1 Introduction

Mixed-valence manganite compounds possessing the general formula M1-xAxMnO3 (where M is a rare earth and A denotes an alkaline-earth) were studied for several decades. But, in recent years, they have aroused more interest, as they exhibit exceptionally novel properties such as colossal magnetoresistance (CMR) [1,2,3] and giant magnetocaloric (MC) effect [4, 5] giving possibility to propose new technological applications. Therefore, many works are conducted on the magnetic and magneto-transport properties of these materials. In addition, the effects of divalent alkaline earth elements have been widely discussed [6,7,8]. These earlier works have shown that the Curie temperature Tc and the magnetoresistance (MR) effects are optimized for a Mn4+ content of about 33%.

Nowadays, magnetic refrigeration has proven to be a promising technology that can replace conventional refrigeration based on gas compression. This new technology, based on magnetic materials, essentially perovskite manganites, notably offers high efficiency, low volume and above all ecological cleanliness [9, 10]. In addition, it can ensure the reduction of greenhouse gases since this new technology does not use toxic gases [11]. Among these manganite materials with high magnetocaloric effect, the La0.6Ca0.4-xSrxMnO3 compounds (x = 0, 0.05 and 0.4) [12], the Pr0.8-xBixSr0.2MnO3 (x = 0, 0.05 and 0.1) [13], La1-xKxMnO3 (x = 0.11, 0.13, 0.15) [14] and (La1-xPrx)0.67Pb0.33MnO3 (0.0 ≤ x ≤ 0.3) [15] compounds.

The LaMnO3 compound has wide applications in magnetic and technological devices due particularly to its colossal magnetoresistance effect (CMR) as well as its tunable structural and magnetic properties. [16,17,18,19,20].

Boujelben et al. [21] and Ammar et al. [22] studied the substitution effect of manganese by iron in Pr0.67Sr0.33Mn1−xFexO3 and Pr0.5Sr0.5Mn1−xFexO3 respectively. They demonstrated that this substitution reduces the Mn4+/Mn3+ ratio and causes a decrease in the Curie temperature in the substituted compounds. Electrical characterizations show that the substitution of manganese by iron in the Pr0.67Sr0.33Mn1−xFexO3 compounds [21] kept the semiconductor–metal transition with a significant reduction in the electrical transition temperature.

Rahmouni et al. have explored the effect of the substitution of manganese by iron (Fe3+) on the electrical properties of the La0.67Ba0.33Mn1-xFexO3 (0 ≤ x ≤ 0.2) compounds [23]. This work showed that electrical conductivity decreases with increasing iron concentration. Indeed, the conductivity in manganites is ensured by the double exchange mechanism but for samples whose manganese is doped with iron, the jump of Mn3+ electrons to Fe3+ is prohibited since the iron eg band is completely filled. The lowering of the conductance is explained by the reduction in concentration of the electrons participating in the jump. In addition, the metal–semiconductor transition is completely suppressed from x = 0.15.

For this reason, the La0.67Ba0.33MnO3 compound was investigated in order to study the effect of substituting lanthanum with europium on the structural, electrical and dielectric properties for x = 0 and 0.1. Then, the La0.67Ba0.33MnO3 was chosen to improve the dielectric properties by doping with Eu in A-site and with Fe in B-site. In addition, we focused on the magnetic properties of these compounds to see if these compounds may have multiple properties.

For these compounds, the introduction of Barium and Europium in site A which have different ionic radii (rLa3+ = 1.36Å, rBa2+ = 1.61Å et rEu3+ = 1.12Å) causes the variation of the cationic disorder in A-site of the studied compounds leading to a variation in the Curie temperature.

The modification of the cationic disorder in A-site causes variations in the average bond length 〈Mn–O〉 and the average angle 〈Mn–O-Mn〉 leading to a tilting of the MnO6 octahedrons.

Concerning manganite materials with ferromagnetic (FM) to paramagnetic (PM) transition, a phenomenological model has been proposed by Mahmoud Aly Hamad [24]. This theoretical model has been tested and has proven its reliability on several manganite compounds. In addition, it is widely used to simulate the temperature evolution of magnetization and to study MC properties, such as magnetic entropy change, relative cooling power (RCP), and heat capacity change [12, 25]. Moreover, this theory has been confirmed for many materials with a magnetic transition such as Pr0.8-xBixSr0.2MnO3 (x = 0,0.05 and 0.1) [26], La0.67Ca0.33MnO3 [27], La0.67Sr0.33MnO3 [27] and La0.7Ca0.3MnO3 [28].

On the other hand, the applied magnetic field required to promote MCE in lanthanum manganites is much lower than that required for gadolinium and its alloys [29].

It is important to note that the reliability of the studied model was also confirmed for a magnetic system with multiple magnetic transitions such as charge ordered Pr0.5Sr0.5MnO3 [30] and Pr0.5-xGdxSr0.5MnO3 (0 ≤ x ≤ 0.1) compounds [31]. On the other hand, other works have demonstrated that the proposed phenomenological model is unable to give a rigorous estimate of the magnetocaloric effect in FM materials contrary to what was expected [32]. The unreliability found is attributed not to the phenomenological method but to the erroneous assumption made in the derivation of the entropy change expression, namely that the model parameters are independent of the strength of the applied magnetic field.

The present work is devoted to the verification of a known theoretical model which is the Hamad model to predict the magnetocaloric data of a magnetic material in order to select materials suitable for magnetic refrigeration.

The advantage of Hamad's phenomenological model is that based on theoretical considerations and from a simple measurement of M(T), one can predict whether or not a material exhibits significant magnetocaloric properties. Previous works has also shown that this model is closer to reality than, for example, the Landau model which deviates at low temperature (T < TC) from the experimental curve derived from M(H, T).

2 Experimental details

Polycrystalline samples of nominal composition La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) were successfully prepared by means of the sol–gel route using high purity (up to 99.9%) dried MnO2, La2O3, BaCO3, Eu2O3 and Fe2O3 precursors on the basis of the following chemical reaction:

$$\begin{array}{c}{\frac{\left(0.67-x\right)}{2}\,La}_{2}{O}_{3}+\frac{x}{2}{ Eu}_{2}{O}_{3}+0.33\;BaC{O}_{3}+0.85\;Mn{O}_{2}+0.075\;{Fe}_{2}{O}_{3}\\\to {La}_{0.67-x}{Eu}_{x}{Ba}_{0.33}\;{{Mn}_{0.85}{Fe}_{0.15}}\;O_{3}+\delta\;C{O}_{2}+\delta^{\prime}{O}_{2}\end{array}$$
(1)

These samples were elaborated in air, then they can be stoichiometric in oxygen [33].

In a previous work [34], the powder X-ray diffraction (XRD) of these samples, determined at room temperature, was verified, using Cu-Kα radiation (λ = 1.54056 Å). The obtained results, analyzed by the Rietveld method based on the Fullprof program [35] demonstrated that our samples are single-phase and crystallize in the orthorhombic structure with the Pbnm space group.

The electrical and dielectric properties of these compounds were also studied using the impedance spectroscopy technique in the temperature range of 80 to 400 K and the frequency range of 40 to 107Hz [34]. DC measurements showed that the studied compounds present a semiconductor (SC) behavior in all the temperature range of study. Further, analysis of AC data shows a transition from SC to metallic behavior at the transition temperature 160 < TSM < 170 K and 220 < TSM < 240 K for x = 0 and 0.1 respectively. On the basis of Nyquist plots, an equivalent circuit involving the contribution of grains and grain boundaries in the conduction process was proposed. The dielectric measurements revealed also that dielectric relaxation is governed by the DC conduction process. The dielectric results further confirm the high permittivity values which confirm the importance of the studied compounds in electronic applications.

In addition, magnetic measurements were performed using a vibrating sample magnetometer in the temperature range of 5–335 K and under applied magnetic fields up to 7 T.

In this research work and in order to continue our investigation, we used the theoretical phenomenological model of Hamad to simulate experimental results and predict the magnetic, the magnetocaloric properties and the specific heat capacity values of these compounds.

3 Theoretical study

Hamad’s theory [26] was based on the magnetization measurements which evolution with temperature is given by:

$$M\left(T\right)=\frac{\left({M}_{1}-{M}_{2}\right)}{2}tanh\left[A\left({T}_{C}-T\right)\right]+BT+C$$
(2)

In this relation:

  1. (a)

    M1 and M2 designate, the initial and the final magnetizations values at FM and PM transition respectively, as shown in Fig. 1.

  2. (b)

    B represents the magnetization sensitivity given by: \(B=dM/dT\) in the FM region (before transition).

  3. (c)

    The value of C is given by: \(C=\frac{\left({M}_{1}+{M}_{2}\right)}{2}-B*{T}_{C}\).

  4. (d)

    \(A=2\frac{\left(B-{S}_{C}\right)}{\left({M}_{1}-{M}_{2}\right)}-B*{T}_{C}\). In this relation, SC is the magnetization sensitivity at TC (\({S}_{C}=dM/dT\)).

Fig. 1
figure 1

Evolution of the magnetization curve with temperature T under constant applied magnetic field

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According to the studied theory, the magnetic entropy change resulting from the spin ordering under isothermal magnetic field change (varying from 0 to µ0Hmax) is given by the following relation [24]:

$$\Delta {S}_{M}(T)=\left[-A\frac{\left({M}_{1}-{M}_{2}\right)}{2}{sech}^{2}\left[A\left({T}_{C}-T\right)\right]+B\right]{\mu }_{0}{H}_{max}$$
(3)

In this relation,\(\text{sech}\left(x\right)=1/\text{cosh}(x)\)

Among others, at Curie temperature TC, \(\Delta {S}_{M}(T)\) reaches its maximum value \(\Delta {S}_{Max}(T)\) given by:

$$\Delta {S}_{Max}(T)=\left[-A\frac{\left({M}_{1}-{M}_{2}\right)}{2}+B\right]{\mu }_{0}{H}_{max}$$
(4)

An important parameter of the isothermal entropy change \(\Delta S\) is its full width at half maximum δTFWHM determined by: \(\Delta S=\Delta {S}_{Max}(T)/2\) which gives [27, 36]:

$$\delta {T}_{FWHM}=\frac{2}{A}{cosh}^{-1}\left[\sqrt{\frac{2A\left({M}_{1}-{M}_{2}\right)}{A\left({M}_{1}-{M}_{2}\right)+2B}}\right]$$
(5)

Then, the relative cooling power (RCP) describing the magnetic cooling efficiency of a manganite material can be given by:

$$RCP=-\Delta {S}_{M}(T,{H}_{Max})\delta {T}_{FWHM}=\left({M}_{1}-{M}_{2}-2\frac{B}{A}\right){\mu }_{0}{H}_{max}\text{x}{cosh}^{-1}\left[\sqrt{\frac{2A\left({M}_{1}-{M}_{2}\right)}{A\left({M}_{1}-{M}_{2}\right)+2B}}\right]$$
(6)

Moreover, the heat capacity which represents also a crucial parameter to assess the importance of a magnetic material, can be determined from the magnetic contribution to ΔSM(T) based on the following expression [27]:

$$\Delta {C}_{p}\left(T,{\mu }_{0}H\right)={C}_{p}\left(T,{\mu }_{0}H\right)-{C}_{p}\left(T,0\right)=T\frac{\partial \left[\Delta {S}_{M}\left(T,{\mu }_{0}H\right)\right]}{\partial T}$$
(7)

This relation can be also written as [13, 37]:

$$\Delta {C}_{p}\left(T,{\mu }_{0}H\right)=-T{A}^{2}\left({M}_{1}-{M}_{2}\right)\text{x }tanh \left[A\left({T}_{C}-T\right)\right]{\mu }_{0}{H}_{Max}{sech}^{2}\left[A\left({T}_{C}-T\right)\right]=T\frac{\partial \left[\Delta {S}_{M}\left(T,{\mu }_{0}H\right)\right]}{\partial T}$$
(8)

4 Results and discussions

Figure 2(a) and (b) illustrates the evolution of magnetization with temperature under an applied magnetic field of 0.05T for the La0.67-xEuxBa0.33Mn0,85Fe0,15O3 (x = 0 and 0.1) studied compounds.

Fig. 2
figure 2

Temperature evolution of the magnetization M(T) determined for the La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0,1) compounds under an applied magnetic field of 0.05 T: (a) x = 0 and (b) x = 0.1. In each figure we have shown the evolution of dM/dT with temperature to determine the Curie temperature TC

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As can be clearly seen, these materials exhibit a FM to PM transition when the temperature increases. On the basis of dM/dT plots, the Curie temperature was estimated for each compound which is 170K and 100K for x = 0 and 0.1 respectively. These determined values will be used to fit the magnetization data as a function of temperature.

Figure 3(a) and (b) demonstrates the evolution of the isothermal magnetization as a function of the applied magnetic field M(µ0H) measured under an applied magnetic fields up to 7T for our samples. Based on these curves, the temperature dependence of the magnetization M(T) for different applied magnetic fields up to 7T was plotted in Fig. 4. Using the values extracted from M(T) at µ0H = 0.05T, the magnetization data M(T) for applied magnetic fields varying from 1 to 7T were simulated on the basis of relation (2). As seen in Fig. 4, a good agreement is obtained between experimental and obtained simulated data. This result confirms the reliability of the phenomenological model proposed by Hamad. The fitted model parameters are gathered in Table 1. This agreement was verified also in our previous work with the Pr0.8−xBixSr0.2MnO3 (x = 0, 0.05 and 0.1) compounds [26].

Fig. 3
figure 3

Isothermal magnetization curves for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds

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Fig. 4
figure 4

Temperature dependence of the magnetization at different applied magnetic fields for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds. Symbols represent the experimental data and solid lines represent the simulated results

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Table 1 Parameters fit from Eq. (2) for the La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds at different applied magnetic fields
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Using the results presented in Table 1, the evolution of the magnetic entropy change (− ΔSM) with temperature at different applied magnetic fields was calculated using Eq. (3). Obviously, this parameter was determined under the same applied magnetic fields in order to compare the predicted values with those calculated from magnetic measurements.

Results, illustrated in Fig. 5, demonstrate a good agreement between the experimental data and those determined theoretically with the proposed model. Besides, the obtained negative values of the magnetic entropy change confirm the FM behaviour of the samples under investigation [13].

Fig. 5
figure 5

Temperature dependence of the magnetic entropy change |-ΔSM(T)| under different applied magnetic fields for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds a x = 0, and b x = 0.1. Symbols represent the experimental data and solid lines show the predicted values of |-ΔSM(T)| using Table 1

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Note especially that the main advantage obtained with this phenomenological model is the prediction of the magnetocaloric properties of any studied compound (possessing magnetic properties) under any applied magnetic field without carrying out the experimental measurements. The obtained results demonstrate that our samples show a large variation in the magnetic entropy change which can be explained by the double exchange mechanism between the Mn3+ and Mn4+ ions via the oxygen ions proposed by Zener [38]. As known, this model favors FM interactions due for example to the jump of electrons. Indeed, Zener's double exchange mechanism consists in the delocalization of eg elctron, which takes place via the 2p orbitals of oxygen ion. The eg electron hopping promotes the FM interactions and favors metallic tendencies.

It can be noticed that, the evolution of the magnetic entropy change ΔSM(T,µ0H) in perovskite magnetic materials, is mainly due to the coupling between the applied magnetic field and the magnetic sublattice. Particularly, this coupling affects the magnetic part of the total entropy due to the evolution of the corresponding magnetic field [39, 40]. Moreover, and as can be seen, ΔSM(T,µ0H) is maximal near the Curie temperature TC corresponding to the studied materials where the M(T) curves show a rapid evolution with temperature [41].

Using the magnetocaloric results shown in Fig. 5 and according to Franco and Conde [42], a universal phenomenological curve can be represented based on the following relations:

$$\theta =\left\{\begin{array}{cc}-\frac{\left(T-{T}_{C}\right)}{\left({T}_{1}-{T}_{C}\right)}, & T\le {T}_{C}\\ \frac{\left(T-{T}_{C}\right)}{\left({T}_{2}-{T}_{C}\right)}, & T>{T}_{C}\end{array}\right.$$
(9)

In these relations, θ designate the rescaled temperature and T1 and T2 are the temperatures of the two reference points obtained for \(\Delta {\text{S}}_{\text{M}}=\left(-\Delta {S}_{M}^{Max}\right)/2\). In fact, at fixed applied magnetic field µ0H, ΔSM(T) is normalized using its peak entropy change \(\left(-\Delta {S}_{M}^{Max}\right)\) and the temperature is rescaled below and above the Curie temperature.

Using relation (9), the variation of the normalized entropy change as a function of θ for different applied magnetic fields was reported in Fig. 6.

Fig. 6
figure 6

Normalized entropy change vs. rescaled temperature θ for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds at different applied magnetic fields

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As clearly illustrated and for each sample, all experimental values collapse into one universal curve which confirms the reversible behavior of the second-ordered magnetic phase transition for the studied compounds.

On other hand and around TC, the evolution of the maximum entropy change with the magnetic field can be described with the following expression [43]:

$$-\Delta {S}_{Max}\left(T\right)=\alpha {\left({\mu }_{0}H\right)}^{n}$$
(10)

In this relation, n depends on the magnetic state of the studied compounds and α is a constant.

Figure 7 illustrates the variation of the maximum entropy change |-ΔSMax| and the related fit with relation (10) for La0.67-xEuxBa0.33Mn0,85Fe0,15O3 (x = 0 and 0.1) compounds at different applied magnetic fields. This fit gives n = 1.0999 and 1.3089 for x = 0 and 0.1 respectively.

Fig. 7
figure 7

Evolution of the maximum entropy change |-ΔSMax| for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0,1) compounds at different applied magnetic fields

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On the basis of the theoretical (− ΔSM) values, we have illustrated in Fig. 8, the evolution versus applied magnetic field of δTFWHM and RCP which is considered as the most important parameter to evaluate the MCE relative to our compounds.

Fig. 8
figure 8

Evolution of a δTFWHM and b RCP as a function of applied magnetic field for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds

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As we can notice, the doping with europium reduces the maximum entropy change as well as the δTFWHM which is not beneficial for magnetic refrigeration.

In fact, the reduction in δFWHM can be explained by the increase in grain size estimated at 22 and 28 nm for x = 0 and 0.1, respectively [30]. As a matter of fact, an increase in the particle size induces a reduction in spins near the surface inducing a FM coupling higher than that of bulk grains. Consequently, a Curie temperature distribution may appear which may be the cause of the decrease in |-ΔSM|, δTFWHM and consequently RCP [30].

In Fig. 9, we have compared the theoretical values of RCP calculated with the studied phenomenological model with those obtained from experimental data. A satisfactory agreement is obtained confirming the reliability of the proposed phenomenological model.

Fig. 9
figure 9

Comparison between the RCP values obtained from experimental data (RCPexp) and those calculated from the phenomenological model (RCPth) for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds

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This result shows that the RCP values can be predicted directly by the M(T) measurements and subsequently the efficiency of a magnetic material can also be predicted.

Figure 10 shows the evolution of Ln(RCP) vs. Ln(μ0H) for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds. The obtained linear fit permits to obtain the critical exponents [23].

Fig. 10
figure 10

Ln(RCP) vs. Ln(μ0H) for La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds

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Another important parameter can be predicted from this model instead of doing the experimental measurements is the heat capacity. Indeed, Fig. 11(a) and (b) illustrates the temperature evolution of the predicted heat capacity under different applied magnetic fields relative to the La0.67-xEuxBa0.33Mn0.85Fe0.15O3 compounds. As shown, the magnetic field evolution of the specific heat ΔCp(T, μ0H) deviates from zero only in the vicinity of the transition temperature TC for the two studied compounds. Besides, ΔCp(T, μ0H) values are negative below TC and are positive above.

Fig. 11
figure 11

Temperature evolution of the predicted heat capacity under different applied magnetic fields for the La0.67-xEuxBa0.33Mn0.85Fe0.15O3 compounds a x = 0 and b x = 0.1

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In Table 2 we have gathered the estimated values of ΔCp,max and ΔCp,min at μ0H = 5 T for our studied compounds.

Table 2 Specific heat values obtained at μ0H = 5T for the La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) compounds
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5 Conclusions

In conclusion, we have synthesized polycrystalline manganites with formula La0.67-xEuxBa0.33Mn0.85Fe0.15O3 (x = 0 and 0.1) using the sol–gel method. Experimental magnetic results show that these compounds exhibit a second order ferromagnetic FM to paramagnetic PM transition with increasing temperature. The magnetic and magnetocaloric properties of these compounds were studied on the basis of the phenomenological model proposed by Mahmoud Aly Hamad.

The present study shows a good agreement between experimental measurements and the magnetic and magnetocaloric results simulated by the studied model.

The full-width at half-maximum, the relative cooling power as well as the specific heat capacity values at several applied magnetic fields is also calculated.

On the basis of the obtained results, the magnetic parameters such as the Curie temperature, around which the application of the magnetic refrigeration is focused, as well as the related parameters of this application can be controlled.

The predicted results are shown to decrease with doping by Europium. Despite that, we can conclude that the prepared compounds may be promising candidates for magnetic refrigeration at low temperatures.