Large magnetocaloric effect in manganese perovskite La_0.67−xBi_xBa_0.33MnO₃ near room temperature

Abstract

La_0.67−xBi_xBa_0.33MnO₃ (x = 0 and 0.05) ceramics were prepared via the sol–gel method. Structural, magnetic and magnetocaloric effects have been systematically studied. X-ray diffraction shows that all the compounds crystallize in the rhombohedral structure with the R3̄c space group. By analyzing the field and temperature dependence of magnetization, it is observed that both samples undergo a second order magnetic phase transition near T_C. The value of T_C decreases from 340 K to 306 K when increasing x from 0.00 to 0.05, respectively. The reported magnetic entropy change for both samples was considerably remarkable and equal to 5.8 J kg⁻¹ K⁻¹ for x = 0.00 and 7.3 J kg⁻¹ K⁻¹ for x = 0.05, respectively, for μ₀H = 5 T, confirming that these materials are promising candidates for magnetic refrigeration applications. The mean-field theory was used to study the magnetocaloric effect within the thermodynamics of the model. Satisfactory agreement between experimental data and the mean-field theory has been found.

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1 Introduction

Over the past few years, the perovskite manganites with ABO₃-type compounds Tr_1−xM_xMnO₃ (where Tr stands for a trivalent rare-earth element such as Bi³⁺, La³⁺ or Pr³⁺, and M for the divalent alkaline earth ions such as Sr²⁺, Ca²⁺ or Ba²⁺), have been extensively studied due to their extraordinary magnetic and electronic properties as well as their promise for potential technological applications.^1,2 A prominent feature of the mixed-valence perovskite manganite materials is an insulator–metal (IM) transition accompanied simultaneously by the paramagnetic–ferromagnetic (PM–FM) transition giving rise to a colossal magnetoresistance (CMR) effect.³ The existence of the observed CMR near the transition temperature was due to the mixed valence state of Mn, evolving from Mn³⁺ (t³_2g↑e¹_g↑, S = 2) in the parent atom LaMnO₃ to Mn⁴⁺ (t³_2g↑e⁰_g, S = 3/2) to the doped element SrMnO₃.⁴ The double exchange interaction of the neighboring spin moment of (Mn³⁺, Mn⁴⁺) coupled through oxygen ions (O²⁺), the small polaron theory and the Jahn–Teller (JT) effect have been proposed to explain the CMR phenomenon near the transition temperature.⁵ In addition, when a field is applied to this material, the unpaired spins are aligned parallel to the field. Since the total entropy of spins plus the lattice remains constant, the magnetic entropy change (−ΔS_M) is removed from the spin system and goes into the lattice, which lowers the magnetic entropy and produces a net heat. On the contrary, when an applied field is removed from a magnetic sample, the spin tends to become random, leading to increment of the entropy and causing the material to cool down. As well known, the maximum of the magnetic entropy change in this kind of material always occurs around its magnetic ordering temperature (i.e., Curie temperature, T_C). Nowadays, there is a need of new advanced magnetics materials with a second order magnetic phase transition, showing a large reversible (−ΔS_M) at low applied fields. Some theoretical works have focused on this subject, for second order phase transition via the molecular mean field theory.⁶ For this it is important to know the field dependence of a given magnetic refrigerant sample. The study of the magnetocaloric effect is not only important from the point of view of potential applications; it also provides a tool to understand the intrinsic properties of a material. In Bi based manganites, the lone pair electrons of Bi³⁺ ion hybridize with oxygen 2p orbitals, which in turn reduces the bond length of d_Bi–O and bond angle of θ_Mn–O–Mn and increases the bond length of d_Mn–O.⁷

From this viewpoint, this paper reports the structural, magnetic and magnetic entropy change of Bi-substituted perovskite manganites La_0.62Bi_0.05Ba_0.33MnO₃. It is found that these materials show quite large magnetic entropy changes induced by low magnetic field changes.

2 Experimental details

Powders of La_0.67−xBi_xBa_0.33MnO₃ (x = 0 and 0.05) were prepared via sol–gel route. In this process, La(NO₃)₃·6H₂O, Ba(NO₃)₂, Bi(NO₃)₂·5H₂O and Mn(NO₃)₂·6H₂O precursors, all with purity of 99.9%, were weighed in the desired proportions and dissolved with small amounts of water. Ethylene glycol (EG) and citric acid (CA) were used as polymerization/complexation agents, respectively, forming a stable solution. 100 cm³ of metallic salts solution was added to 300 cm³ of a solution containing a mixture of citric acid (60 g) and ethylene glycol (13 mL). This solution was then heated on a thermal plate under constant stirring, where polymerization occurs in the liquid solution and leads to a homogeneous sol. When the sol is further heated to remove the excess of solvent, an intermediate resin is formed. Calcination of the resin at 573 K in air was performed and sintering at 1073 K for 10 hours. These procedures are outlined in the flow chart of Fig. 1.

Fig. 1 A flow chart illustrating the processing procedure for La_0.67−xBi_xBa_0.33MnO₃ powders preparation.

The phase purity and structure of sample were identified by X-ray powder diffraction at room temperature using a Siemens D5000 X-ray diffractometer with a graphite monochromatized CuKα radiation (λ_CuKα = 1.5406 Å) and 20° ≤ 2θ ≤ 90° with steps of 0.02° and a counting time of 18 s per step. According to our measurements, this system is able to detect up to a minimum of 3% of impurities. The structure analysis was carried out using the Rietveld method with FULLPROF software (version 0.2-Mars 1998-LLB-JRC).⁸ Scanning electron microscopy (SEM) using a Philips XL30 equipped with a field emission gun at 20 kV was used to characterize La_0.67−xBi_xBa_0.33MnO₃ morphologies.

Magnetization (M) versus temperature (T) and magnetization versus magnetic field (μ₀H) were performed by using BS1 and BS2 magnetometers developed in Louis Neel Laboratory at Grenoble. The isothermal curves were determined in the magnetic field range of 0–5 T. The temperature interval is fixed to 2 K in the vicinity of the Curie temperature (T_C). The temperature steps were smaller near T_C and larger further away.

3 Results and discussion

3.1 X-ray diffraction and microstructure analysis

The X-ray diffraction pattern for the samples (x = 0.00 and 0.05) is shown in Fig. 2. The samples of La_0.67−xBi_xBa_0.33MnO₃ are a single phase without detectable secondary phase, within the sensitivity limits of the experiment (a few percent). The Rietveld refinements was successful considering the R3̄c (no. 167) rhombohedral and centro symmetric space group (inset (a) of Fig. 2, for x = 0.0 for example). Standard hexagonal setting of the R3̄c space group (with a_H and c_H cell parameters) was used. The manganite structure (LaAlO₃ type) is described in this hexagonal setting, with (La/Bi/Ba) atoms at 6a (0, 0, 1/4) position, Mn at 6b (0, 0, 0) and O at 18e (x, 0, 1/4) position. This distorted manganite is characterized by ā ā ā antiphase oxygen octahedral tilt system (Glazer notation⁹) corresponding to rotations along the three pseudo cubic directions of the manganite. Detailed results of the structural refinements are regrouped in Table 1. It can be observed from the inset (b) in Fig. 2 that the position of the most intense peak shows a slight shift towards low angles with the increase of Bi, indicating that the cell volume of the La_0.67−xBi_xBa_0.33MnO₃ specimens increases.

Fig. 2 X-ray powder diffraction patterns and Rietveld refinement for the compounds La_0.67−xBi_xBa_0.33MnO₃ (x = 0 and 0.05) at room temperature.

Table 1 Physical properties of the sample La_1−xBi_xBa_0.33MnO₃ system, prepared by sol–gel method at room temperature^a

Crystallographic data			Refinement conditions				Average crystallite size		Strain ε (%)	Average grain size D (nm)
Lattice parameters		Cell volume	Discrepancy factors				Debye–Scherrer technique Ds (nm)	Williamson–Hall technique Dw (nm)
a = b (Å)	c (Å)	V (Å^3)	Rwp %	Rp %	RF %	χ^2 %
a a and c: hexagonal cell parameters; V: cell volume; Rwp, Rp; RF: the residuals for, respectively, the weighted pattern, the pattern and the Bragg structure factor; χ^2: the goodness of fit. The numbers in parentheses are estimated standard deviations to the last significant digit.
x = 0
5.5160 (3)	13.5023(1)	355.78(2)	4.12	3.43	2.78	1.85	38	49	0.16	220
x = 0.05
5.50184 (2)	13.5141(4)	356.40(1)	5.23	3.42	3.42	1.25	35	40	0.14	260

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In order to quantitatively discuss the ionic match between A and B sites in perovskite compounds, a geometrical quantity, noted Goldschmidt tolerance factor (t), is usually introduced and is defined as:¹⁰

Here r_La+Bi+Ba; r_Mn and r_O are the average ionic radii of A, B and oxygen, respectively in the perovskite ABO₃ structure. Manganite compounds have a perovskite structure if their tolerance factor lies in the limits of 0.75 < t < 1 and in an ideal case when the value must be equal to unity. In the present work, the tolerance factor of La_0.62Bi_0.05Ba_0.33MnO₃ is calculated from Shannon's ionic radii (r_La³⁺ = 1.22 Å, r_Bi³⁺ = 1.24 Å, r_Ba²⁺ = 1.47 Å, r_Mn³⁺ = 0.645 Å, r_Mn⁴⁺ = 0.53 Å, r_O²⁻ = 1.35 Å)^11,12 and it is found to be t = 0.9595 and 0.9599 for x = 0 and x = 0.05, respectively, which is within the range of stable perovskite structure.

The value of average crystallite size was estimated from the full width at half maximum (FWHM) of X-ray diffraction peaks. The effects of synthesis, instruments and processing conditions were taken into consideration while making the calculation of crystallite size. The dependence of the size effect is given by Scherrer's formula: , where λ is the wavelength of CuKα radiation (λ = 1.5406 Å), K is grain shape factor (=0.89) and D_s is the thickness of the crystal. Using the Williamson–Hall (W–H) method,¹³ the average values of both D_w and lattice strain (ε) can be obtained from the intercept and the slope of the following relation, respectively,

where β is the full-width at half-maximum of an XRD peak, θ is the Bragg angle, and K = 0.9 is the shape factor. The values of average crystallite size D_S, D_W and micro-strain of La_0.67−xBi_xBa_0.33MnO₃ compounds are tabulated in Table 1. The particle size, calculated in the present system using Williamson–Hall technique, is larger than the particle size obtained from Debye–Scherrer method because the broadening effect due to strain is completely excluded in Debye–Scherrer technique.¹⁴

Fig. 3 shows the SEM photograph of the compounds. The samples contained connected particles with hexagonal shape and clear grain boundaries. These particles are largely agglomerated with a broad size distribution. The average value of thickness of both compounds is listed in Table 1.

Fig. 3 Scanning electron micrograph for La_0.67−xBi_xBa_0.33MnO₃ (x = 0 and 0.05) manganite. The inset: dispersion histogram.

After measuring the diameters of all the particles in SEM image, the size distribution histogram is fitted with the log-normal function expressed as:

Here D₀ and σ are the average diameter obtained from the SEM results and the data dispersions, respectively. The inset of Fig. 3 shows the dispersion histogram. The mean diameter 〈D〉 and standard deviation σ_D were determined using these relations:^15,16

〈D〉 = D₀ exp(σ²/2)

σ_D = 〈D〉[exp(σ²) − 1]^1/2

The results analysis showed 〈D〉 ≥ 397.48 μm and σ_D = 291.53 μm.

3.2 Bulk magnetization

Low-field magnetization (M) versus temperature was first measured for the samples, in order to have an estimation of the transition temperature (T_C). The result is presented in Fig. 4. The M(T) curve reveals that when increasing temperature, the samples exhibit a magnetic transition from paramagnetic (PM) state to ferromagnetic (FM) one. This transition occurs at the Curie temperature (T_C) which is obtained from the peak of dM/dT curve. The Curie temperature decreases from 340 K to 306 K when increase x from 0.00 to 0.05, respectively for μ₀H = 0.05 T. The Curie temperature of the Bi-doped compound was found to be lower than that of the undoped sample. This indicates that Bi substitution appears to weaken the magnetic interaction in the sample. Theoretical calculation has shown that off-center shifts of the ions with ns² electronic configuration results in the structural distortion and minimization of the Coulombic energy.¹⁷ An orientation of the 6s² lone pair toward a surrounding anion (O-2p) can produce a local distortion or even hybridization between Bi-6s-orbitals and O-2p orbitals,¹⁸ leading to the block of the movement of e_g electrons through the Mn–O– Mn bridges (stronger localization).

Fig. 4 Temperature dependence of the magnetization measured at μ₀H = 0.05 T for La_0.67−xBi_xBa_0.33MnO₃ sample (x = 0 and 0.05). Inset (a) show temperature dependence of the inverse dc magnetic susceptibility from magnetization measurements. Inset (b) variation of the magnetization as a function of applied magnetic field at 10 K.

The inset of Fig. 4 shows the temperature dependence of the inverse magnetic susceptibility of x = 0 and x = 0.05. It could be fitted to the Curie–Weiss law just above T_C (the PM region): χ = C/T − θ_CW, where θ_CW is the Weiss temperature and C is the Curie constant defined as: , where μ₀ = 10⁷ H m⁻¹ is the permeability, g is the Landé factor, μ_B = 9.27 × 10²⁴ J T⁻¹ is the Bohr magneton, k_B = 1.38 × 10²³ J K⁻¹ is the Boltzmann constant, J = L + S is the total moment and μ_eff is the effective paramagnetic moment. We can determine the effect of paramagnetic moment (μ^exp_eff) from the curie constant. The theoretical μ^calc_eff is estimated using the following expression: . The parameters μ^exp_eff and μ^calc_eff are summarized in Table 2. It is found that the μ^exp_eff is greater as compared to μ^calc_eff. This discrepancy validates the formation of ferromagnetic spin clusters within the paramagnetic state.¹⁹ A linear fit yields positive Curie–Weiss temperature θ_CW = 312 K (x = 0.05). This result confirms a mean FM interaction between spins for all samples (Table 2). Moreover, this value is higher than T_C, which may be due to the existence of short range FM ordering.¹⁹

Table 2 Transition temperature T_C, W/W₀ and Curie Weiss temperature θ_W as a function of x content for La_1−xBi_xBa_0.33MnO₃ samples

x	TC (K)	W/W0 (10^−2)	θCW (K)	θMn–O–Mn (°)	dMn–O (Å)	μ^expeff (μB)	μ^calceff (μB)
0	340	4.73	348	165.32	1.959	5.32	4.586
0.05	306	4.66	312	165.12	1.965	5.81	4.586

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The structure analysis shows that the unit cell becomes slightly larger as the 6s² lone pair character becomes dominant, it has been shown that the Bi–O bond is shorter than the La–O, despite of the similar ionic radius of La³⁺ and Bi³⁺ ions.²⁰ This can be interpreted as arising from the rather covalent character of the Bi–O bonds. The electronegativity of Bi enhances hybridisation between 6s² of Bi³⁺ orbitals and 2p of O²⁻ orbitals and this hybridisation produces a local distortion. It is observed that transition temperature T_C decreases with increase in Bi ratio. This is presumably due to tilts the MnO₆ octahedra, resulting in a reduced overlap between the Mn-3d and O-2p orbitals.²¹ It should also be noted that the La_0.67Ba_0.33MnO₃ sample is ferromagnetic while Bi_0.67Ba_0.33MnO₃ is antiferromagnetic, indicating a competition between the double exchange and the antiferromagnetic super exchange in these compounds can decrease T_C. This phenomenon has been observed in the compound Bi_0.6−xLa_xCa_0.4MnO₃.²²

3.3 Effect of Bi on magnetocaloric properties

The change of magnetic entropy of magnetic compounds has the largest value near a phase transition. According to the classical thermodynamic theory, the isothermal magnetic entropy change (–ΔS_M) produced by the variation of a magnetic field from zero to μ₀H_max is given by:^23,24

The magnetic entropy is related to the magnetization M, magnetic field strength μ₀H and absolute temperature T through the Maxwell relation:

In the case of magnetization measurement in small discrete magnetic fields and temperature interval ΔT, ΔS_M can be approximated to:

The −ΔS_M(T) data calculated from the M(μ₀H) curves (inset in Fig. 5) at different magnetic fields for the La_0.67−xBi_xBa_0.33MnO₃ (x = 0 and 0.05) are plotted in Fig. 5. The compounds exhibit large changes in magnetic entropy around Curie temperature (T_C), which is a characteristic property of simple ferromagnets due to the efficient ordering of magnetic spins at the temperature induced by magnetic field.²⁵ Large magnetic entropy changes ΔS^max_M are reported for all the samples and are summarized in Table 3. The magnitude of (−ΔS^max_M(T)) for all samples increases with increasing the applied magnetic field (inset of Fig. 5). For example, the maximum magnetic-entropy value increases from 2.37 J kg⁻¹ K⁻¹ for x = 0.00 to 2.8 J kg⁻¹ K⁻¹ (2T) and 5.8 J kg⁻¹ K⁻¹ for x = 0.00 to 7.3 J kg⁻¹ K⁻¹ for x = 0.05 respectively (5T). Guo et al.²⁶ indicated that the large magnetic entropy change in perovskite compounds could originate from the spin–lattice coupling in the magnetic ordering process. Strong coupling between spin and lattice is corroborated by the observed significant lattice change accompanying magnetic transition in perovskite manganites.²⁷ The lattice structural change in the Mn–O bond distance as well as in the 〈Mn–O–Mn〉 bond angle would in turn favor the spin ordering. Thus a more abrupt variation of magnetization near Curie temperature (T_C) occurs, resulting in a large magnetic entropy change as a large MCE.

Fig. 5 Temperature dependence of magnetic entropy change under different external fields for La_0.67−xBi_xBa_0.33MnO₃ samples. Inset: magnetization versus applied magnetic field μ₀H, measured at different temperatures, for example the sample La_0.62Bi_0.05Ba_0.33MnO₃.

Table 3 ΔS_M and RCP values for the present samples and for some previous works

Composition	TC (K)	(−ΔS^maxM)  (J kg^−1 K^−1)	RCP (J kg^−1)	μ0H (T)	Ref.
Gd5(Sr2Ge2)	275	18.5	535	5	24
Gd	294	10.2	410	5	25
La0.5Sm0.1Sr0.4Mn0.95In0.05O3	308	4.50	193.48	5	26
La0.67Sr0.33Mn0.9Cr0.1O3	328	5	—	5	27
La0.67Ba0.33MnO3	346	5.80	151	5	This work
La0.62Bi0.05Ba0.33MnO3	310	7.30	209	5	This work
Gd	297	4	120	2	28
MnFeP0.45As0.5	300	14.5	188	2	28
La0.7Sr0.3Mn0.95Ti0.05O3	308	2.2	90	2	29
La0.7Sr0.3Mn0.9Fe0.1O3	260	1.7	83	2	30
La0.67Ba0.33MnO3	343	2.37	39	2	This work
La0.62Bi0.05Ba0.33MnO3	308	2.80	80	2	This work

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The change of magnetic entropy can be also calculated from the field dependence of the specific heat by the following integration:

From this equation, it determine the change of specific heat induced by the external magnetic field zero to μ₀H as:

Fig. 6 shows the temperature dependence of ΔC_p(μ₀H, T) under different field variations for the samples (for example x = 0.05) calculated from the ΔS_M(μ₀H, T). The ΔC_p(μ₀H, T) 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 maximum/minimum value of ΔC_p(μ₀H, T) observed at 320/300 K, exhibits an increasing trend with applied field and is obtained to be 122.4/−115.43 J kg⁻¹ K⁻¹ for x = 0.05 at 5 T.

Fig. 6 Change of specific heat of the samples as a function of temperature at different magnetic field.

It should be noted that (−ΔS^max_M) is not the only parameter deciding about an applicability of material. To estimate if a material can be a good candidate for magnetic refrigeration (MR), Gschneidner and Pecharsky²⁸ defined the relative cooling power (RCPS), which is the important index which is used to evaluate the cooling efficiency of a magnetic refrigerant. It is defined as the product between the maximum values of the magnetic entropy change (−ΔS^max_M) and the full width at half maximum δT_FWHM of the magnetic entropy change curve (RCP(S) = −ΔS^max_M × δT_FWHM).²⁹ This parameter corresponds to the amount of heat that can be transferred between the cold and hot parts of the refrigerator in one ideal thermodynamic cycle. The results are summarized in Table 3. Fig. 7 shows the absolute value of RCPS and δT_FWHM for the sample (x = 0.05) versus applied field at 310 K. It can be seen from this figure that RCPS (δT_FWHM) increases monotonically as the field increases. The value of RCP is about 51% of Gd at 294 K for μ₀H = 5 T.³⁰ To evaluate the applicability of ours samples as a magnetic refrigerant, the obtained values of ΔS_M in our study are compared in Table 3 with those reported in the literature for several other magnetic compounds.^31–37

Fig. 7 Magnetic field dependence of RCP and δT_FWHM at 310 K for x = 0.05 for example.

3.4 Modeling the magnetic properties

The Weiss molecular mean field model is a standard model in magnetism. Because of its simplicity, this model is still used in current research for a wide range of magnetic materials, although its limitations are well known. This concept of a molecular field assumes that the magnetic interaction between magnetic moments is equivalent to the existence of an exchange interaction depending on the magnetization M:³⁸

H_eff = H + H_exch = H + λM

where H and H_exch are the external magnetic field, the exchange magnetic field and λ the mean field exchange parameter respectively.

Amaral et al. proposed a model based on mean field theory and presented an approach of applying this method scenario to isotherm magnetization M(T, H) measurements.³⁹ In our study, it consider the general mean field law:⁴⁰ M(H, T) = B_J[(H + H_exch)/T], the Brillouin function B_J is written as: , where , J is the total angular momentum in the lattice, g is the gyromagnetic factor (landé factor), μ_B is the Bohr magnetron and k_B is the Boltzmann's constant. The mean field exchange parameter λ is not predetermined. Then for corresponding values with the same (H + H_exch)/T, M is also the same, the value of the inverse B_J⁻¹(M) function,⁴¹

The study of the exchange field induced by the magnetization change makes it possible to find the value of the average field exchange parameter λ. Fig. 8 shows H/T versus 1/T for some of the values of M (5 emu per g per step) from 266 K to 342 K for x = 0.05. According to the mean scaling method such H/T versus 1/T curves should show a series of straight lines at different temperature. The linear relationship between H/T and 1/T is kept. Linear fits are then easily made to each isomagnetic line. Typically, the interpolation step was of 1 emu g⁻¹. The slope of this isomagnetic line, will then give the exchange field (H_exch).

Fig. 8 H/T versus 1/T curves with constant values of magnetization per curve for La_0.62Bi_0.05Ba_0.33MnO₃ compound.

For all compounds in the paramagnetic domain or the materials of domain ordered such as anti-ferromagnetic, it can always expand increasing M in powers of H, or H in powers of M. In this latter approach it stop at the third order and considering that the magnetization is an odd function of field, it can write:⁴²

H_exch = λ₁M + λ₃M³

Fig. 9 shows the evolution of the exchange field versus the magnetization for the La_0.67−xBi_xBa_0.33MnO₃ (x = 0.05 for example). The experimental data should be included for the fit by eqn (H_exch = λ₁M + λ₃M³). The results show a very small dependence on M³ (λ₃ = −1.3984 × 10⁻⁵ (T g emu⁻¹)³), is found for this second order transition system, thus H_exch = λ₁M with λ₁ = 1.25 T g emu⁻¹. After obtaining the mean field exchange parameter the next step of this method consists on building the scaling plot of M vs. (H + H_exch)/T (Fig. 10). It has successfully fitted the scaled magnetization data with the Brillouin function. From the scaling plot and the subsequent fit with the saturation magnetization equal to 72 emu g⁻¹ (this value is close to the experimental one (M_s = 69 emu g⁻¹ at 10 K) (inset b of Fig. 4), and the value of the total angular momentum of the manganite is J = 1.9.

Fig. 9 Exchange field versus magnetization for La_0.62Bi_0.05Ba_0.33MnO₃ sample, with the function λ₁M + λ₃M³ fit.

Fig. 10 Mean-field scaling plot and Brillouin function fit. Inset 1/χ versus T of La_0.62Bi_0.05Ba_0.33MnO₃ compound.

The magnetization measurements in the law temperature range show that the saturation magnetization is about M^theor_s = Jgμ_B = 3.67μ_B/f.u. This value is close to the experimental value M^exper_s = 3.7μ_B at 10 K. The value of M^theor_s per formula unit is given by: M^theor_s = (n_Mn³⁺)M_sMn³⁺ + (n_Mn⁴⁺)M_sMn⁴⁺, where M_sMn³⁺ = 4μ_B and M_sMn⁴⁺ = 3μ_B are the magnetic moments, n_Mn³⁺ = 0.67 and n_Mn⁴⁺ = 0.33 are the contents of Mn³⁺ and Mn⁴⁺ ions respectively. It deduce that the total angular momentum of the compound is J = 1.835 (it was assumed that g = 2).

From a linear approximation of the susceptibility . Inset of the Fig. 10 shows the evolution of the inverse susceptibility (1/χ) versus temperature for the La_0.62Bi_0.05Ba_0.33MnO₃ compound. The straight line represents the Curie–Weiss law with λ = 1.25 T g emu⁻¹ and T_C = 306 K. The intersection is obtained at a temperature value equals to the critical one. At this point, the susceptibility becomes infinite, which corresponds to the ferromagnetic–paramagnetic transition. Fig. 11 shows how experimental data can be described using the mean-field method. A good agreement between the experimental M(T, μ₀H) curves and the mean-field generated curves with the obtained parameters, except near the paramagnetic-ferromagnetic transition (T_C), which are not adequately described. This result is probably due to the formation of magnetic domains and critical effects.

Fig. 11 Experimental magnetization versus μ₀H (black symbols) of La_0.62Bi_0.05Ba_0.33MnO₃ sample and the interpolation using the mean field method (red lines).

3.5 Estimating the magnetocaloric effect (MCE)

The mean field approach allows us to estimate magnetic entropy variation ΔS_M within the thermodynamics of the model and without using the usual numerical integration of a Maxwell relation.⁴³ Within the mean field approach, ΔS_M between magnetic fields H₁ and H₂ can be calculated using the following general expression, which also accounts for a possible dependence of λ on T:

The linearity relation between H/T and 1/T in Fig. 8 improves that the mean field exchange parameter λ is independent of the temperature T. So the following equation will be simplified:

Fig. 12 shows the evolution of the magnetic entropy change (−ΔS_M) data as a function of temperature at several magnetic applied fields for the La_0.62Bi_0.05Ba_0.33MnO₃ compound, by using the Maxwell relation and that basing on mean field theory. Both results are in good agreement, except close to T_C, here an excepted small difference appears, due to the formation of magnetic domains and critical effects.

Fig. 12 Experimental and calculated magnetic entropy as a function of temperature for x = 0.05.

4 Conclusion

In summary, single phase La_0.67−xBi_xBa_0.33MnO₃ (x = 0.00 and 0.05) compounds were prepared by the sol–gel technic. Using the magnetization measurements, magnetic and magnetocaloric effect have been studied. These compounds show second order ferromagnetic–paramagnetic phase transition, with a large magnetic entropy change. It have studied the mean-field scaling method for these samples. The insight that can be gained from the use of this methodology for a given magnetic system can be of great interest. In a simplistic approach, it can say that if this scaling method does not follow a molecular mean-field behavior, other methods must be pursued in order to interpret the magnetic behavior of the system. The mean-field scaling method is able to determine the exchange parameters J, λ and g of ours samples. These factors allow estimating some magnetic properties.

Conflicts of interest

There are no conflicts to declare.

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