1 Introduction

All magnetic materials exhibit the magnetocaloric effect, which opens a wide range of research opportunities to develop active materials that are ideal for all applications. Gadolinium is the reference material for magnetic cold at ambient temperature. Due to its purity, this element also has the benefit of being a combination that is simple to acquire. It develops technologically as a result of the simplicity of implementation work made possible by its excellent ductility and malleability. As a result, it is utilized in the most. It is not, however, a viable magneto-refrigerating material for consumer applications because to its prohibitive cost (up to 3500 €/kg) and limited reserves. That is why, it is crucial to locate another magnetocaloric substance, and hence the potentials of Manganites as an energetically clean alternative to remedy this problem and to be an effective alternative for magnetic refrigeration. Some of the properties of magnetic materials are highly dependent on the applied mechanical and thermal stresses, as well as the magnetic field. Also, the key requirement of magnetic refrigeration is to have proper magnetocaloric materials, which can produce a large magnetic entropy change [1] and a good magnetic reversibility at low magnetic fields and with a wide temperature range [2].

These compounds are characterized by a fairly simple crystallographic structure as well as a flexibility of substitution. Among the compounds with perovskite structures, titanate and its derivatives are the subject of several studies thanks to their giant magnetic, magnetocaloric and dielectric properties. These properties encourage researchers to use it in certain applications such as industries due to the high values of the colossal magnetoresistance [3, 4] and giant magnetocaloric. These parameters are essential to achieve high magnetic refrigeration efficiency [5]. All this proprieties make them likely to be used in the field of information storage (magnetic memories and reading heads) or the development of electrical components (temperature reversible circuit breakers) [6,7,8,9,10] or used for development of material catalysts for electronic devices with low power consumption [11]. In addition, the titanium structure is very sensitive to substitution at site A and/or at B site level, which is manifested by a great richness in the magnetic and structural phase diagrams assembling between the variations of macroscopic parameters such as temperature, pressure, size of the doping ions, elaboration method and annealing temperature. What is trusty to mention is that it is crucial to understand and master the ferromagnetic processes, microstructural evolution, and the impact of atomic flaws on the characteristics of magnetic materials. The dependencies of the universality class, which are directly tied to the dimensionality of the space and the order parameter, are a fundamental concern that is always raised and related to the dimensionality of the space and the order parameter.

The major interest of the present research work is the study of the effects of 5% of titanium substitution on the physical properties of the compound formula La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3 prepared by the solid–solid method, drawing a special attention to magnetic properties. This compound has been characterized by SQUID magnetometer for magnetization measurements at different temperatures and in different applied magnetic fields. Indeed, the variation of the specific heat was determined and that of the adiabatic temperature ΔTad is carried out by exposing a thermally insulated material to the magnetic field [12] and was calculated to measure the Relative Cooling Power RCP (T) to determine the magnetic properties of our material and mark it’s future applicability in magnetic refrigeration technology [13]. We made a very low doping rate (5%) and we recorded very interesting magnetic properties. A very important relative cooling power (33.2617% J kg−1 per compared to Gd) which makes our sample useful in magnetic refrigeration technology although the doping rate is very low. Besides, the local exponent n was calculated from the critical exponents which are lower than those deduced from the power law (|ΔS (H)|∞ Hn).

Finally, a comparison between the magnetocaloric effect of the parent compound and the studied sample was carried out to develop a single sample with a significant magnetocaloric effect. This allows us to manufacture composites from these two samples to have a large Curie temperature range that covers room temperature and with comparable RCP values.

2 Experimental details

The powder sample of La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3 was synthesized using the standard solid state reaction method at high temperature. All the detailed preparation procedure and basic physical properties are reported in Ref. [11]. The magnetization measurements were conducted using a SQUID magnetometer developed at Neel institute Laboratory in Grenoble.

3 Results and discussion

3.1 Magnetic characterization

According to Oesterreicher et al. [12], the dependence of the experimental data of ΔSM of sample studied as a function of applied magnetic field can be determined by the relationship:

$$\left| {\Delta S \, \left( H \right) \, } \right|\infty \, H^{n}$$
(1)

with n as the local exponent depending on the magnetic state of the material.

The obtained values of n by a linear fit of the curves are shown in Fig. 1. The local exponent was found to be 0.78. Moreover, n can be determined locally as follows [13]:

$$n \, = \frac{{{\text{d }}\;{\text{Ln}} \left| {\left( {{\Delta }S_{M} } \right)} \right|}}{{{\text{d Ln}}H}}$$
(2)
Fig. 1
figure 1

Variation of Ln(ΔSM) as a function of Ln(H) for La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3

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Figure 2 represents the temperature dependence of the exponent n for the different values of the applied field. It should be noted that n evolves with the field in the entire temperature studied range. In agreement with previous experimental data [14,15,16,17], the general behavior of n(T) follows the following pattern: n close to 1 for T < TC, which indicates that at these temperatures the magnetization variation does not depend on the magnetic field. Furthermore, for T = TC, n expects a minimum. In the mean field approximation, this is 2/3 as predicted by Osterreicher and Parker [18], but in the general case, n is different from this value and depends on the critical exponents. First, n gradually decreases for values close to 0.764 (for H = 5 T) at Tpeak, after that it increases for values close to 2 in the PM region, which is in accordance with the Curie–Weiss law in the paramagnetic domain. Consequently, in their general behavior, these variations of n (T, H) accord swell with those found in previous studies. This result indicates that the usefulness of the local exponent n gives precise information on the magnetic state, showing the inhomogeneous character existing in our material. Indeed, manganites are inhomogeneous materials even for good quality crystals [19]. Besides, for the studies of the local exponent n, it is very important to study the Gibbs free energy (G) to classify the transitions of the sample under study. The free energy G as a function of the total magnetization can be developed, while neglecting the terms having the very high powers in the equation that follows:

$$G \, (M, \, T) \, = \, G_{0} + \frac{1}{2} A \, \left( T \right) \, M^{2} + \frac{1}{4}B \, \left( T \right) \, M^{4} + \frac{1}{6}C \, \left( T \right) \, M^{6} {-}MH$$
(3)
Fig. 2
figure 2

Variation of the local exponent n as a function of the temperature of La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3 under different magnetic fields

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The coefficients A (T), B (T) and C (T) express the Landau parameters that were determined from the equation quoted above by plotting μ0H as a function of the magnetization M [20].

Figure 3 shows the evolution of Landau parameters A (T), B (T) and C (T) as a function of temperature for our compound. As expected, A (T) is positive with a minimum at TC corresponding to a maximum at the level of susceptibility. B (TC) = 0, which allows us to conclude that the transition is of a second order. C (TC) is > 0, which is in good agreement with the Landau model [11]. Moreover, the variation of magnetic entropy ΔS (T) can be determined differently using the parameters A (T), B (T) and C (T), according to the following equation:

$$\Delta S_{M} \left( T \right) \, = \, S_{M} \left( {T, \, 0} \right) \, - \, S_{M} \left( {T, \, H} \right) = \, - \frac{1}{2}A^{{\prime }} \, \left( T \right)(M_{0}^{2} - \, M^{2} ) - \frac{1}{4}B^{{\prime }} \, \left( T \right)(M_{0}^{2} - \, M^{4} ) - \frac{1}{6}C \, \left( T \right)(M_{0}^{2} - \, M^{6} )$$
(4)

where A′ (T), B′ (T), C′ (T) are the first derivatives of the coefficients A (T), B (T), C (T) with respect to the temperature and M0 is the value of the magnetization obtained at zero field. To complete the studies on the physical properties of our compound, we carried out measurements of the specific heat Cp carried out between different temperatures at two applied magnetic fields 1 T and 5 T (Fig. 4). It should be mentioned that the studied material is in the solid state, so one can neglect its compressibility and acquire its specific heat at constant pressure (Cp) to that at constant volume (Cv) shown in Fig. 4. The temperature dependence of the Cp(T) curve shows rapid crossing at specific temperatures. Inside this curve, we have enlarged these areas. This transition confirms the Curie temperature given by the magnetic measurements of the magnetization [11] where the rate of change of the magnetization is quite large. This singularity is known as a lambda-like anomaly [21, 22] which can be found in ferromagnetic materials where there are many magnetic interactions. The absence of extended magnetic interactions prevents the occurrence of phenomena related to magnetic properties such as lambda type transition. This peak expands with increasing magnetic field. We therefore observe a change in temperature from 206 K [11] to 215 K when the field goes from 1 to 5 T. This same phenomenon has already been reported for La Sr-based manganites [19]. In addition, the height of the peak is reduced and its width is enlarged with increasing applied field. This is a consequence of the widening of the transition zone, and the rapid decrease in the rate of change of magnetization with the applied field at the transition temperature [23]. This peak is also due to the particularly high values around the PM-FM transition temperature, where the rate of variation of the magnetization is quite large. This anomaly has also been attributed to the possible chemical inhomogeneity of the sample [24]. Moreover, the large increase in the transition region and the reduction in the rate of change of the magnetization temperature with doping are consistent with the behavior of the specific heat anomaly. Also, this peak is explained by the fact that before the transition, the species are mainly in the form of pairs (spins, magnetic dipoles and dissimilar atoms); after the transition, they are mainly in the form of singles. Before transition, couples and singles are in balance; after the transition the pairs and singles are in balance. But during the transition, a state composed mainly of pairs changes to a state composed mainly of singles. So there is a change in the energies associated with a chemical reaction in which one mole of pairs is converted into two moles of singles. The change in magnetic entropy ΔSCp was calculated according to the following equation [25]:

$$\Delta S_{{{\text{Cp}}}} (T, \, H) = \mathop \int \limits_{0}^{T} \frac{{{\text{Cp}}\left( {T,H} \right) - {\text{Cp}}\left( {T,0} \right)}}{T} {\text{d}}T$$
(5)
Fig. 3
figure 3

Variation of Landau parameters A(T), B(T) and C(T) whose units are T2 kg/J, T4 kg3/J3 and T6 kg5/J5, respectively

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Fig. 4
figure 4

Variation of specific heat as a function of temperature under two applied magnetic fields 2 T and 5 T for La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3

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The magnetocaloric behavior of our system given in Fig. 5 is identical to that determined by magnetic measurements. By combining the variation of the magnetic entropy ΔSM and the specific heat Cp at a constant magnetic field μ0H, we can determine the variation of the adiabatic temperature ΔTad (Fig. 6) ΔTad which is determined by [26, 28]:

$$\Delta T_{{{\text{ad}}}} = \, - \mathop \int \limits_{0}^{H} \frac{T}{{{\text{Cp}}\left( {T,H} \right)}}\left( {\frac{\partial M}{{\partial T}}} \right)_{{\text{H}}} {\text{d}}H \, = \, - \, T\frac{{{\Delta \text{SM}}\left( {T,H} \right)}}{{{\text{Cp}}\left( {T,0} \right)}}$$
(6)
Fig. 5
figure 5

Variation of magnetic entropy as a function of temperature under two magnetic fields 2 T and 5 T of La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3

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Fig. 6
figure 6

Variation of adiabatic temperature as a function of temperature under different magnetic fields of La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3

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The ΔTad variation at 5 T is 1.9 K (Table 1).

Table 1 Values of ΔTad, RCP (T) deduced from ΔTad and the RCP (S) deduced from magnetic measurements, β, γ and σ2 for La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3
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Our results are comparable with the indicated adiabatic temperature change for La0.7Sr0.3MnO3 which is around 3 K at 5 T [26], 3.3 K for the compound La0.8Sr0.2MnO3 at 7 T [25], and 3.33 K for La0.67Sr0.33MnO3 [28]. Additionally, Lin et al. [29] reported the case of Gd-based compounds. Ban et al. [30] reported that ΔTad for Gd45Tb55 is 5.8 K at TC = 256 K, while for Gd, ΔTad = 5.6 at 5.8 K. In Fig. 7, we represent the variation of the magnetic entropy and the adiabatic temperature as well as the specific heat for the compound at 5 T. There is a good agreement between the shapes of ΔTad and ΔSM. While the FM-PM transition is well visualized even for Cp, there is a small deviation in the TC temperature. This shift is due to the measurement method and experimental error. Figure 8 indicates the Variation of \(\Delta T_{{{\text{ad}}}}^{\max }\) as a function of the applied magnetic field of the compounds. It can be noticed that the peak of the maximum of the variation of the adiabatic temperature increases with the applied magnetic field. A curve fit allows us to determine the parameters b and p of the power law defined by [31]:

$$\Delta T_{{{\text{ad}}}}^{\max } = \, b\left( {\mu_{0} H} \right)^{p}$$
(7)
Fig. 7
figure 7

The variation of the magnetic entropy ΔSM, of the adiabatic temperature ΔTad and specific heat CP of La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3 under a field of 5 T

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Fig. 8
figure 8

Representation of Ln (M) vr. Ln (H) for La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3

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The obtained values of b and p are 0.377 and 0.967, respectively. The values of the exponent b seem to be deviated by 2/3, this is the predicted value for the mean field theory [11].

3.2 Critical isotherm

For a temperature equal to Curie, the variation of the magnetization as a function of the field can be given by a power law characterized by a critical exponent δ indicated in Eq. 8:

$$M \, = {\text{ DH}}^{1/\delta }$$
(8)

This critical exponent can be obtained from the Ln (M) versus Ln (H) diagram as shown in Fig. 9. According to Eq. 8, the variation of Ln (M) as a function of Ln (H) must be a straight line with a slope equal to 1/δ. The values of δ deduced from this method are in good agreement with those found by the mean field model for our compound [11]. The critical exponent δ is also determined from Widom's relation [32] which gives a relation between the three critical exponents β, γ and δ as shown in Eq. 9 [33]:

$$\delta = 1 \, + \, \gamma / \, \beta$$
(9)
Fig. 9
figure 9

Scale curve indicating the universal aspect one below and above TC for La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3. Insets show the same data for a double logarithmic scale

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Using the values of β and γ estimated from the method of modified Arrott plots [11], we obtained new values of δ, which are presented in Table 1. It is necessary to point out that there is a good agreement between the experimental and calculated values of δ. Furthermore, the critical exponents found in this study obey Widom's scaling relation, confirming that the estimated values are self-consistent and acceptable. The value of the critical exponent γ is illustrated in Table 1.

3.3 Scale behavior

The estimation of the credibility of the critical parameters can be carried out by means of the scaling law, which predicts that in the critical region, the internal field and the magnetization must confirm to universal scale behavior. Based on the equation below:

$$\left( \frac{H}{M} \right)^{1/\delta } = \frac{T - Tc}{{T1}} + \left( \frac{M}{M1} \right)^{1/\beta }$$
(10)

We plot the variation of M/|ε|β as a function of H/|ε|β+γ for temperatures around TC, where ε = (T − TC)/TC is the reduced temperature. We have arrived at two universal branches, one for temperatures below TC and the other for those above TC, either on a linear or logarithmic scale as shown in Fig. 9. The scaling behavior is well verified for our compound and all points are formed on two curves. This demonstrates the accuracy and good agreement of the calculated values of the critical and TC exponents with the scaling hypothesis. The substitution of the 5% titanium in the manganese site results from a change of the critical exponents i.e., while the mother sample belongs to the 3-dimensional Heisenberg model, our system is compatible with the model of mean field. To correlate disorder with critical behavior, Fisher et al. [34, 35] have evaluated a computation using renormalization group theory [34, 35], predicting that in the case of a ferromagnetic system with an exchange interaction of the form:

$$J\left( r \right) \, = \, 1/r^{d + \sigma }$$
(11)

with d as a dimension of the space and σ the gap of the interaction. It is to be noted that the variance σ designates the effect of the distribution of the different cations in the Site B. This effect is called the “mismatch” effect and it is defined by [36]:

$$\sigma^{2} = \, \Sigma r_{i} \chi_{i}^{2} - \left\langle {r_{B} } \right\rangle^{2}$$
(12)

Here ri indicates the radius of the cation i in the B site (Mn3+, Mn4+ and Ti3+) and xi is fraction occupancy (Σxi = 1). The calculated values of the variance σ2 are grouped in Table 1. We notice that the critical exponents of our sample belong to the mean field model.

3.4 Study of disorder

Harris [37] discussed the effect of disorder on the critical behavior of magnetic systems When the pure critical exponent αpure of the pure system is positive, the disorder acts on the critical exponents and results in a modification of the critical behavior governed by a new random fixed critical point. Conversely, if αpure is negative, the disorder is irrelevant. The exponent for the αpure system is determined by the Rushbrooke scaling relation [38] given by:

$$\alpha_{{{\text{pure}}}} + \, 2 \, \beta \, + \, \gamma = \, 2$$
(13)

Based on this relationship, we found that the value of αpure is negative (αpure = − 0.19). So, clutter is irrelevant, which explains the mean field class obtained for our compound. Besides, the value of the local exponent n can be determined from the critical exponents β and γ using the following equation [38]:

$$n \, \left( {T_{{\text{C}}} } \right) = \, 1 \, + \frac{\beta - 1}{{\beta + \gamma }}$$
(14)

and the relation (14) can be written as follows

$$n \, \left( {T_{{\text{C}}} } \right) \, = \, 1 \, + \frac{1}{\delta }\left( {1 \, - \frac{1}{\beta }} \right)$$
(15)

Some researchers have reported that for materials exhibiting second-order phase transitions n, they should be described by the mean-field model, such as n = 2/3 [14, 39], and any deviation from this value must be justified by the Curie temperature distribution of the sample [38, 40]. The values of critical exponents for the parent sample agree with those of a 3D-Heisenberg. However, with the substitution of Ti4+ ions, the critical exponent β increased up to the value characteristic of the mean field model, and the critical exponent γ decreased and became closer to the mean-field value. These results indicate that Ti-doping changes the universality class of the manganite perovskites. In fact, the obtained value of n is 0.71. From Fig. 10, we can deduce that with the increase of Ti substitution, the maximum of − ΔSM versus T decreases. Previous results have confirmed that the change in maximum entropy decreases with doping and followed by a decrease in the critical transition temperature TC. Furthermore, this decreases in \(- \Delta S_{M}^{{{\text{Max}}}}\) remains relatively high, which favors the possibility of making compounds useful for magnetic refrigeration in different temperature. This decrease is linked to the decrease in double exchange interactions between Mn4+ and Mn3+ ions. According to Othmani et al. [41] annealing at low temperature increases the maximum value of the variation of entropy. Besides, from the curve − ΔSM (T) we can deduce the type of the magnetic transition. In fact, a first-order transition is characterized by a narrow magnetic transition and a large value of \(- \Delta S_{M}^{{{\text{Max}}}}\). Nevertheless, a second-order transition shows a large magnetic transition (high value of ΔTFWHM), yet with a small maximum of the magnetic entropy variation (Fig. 10). The obtained results confirm those obtained in the studies of isotherms using the Banerjee criterion [42].

Fig. 10
figure 10

Temperature dependence of the magnetic entropy change (− ΔSM) at different measured for an applied field 5 T for compounds La0.67Ba0.22Sr0.11MnO3 and La0.67Ba0.22Sr0.11Mn0.95Ti0.05O3

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It is known that magnetic refrigeration works most efficiently near the Curie temperature of the working substance. In order to increase the temperature range of magnetic refrigeration, a refrigerator can be constructed from a variety of substances with comparable physical and chemical properties. Next, we can use composites that have magnetocaloric compound properties, where a series of magnetocaloric materials with comparable RCP value are combined for the purpose of forming a composite refrigerant. This situation can be well satisfied by considering a mixture of the same compounds annealed at two different temperatures (1373 K [11] for father sample and 1400 K for our compound) for refrigeration over the temperature range. This new method of preparing composites with magnetocaloric properties compounds presents an important advantage based on the synthesis of a single series of compounds annealed at different temperatures or in the same annealing temperature. Additionally, it should be noted that the RCP values of the mother and studied compound are comparable, which prompts us to reconsider the possibility of creating a multilayer composite that functions as a magnetic cooler over a large temperature range. To create a single sample with a strong magnetocaloric effect, we will also compare the magnetocaloric effects of two different substances. This enables us to create composites from these samples to have a wide Curie temperature range that covers the ambient and with comparable RCP values (Fig. 11). Our initial goal was to create a single sample with a significant magnetocaloric effect. However, by annealing at various temperatures, we were able to produce a set of samples with the same composition, but with different magnetic transition temperatures (TC). This allows us to fabricate multilayers from these samples to have a wide Curie temperature range that covers room temperature and with comparable RCP values.

Fig. 11
figure 11

Temperature dependence of the magnetic entropy change (− ΔSM) at different measured for an applied field 5 T for composites

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4 Conclusions

The present research work was intended to develop and characterize new materials with potential of application in the field of magnetic cold, information storage. We then developed a system using the classic ceramic production method based on the sintering technique. The physical properties of these materials strongly depend on the elaboration conditions (sintering, annealing, etc.) on the one hand and on the purity of the samples elaborated on the other. In a subsequent step, it is noted that the specific heat variation is positive prior to the transition and negative following it, and that it only deviates from zero in the vicinity of TC.

Besides, we have calculated the variation of the adiabatic temperature ΔTad and the cooling power RCP (T), which reveal values slightly greater than those determined from the measurements of the magnetic isotherms (RCP(S)). However, we found that the values of the local exponent n derived from the critical exponents are less than those deduced from the power law (|ΔS (H) |∞ Hn). In addition, using the Harris criterion, we discovered that αpure is negative (disorder is irrelevant). These findings confirm that the critical exponents and universality class are not significantly impacted by disorder.

Moreover, by calculating the variance σ2 and following the rules of selection of the universality class according to the value of σ, we found that the critical exponents of our system belong to the mean field model. These results indicate that Ti-doping changes the universality class of the manganite perovskites. Finally, in future research, we intend to conduct a comparison between the magnetocaloric effects of two compounds to develop a single sample with a significant magnetocaloric effect.