x)₁₃.⁵ They undergo a simultaneous first-order structural and magnetic-phase transition that is believed to be responsible for the large MCE. These compounds exhibit a MCE about twice as large as that exhibited by gadolinium, the best known magnetic refrigerant material for near room-temperature applications.¹ Further efforts to find new materials, especially the materials without rare-earth elements, have revealed that some of perovskite manganese oxides are also expected to be promising candidates in magnetic refrigeration technology.^6,7 This is attributed to their colossal magnetoresistance, as well as the large change in temperature in adiabatic conditions under moderate external magnetic field. Perovskite manganites with a general formula R_1−xT_xMnO₃ where R is a rare earth atom (La, Pr, Nd, etc.) and T is an alkaline earth element (Sr, Ca, Ba, etc.), present a wide spectrum of structural and functional phases associated to the complex interplay between charge, spin, orbit and structural aspects which are sensitive to the arrangement of the 3d electrons of Mn in their orbit.⁸ Recent studies suggest that multifunctional behavior can be achieved through a careful chemical manipulation explained by the double-exchange (DE) mechanism and the strong electron-phonon coupling caused by lattice polarons and dynamic Jahn–Teller (JT) distortions.⁹ Following this mechanism, a number of works reported on the MCE in lanthanum manganites wherein a large magnetocaloric effect has been found. Among the lanthanum-based manganites, the strontium-doped lanthanum manganites (La_1−xSr_xMnO₃, LSMO) have secured a prominent position based on the high potential for magnetic sensor applications magnetoresistance effect and the MCE as they possess high magnetic moments.¹⁰ These manganites exhibit a metal-to-insulator transition accompanied by a ferromagnetic-to-paramagnetic transition near the Curie temperature T_C, which directed the focus towards magnetic entropy studies. Experimentally, it has been revealed that the La_0.67Sr_0.33MnO₃ compound is characterized by a maximum magnetic entropy change of ΔS_Max equal to 2.68 J kg⁻¹ K⁻¹ at a 370 K and under a magnetic field of 2 T which is beneficial for technological applications.¹¹ The new investigation field is focused on the manganites doped with monovalent elements. In fact, the substitution of the monovalent cations for the rare earth cations in manganites is of a particular interest. This is because it leads to converting twice more Mn³⁺ to Mn⁴⁺ ions to give rise to the double exchange interactions, which are the origin of the ferromagnetic character and the CMR effect. Therefore, even a small amount of doping results in a large number of charge carriers and as a consequence an increase in the conductivity can be achieved and the crystal structure may be modified depending on the radius of monovalent ions. The Curie temperature is, therefore, expected to be reduced as compared to that reported for the parent undoped manganite. Besides, our research group has widely studied the effect of monovalent elements on the physical properties of La_0.67Sr_0.33MnO₃. In addition, we have found an important value of ΔS_Max = 2.32 J kg⁻¹ K⁻¹ and T_C = 363 K under a magnetic field of 2 T in the La_0.7Sr_0.25Na_0.05MnO₃ sample.¹² These values of ΔS_Max are significant but the Curie temperatures are higher at room temperature. To get more insights into the effect of the monovalent cations on manganite, we turn our attention to potassium-doped perovskite manganites to see if it improves the magnetic and magnetocaloric properties of our sample.

The purpose of this work is to investigate in details the field induced magnetic phase transition and the MCE of La_0.7Sr_0.25K_0.05MnO₃ compound and its possibility for application as an active magnetic refrigerant (AMR) material. A phenomenological model was used for the simulation of the dependence of magnetization on temperature variation to predict the magnetocaloric properties such as magnetic entropy change, heat capacity change, and relative cooling power.

2. Experimental details

The polycrystalline sample L_0.7Sr_0.25K_0.05MnO₃ was synthesized by solid state reaction. Pure (≥99.99%) raw powders with appropriate amounts of La₂O₃, SrCO₃, K₂CO₃ and MnO₂ were thoroughly ground and mixed, then pressed into pellets. These were calcined at several processing steps with increasing temperatures from 900 °C to 1100 °C and with intermediate grindings and pelletizations. The product was then sintered at 1200 °C for 24 h in ambient atmosphere. Finally, these pellets were rapidly quenched to room temperature in order to keep the structure at an annealing temperature.¹³

Phase purity, homogeneity and crystal properties were determined by powder X-ray diffraction (XRD) data, recorded at room temperature on a PANalytical X'PERT Pro MPD diffractometer, using θ/2θ Bragg–Brentano geometry with diffracted beam monochromatized CuK_α radiation (λ = 1.5406 Å). The surface morphology was investigated by a field-emission scanning electron microscopy (SEM) attached with energy dispersive analysis through X-rays (EDAX). Magnetizations (M) vs. temperature (T) were measured using BS1 and BS2 magnetometers developed in Louis Néel Laboratory of Grenoble. BS1 (300–900 K) and BS2 (1.5–300 K) magnetometers are used respectively for magnetic measurements at high and low temperatures equipped with a super conducting coil. These two instruments are automated by a computer system that allows the registration of digital data for each successive measurement. Magnetization isotherms were measured in the range of 0–5 T and with a temperature interval of 3 K in the vicinity of Curie temperature (T_C). These isothermals were corrected by a demagnetization factor D that was determined by a standard procedure from low-field dc magnetization measurement at low temperatures (H = Happ-DM). Finally, the magnetocaloric effect (MCE) was characterized by an isothermal change of the magnetic entropy and the adiabatic change of temperature. In fact, the magnetocaloric effect was estimated in terms of isothermal magnetic entropy change (ΔS_M) using M–μ₀H-T data and Maxwell's relation:^14,15

On the other hand, the relative cooling power RCP was calculated using the relation:

here T₂ and T₁ are respectively, the temperatures of hot and cold sinks, which correspond to the full-width at half-maximum points (FWHM). The material with the highest RCP value usually represents the best magnetocaloric substance due to its high cooling efficiency.

3. Results and discussion

3.1. X-ray analysis

Fig. 1 shows the room temperature X-ray diffraction (XRD) pattern of the synthesized LSKMO_0.05 sample with a fit of whole profile. The diffraction peaks are sharp, indicating well crystallized phases. No impurity phases were detected within the X-ray diffraction limits.

Fig. 1 Observed (open symbols) and calculated (solid lines) X-ray diffraction pattern for La_0.7Sr_0.25K_0.05MnO₃. Position for the Bragg reflections is marked by vertical bars. Differences between the observed and the calculated intensities are shown at the bottom of the figure.

The structural parameters of the samples are refined by the standard Rietveld technique¹⁶ using the FullProf program.¹⁷ As can be seen from this figure, the sample is of a single phase without any detectable secondary phase and can be indexed according to the rhombohedral perovskite structure with a space group R3̄c (no. 167), in which the (La, Sr, K) atoms are at 6a (0, 0, 1/4) positions, Mn at 6b (0, 0, 0) and O at 18e (x, 0, 1/4). The difference observed between the intensities of the measured and calculated diffraction pattern can be attributed to the existence of a preferential orientation of the crystallites in the sample. The quality of the agreement is evaluated through the adequacy of the fit indicator χ². The related results are listed in Table 1. The structural stability of LSKMO_0.05 sample was estimated by Goldschmidt's tolerance factor t_G, which has been widely admitted as a criterion for the formation of a perovskite structure, expressed as follows:¹⁸

where r_B, r_A and r_O are the ionic radii of A, B, and O site atoms in ABO₃, respectively.

Table 1 Results of Rietveld refinements, determined from XRD patterns measured at room temperature for La_0.7Sr_0.25K_0.05MnO₃ compound

Phase R3̄C
Structural model cell parameters
a (Å)	5.524 (2)
c (Å)	13.376 (1)
Cell volume (Å^3)	353.52 (3)
Isotropic thermal parameters
B(La/K/Sr) (Å^2)	0.24 (2)
B(Mn) (Å^2)	0.73 (3)
B(O) (Å^2)	0.47 (3)
Discrepancy factors
RP (%)	6.8
Rwp (%)	6.4
RF (%)	5.2
χ^2	2.7

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In general, the perovskite structure is stable in the region 0.75 < t_G < 1.06, and the symmetry is higher as the t_G value is closer to 1. This parameter is an indication of how far the atoms can move from the ideal packing and be still belonging to the perovskite structure. In the present study, the tolerance factor of LSKMO_0.05 is calculated from Shannon's ionic radii r_La³⁺ = 1.22 Å, r_Sr²⁺ = 1.44 Å, r_K⁺ = 1.55 Å, r_Mn³⁺ = 0.65 Å, r_Mn⁴⁺ = 0.53 Å and r_O²⁻ = 1.4 Å (ref. 19) and it is found to be 0.963 which improves a stable perovskite structure.

The inset of Fig. 2 shows the scanning electron microscopy (SEM) photograph for the LSKMO_0.05 sample. We can see that the grains exhibit spheroid-like shapes and a good connectivity between each other. Using the linear intercept method, the grain size was estimated to be mostly within 208 nm. In order to check the cationic composition of the sample, the Energy Dispersive X-ray Analysis (EDAX) was performed at room temperature (Fig. 2). EDAX plot reveals the presence of all expected chemical elements introduced in the starting oxide powders, which confirms that there was no loss of any integrated element during the process of preparation.

Fig. 2 Plot of EDX analysis of chemical species of La_0.7Sr_0.25K_0.05MnO₃. The inset shows typical scanning electron micrography (SEM).

3.2. Magnetization investigation

Magnetization measurements versus temperature, recorded under a magnetic field of 0.05 T, are shown in Fig. 3a. We observe a sharp transition from high-temperature paramagnetic state to low temperature ferromagnetic state. This curve reveals a strong variation of magnetization around the ferromagnetic Curie temperature (T_C = 332 K) defined as the inflection point of dM/dT curve (inset of Fig. 3a). It indicates that there is a possible large magnetic entropy change. For a further understanding of the magnetic interaction at a high temperature, we have investigated the temperature dependence of the inverse magnetic susceptibility χ for 0.05 T (Fig. 3b). For a ferromagnet, it is well known that in the PM region, the relation between χ and the temperature T should follow the Curie–Weiss law, i.e. χ = C/T − θ_CW, where θ_CW is the Weiss temperature and C is the Curie constant defined as:N_A = 6.023 × 10²³ mol⁻¹ is the number of Avogadro, μ_B = 9.274 × 10⁻²¹ emu is Bohr-magneton and k_B = 1.38016 × 10⁻¹⁶ erg K⁻¹ is Boltzmann's constant.

Fig. 3 (a) Variation of the magnetization vs. temperature for La_0.7Sr_0.25K_0.05MnO₃ sample at 0.05 T, the inset shows the plot of dM/dT curve as a function of temperature. (b) The temperature dependent of the inverse susceptibility for La_0.7Sr_0.25K_0.05MnO₃ and the solid line is the fitting result following the Curie Weiss law.

From the slope of the linear 1/χ vs. T curve (see Fig. 3b), the experimental effective paramagnetic moments can be determined and the temperature at which 1/χ intercepts the temperature axis is the Curie Weiss temperature θ_CW. Assuming orbital momentum to be quenched in Mn³⁺ and Mn⁴⁺, the theoretical paramagnetic effective moment μ^th_eff can be written as:

where g = 2 is the gyromagnetic factor and S is the spin of the cation (the S values are 3/2 for Mn⁴⁺ and 2 for Mn³⁺). Thus, the theoretical values are μ^th_eff(Mn³⁺) = 4.9 μ_B and μ^th_eff(Mn⁴⁺) = 3.87​ μ_B.²⁰ Using the spin-only moment of the free Mn ions, the effective paramagnetic moment per formula unit that can be written as:

From the high temperature linear fit, the effective magnetic moment evaluated is equal to 3.83 μ_B, which is slightly smaller than the calculated effective paramagnetic moment 4.04 μ_B. The discrepancy between the experimental and the theoretical values implies that some short-range FM couplings might have been developed in the PM region contributing to the additional magnetic moments. This phenomenon is generally termed as “magnetic phase separation” and has been extensively reported in manganites.²¹ It can also be remarked that the paramagnetic Curie–Weiss temperatures θ_CW value is slightly higher than T_C. Generally, the difference between θ_CW and T_C (θ_CW > T_C) depends on the substance and is associated with the presence of short-range magnetic interaction ordered slightly above T_C, which may be related to the presence of a magnetic inhomogeneity.²²

To evaluate the MCE of the present sample, the isothermal magnetic field dependence of magnetization M–μ₀H was measured at different temperatures in the range of 296–398 K, as shown in Fig. 4a (magnetic field was varying from 0 to 5 T). We found a clear ferromagnetic behavior at all temperatures. This reveals a strong variation of magnetization around Curie temperature indicating that there is a possible large magnetic entropy change associated with the FM-PM transition temperature occurring at T_C. Fig. 4b show the Arrott plot M² vs. μ₀H/M. However, all the curves in the Arrott plots are nonlinear and show upward curvature even at high field, indicating that the present phase transition will not satisfy the mean field theory. Moreover, the concave downward curvature clearly indicates a second order phase transition according to the criterion suggested by Banerjee.²³ According to the criterion, the negative and positive slopes of μ₀H/M vs. M² isotherms curves (Arrott plots) suggest a first-order and a second-order phase transition, respectively. For the present compound, a positive slope of the Arrott plots can be observed, which indicates the second-order nature of phase transition.

Fig. 4 (a) Isothermal magnetization versus magnetic field M vs. μ₀H at different temperatures for La_0.7Sr_0.25K_0.05MnO₃ sample. (b) The standard Arrott plot M² versus μ₀H/M.

3.3. Theoretical considerations

3.3.1. Model. In order to determine the MCE especially in these materials, both the experimental and theoretical approaches were adopted. For the experimental evaluation, heat capacity and/or magnetization data are required. Change in entropy (ΔS_M), full width at half maximum (δT_FWHM), change in specific heat (ΔC_P,H) and relative cooling power (RCP) etc., are important parameters in the investigation of MCE. Therefore, in the present investigation, a simple approach was adopted for the evaluation of all these parameters. Based on the phenomenological model,^24,25 the dependence of magnetization on the variation of temperature and Curie temperature T_C may be written as:where M_i/M_f is an initial/final value of magnetization at a ferromagnetic–paramagnetic transition as shown in Fig. 5. ; B is the magnetization sensitivity dM/dT at ferromagnetic state before transition; is the magnetization sensitivity dM/dT at Curie temperature T_C and .

Fig. 5 Temperature dependence of magnetization in constant applied magnetic field.

Differentiating eqn (7) gives:

Using eqn (1) and (8), the magnetic entropy change of a magnetic system under adiabatic magnetic field variation from 0 to final value μ₀ H_max can be rewritten as follows:

The presence of a large magnetic entropy change is accredited to a high magnetic moment interaction and a rapid change of magnetization at T_C. The result of eqn (9) is a maximum magnetic entropy change ΔS_max (where T = T_C) can be evaluated as the following equation:

Furthermore, the determination of full-width at half-maximum δT_FWHM can be performed as follows:

Relative cooling power (RCP) is a useful parameter, which decides the effectiveness of magnetocaloric materials based on the magnetic entropy change.²⁵ The RCP is defined as the product of the maximum magnetic entropy change ΔS_Max and full width at half maximum (δT_FWHM) in ΔS_M (T) curve.

The heat capacity ΔC_P,H can be calculated, from the magnetic contribution to the entropy change induced in the material, by the following expression:²⁶

From eqn (9) can be inferred as follows:

From eqn (13) and (14), ΔC_P,H can be rewritten as:

ΔC_P,H = −TA²(M_i − M_f)sech²[A(T_C − T)] tanh[A(T_C − T)]μ₀H_max (15)

From this phenomenological model, ΔS_M, δT_FWHM, RCP and ΔC_P,H can be simply evaluated for La_0.7Sr_0.25K_0.05MnO₃ under magnetic field variation.

3.3.2. Model application. In the present investigation, we derived the values of M_i, M_f and T_C directly from our experimental data and by referring to the values of magnetization sensitivity (dM/dT) in the ferromagnetic region, B and the magnetization sensitivity (dM/dT) at the transition temperature, a numerical calculation was made to estimated the value of A and C. To this end we have tabulated all parameters in Table 2. Fig. 6 shows the thermo-magnetization curves of the sample LSKMO_0.05 recorded every 3 K in the field-cooling process. The M(T) curves reveal that the LSKMO_0.05 sample exhibits a magnetic transition from a ferromagnetic state to a paramagnetic one, without any anomalies detected, when increasing temperature. On the other hand, upon μ₀H = 1 T applied field, along with the increase of magnetization, T_C is slightly shifted towards a higher temperature, while the shape of the M–T curve remains almost unchanged. Similar behaviors were observed in active magnetic refrigerant materials.²⁷ Still in Fig. 6, the symbols represent experimental data, while the solid line represents modeled data using parameters given in Table 2. Overall, the effectiveness of the procedure is illustrated by the very good agreement between measured and calculated thermal-magnetization data of LSKMO_0.05 sample.

Table 2 Model parameters for LSKMO_0.05 sample in different applied magnetic fields

μ0H(T)	TC (K)	Mf (emu g^−1)	Mi (emu g^−1)	B (emu g^−1 K^−1)	SC (emu g^−1 K^−1)
1	332.3	62.3	3.3	−0.014	−1.54
2	335.5	64.1	6.2	−0.033	−1.24
3	336.7	67.5	9.6	−0.036	−0.05
4	337.6	68.7	11.8	−0.045	−0.92
5	338.2	69.4	14.7	−0.058	−0.79

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Fig. 6 Magnetization versus temperature for the La_0.7Sr_0.25K_0.05MnO₃. The solid lines are modeled results and symbols represent experimental data.

Fig. 7 shows the dependence of the magnetic entropy change on temperature for different applied magnetic fields for La_0.7Sr_0.25K_0.05MnO₃ sample. The solid lines are modeled results and the symbols represent experimental data. The results of calculations are in accordance with the experimental results. It is interesting to note that the maximum of the magnetic entropy change is obtained near Curie temperature (T_C = 332 K) of the sample. It is found to be positive in the entire temperature range for all magnetic fields, which confirmed the ferromagnetic character.²⁸ We can see from Fig. 7 that the magnitude of ΔS_M increases with increasing the strength of μ₀H and reaches a maximum value of 1.34 J kg⁻¹ K⁻¹ and 4.85 J kg⁻¹ K⁻¹ under a magnetic field change of μ₀H = 1 T and μ₀H = 5 T, respectively. In fact, the intensity of the magnetic field plays an important role in magnetic cooling. The maximum magnetic entropy change ΔS_max exhibits a linear rise with increasing field, which indicates a much larger entropy change to be expected at a higher magnetic field, signifying, therefore, the effect of spin-lattice coupling associated to changes in the magnetic ordering process in the sample.²⁹ Another interesting feature to be noted is that for the LSKMO_0.05 sample, the maximum of the magnetic entropy, 2.37 J kg⁻¹ K⁻¹ for a 2 T applied magnetic field, which is about 45% of the pure Gd³⁰ is larger than the ΔS_max value of many perovskite materials (Table 3). These results clearly suggest that this compound could be a suitable candidate as a working material in refrigeration devices at near room temperature. To explain the large magnetic entropy change in perovskite manganites, Zener's double exchange model is strongly recommended.³¹ The effect of the double-exchange interaction between Mn³⁺ and Mn⁴⁺ ions would be closely related to the change in Mn³⁺/Mn⁴⁺ ratio, under the doping process. In addition, Guo et al. suggest that the large magnetic entropy change in perovskite manganites could be originating from the spin-lattice coupling in the magnetic ordering process.³² Because of the strong coupling spin and lattice, a significant change in the lattice accompanying the magnetic transition in perovskite manganites has been observed.^33,34 The lattice structural change in 〈Mn–O〉 bond distance as well as in the 〈Mn–O–Mn〉 bond angle would in turn favor the spin ordering. As a result, a more abrupt variation of magnetization near T_C occurs, which leads to a large magnetic entropy change, resulting in a large MCE.

Fig. 7 Magnetic entropy change (ΔS_M) of La_0.7Sr_0.25K_0.05MnO₃ sample as a function of temperature upon different magnetic field intervals (μ₀ΔH). The symbols represent experimental data and the solid lines are modeled results.

Table 3 Maximum entropy change ΔS_M^max and relative cooling power (RCP), occurring at the Curie temperature (T_C) for La_0.7Sr_0.25K_0.05MnO₃ compounds. The present results are compared with some of the divalent and the mono-valent hole-doped manganites as well as pure Gd

Materials	μ0H (T)	−ΔSM (J kg^−1 K^−1)	δTFWHM (K)	RCP (J kg^−1)	ΔCP,Hmax (J kg^−1 K^−1)	ΔCP,Hmin (J kg^−1 K^−1)	Refs.
La0.7Sr0.25K0.05MnO3	1	1.34	36.05	48.32	5.79	−5.48	Present
	2	2.37	44.64	105.37	11.64	−10.97	Present
	3	3.15	51.54	162.38	19.56	−18.41	Present
	4	4.04	55.38	223.75	27.66	−25.93	Present
	5	4.85	58.15	282.03	34.81	−32.45	Present
Gd	2	5.5	—	164	—	—	47
La0.95Ag0.05MnO3	1	1.1	—	44	—	—	48
La0.925Na0.075MnO3	1	1.32		93	—	—	49
Pr0.55Sr0.4K0.05MnO3	2	2.19	—	87			50
La0.8Na0.05K0.15MnO3	2	2.2		107			51
Pr0.6Sr0.35Na0.05MnO3	2	1.84		92			52
La0.7Sr0.25Na0.05MnO3	2	2.32		82			12
La0.7Ca0.3MnO3	3	2.2		55			53
La0.60Y0.07Ca0.33MnO3	3	1.46		140			54
Nd0.55Sr0.45MnO3	3	3.12		71			55
Gd	5	9.5		410			1
La0.7Sr0.3Mn0.93Fe0.07O3	5	4		255			11
La0.7Sr0.2Ag0.1MnO3	5	4		244			56
La0.7Sr0.25Na0.05MnO3	5	4.34		298			12
La0.6Pr0.1Sr0.3MnO3	5	3.32		227			57

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Another relevant quantity to evaluate the applicability of a magnetocaloric material is the relative-cooling-power (RCP), which is defined as:³⁵

RCP = −ΔS_max × δT_FWHM (16)

where ΔS_max present the maximum-entropy change at T_C and δT_FWHM is the full-width at the half maximum of the ΔS_M. The values of maximum magnetic entropy change, full width at half maximum (δT_FWHM) and relative cooling power (RCP) at different magnetic fields for LSKMO_0.05 are calculated using eqn (10) and (11). They are summarized in Table 3. It is clear that the RCP values are also sensitive to the A-site-cation mismatch σ². A large value of RCP is 104 J kg⁻¹ at a relative low field change of 2 T for the LSKMO_0.05 sample. The obtained RCP values is about 64% of the standard metallic Gd and much larger than those found in other manganese-site modified perovskites based on the La_0.7Sr_0.3MnO₃ and La_0.7Ca_0.3MnO₃ composition. Since the RCP factor represents a good way for comparing magnetocaloric materials, our compound can be considered as a potential candidate thanks to the high RCP compared to conventional refrigerant materials.

Fig. 8 shows the temperature dependence of ΔC_p under different field variations for the sample calculated from ΔS_M − T curves of Fig. 7 by using relation:

and predicted values using eqn (15).

Fig. 8 Heat capacity changes (ΔC_p) as function of temperature for La_0.7Sr_0.25K_0.05MnO₃ in different applied magnetic field variations.

The ΔC_p undergoes a sudden change from positive to negative around T_C with a positive value above T_C and a negative value below T_C. The positive or negative values of ΔC_p closely above or below T_C may strongly alter the total specific heat. Furthermore, the maximum and minimum values of specific heat change for this sample are estimated and summarized in Table 3. Finally, in this model, a few necessary parameters are required and the processing time for the simulation is limited. The used procedure does not add any supplementary computational effort to the numerical simulation and it is suitable to be coupled with field computation.

Based on the standard scaling theory,³⁶ it has been found that the magnetic materials with a second order phase transition follow a universal behavior as far as the MCE variation is concerned. However, Franco et al.³⁷ prescribed that the measured curves of ΔS_M(T) at different applied magnetic fields should collapse on to the same universal curve. This procedure was performed by normalizing the magnetic entropy change curves ΔS_M with respect to their peak ΔS_max and rescaling the temperature axis using a two reference temperatures in a different way below and above T_C. A new temperature variable θ is defined by the following expression:

The temperatures T_r1 and T_r2 correspond to the temperatures for which ΔS_M(T_r) = 0.5ΔS_max. Noticeably, from Fig. 9 the scaled curves overlap into the same curve; indicating the validity of our data treatment for compounds with a second-order transition. Furthermore, the universal curve can be well fitted by a Lorentz distribution function:

where a, b, and c are the asymmetry parameters. By taking into consideration the asymmetry of the distribution (inset of Fig. 9), two different sets of constants have to be used: for T ≤ T_C, a = 1.62 ± 0.03, b = 1.48 ± 0.01 and c = 0.41 ± 0.01; and for T > T_C, a = 1.16 ± 0.03, b = 1.10 ± 0.03 and c = −0.02 ± 0.02.

Fig. 9 The master curve behavior of the curve as a function of the rescaled temperature. The inset shows the collapse of the experimental data onto the average curve: the solid line is the fit to eqn (18).

Referring to eqn (18), just three main parameters are required to characterize the entropy change: the magnitude (ΔS_max), the position of the peak (T_C) and two reference temperatures T_cold and T_hot, where T_cold < T_C and T_hot > T_C. These parameters can be considered as system resources. Indeed, the “real” ΔS_M (T) can be translated from ΔS′ (θ) by the knowledge of these parameters. Thus, incomplete ΔS_M (T) curves, which are experimentally determined from a small temperature span in the vicinity of T_C for the isothermal magnetization measurements, can be easily transformed into complete curves. This is a helpful tool for the evaluation of material properties such as the refrigerant capacity (RC). Furthermore, engineers can use this function to analyze the influence of material parameters on the design of a magnetic refrigerator.

Besides the urgent need for low cost MCE materials, the analysis of the critical behavior provides an important reference point since it is directly related to the MCE.^38,39 It has been recently experimentally shown and theoretically demonstrated, based on Arrott–Noakes equation of state, that the field dependence of the magnetic entropy change can be expressed by the following relations:^40,41

where (a and b) are constants, A is the function of the critical exponents and the exponent n depends on the magnetic state of the sample. It can be locally calculated as follows:

In the particular case of T = T_C or at the temperature of the peak entropy change, the exponent n becomes field independent:

This can be transformed using the relation:⁴² βδ = β + γ

On the basis of a mean field approach, the field dependence of the magnetic entropy change at the Curie temperature corresponds to n = 2/3.^43,44 However, by fitting the data of ΔS_M vs. μ₀H on the ln–ln scale, the values of n obtained from the slope around T_C (Fig. 10a) are 0.60. These values perfectly agree with the values reported in other soft magnetic materials.⁴⁵ On the other hand, the field dependence of RCP for our samples can be expressed as a power law by taking the field dependence of entropy change ΔS_M and reference temperature into consideration:

where δ is the critical exponent of the magnetic transition. By combining the value of n and δ (which is obtained from the slope of high-field region of RCP vs. μ₀H plot at T_C given in Fig. 10b) the obtained values of the critical exponents β and α obtained are 0.37 (3) and 1.21 (2), respectively for LSKM_0.05 compound. The obtained critical exponents were very close to the 3D-Heisenberg model. This behavior is similar to that of other CMR manganites belonging to the universality of 3D isotropic ferromagnets with short-range exchange couplings.⁴⁶ Insights into the magnetocaloric properties with an applied magnetic field can be delivered from knowing which of the theoretical models matches the experimental observations.

Fig. 10 (a) Temperature dependence of the local exponent n for La_0.7Sr_0.25K_0.05MnO₃ at different magnetic field. (b) Field dependence of RCP for La_0.7Sr_0.25K_0.05MnO₃ sample.

4. Conclusion

The results of the present investigation concerning the structural, magnetic and magnetocaloric properties of La_0.7Sr_0.25K_0.05MnO₃ manganite can be summarized as follows. The single phase of LSKM_0.05 compound, synthesized using the solid state reaction method, crystallizes in the rhombohedral structure with R3̄c space group. This sample exhibits a second-order ferromagnetic to paramagnetic phase transition around Curie temperature T_C = 332 K. A large magnetocaloric effect is observed under an applied magnetic field of 2 T, and the maximum of the magnetic entropy change ΔS_max is found to be 2.37 J kg⁻¹ K⁻¹, which is comparable to other manganites. The dependence of magnetization on temperature variation for LSKM_0.05 upon different magnetic fields was simulated. In general, this allows predicting the magnetocaloric properties such as magnetic entropy change, full-width at half-maximum, relative cooling power and magnetic specific heat change, upon different magnetic field variations. The obtained results indicate that the La_0.7Sr_0.25K_0.05MnO₃ manganite could be a potential candidate for magnetic refrigeration applications at room temperature.

Acknowledgements

This work is supported by the Tunisian National Ministry of Higher Education, Scientific Research, and the French Ministry of Higher Education and Research of CMCU 10G1117 collaboration.

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