Critical Field Analysis and Magnetocaloric Effect of A-Site Double-Doped Manganese Oxide La_0.9EuSr_0.1MnO₃

Abstract

Polycrystalline La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples were prepared using a traditional high temperature solid-state reaction method. The effects of double doping of cationic Eu³⁺ and Sr²⁺ on the preformed clusters, critical field behavior, and change of magnetic entropy of LaMnO₃ were studied systematically. The results show that polycrystalline La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples are orthogonal crystal structures with a good unidirectionality and the space group of the polycrystalline samples is Pbnm. As the temperature increased from 15 to 380 K, the polycrystalline La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) sequentially displayed the characteristics of cluster spin glass, ferromagnetic phase, preformed cluster, and paramagnetic phase. For both La_0.9-xEu_xSr_0.1MnO₃ samples, the second-order phase transition occurs near Curie temperature. The critical index of the undoped parent compound and doped sample fitted best to the mean-field model and 3D-Heisenberg model, respectively. At an applied magnetic field of 7 T, the maximum absolute values of the magnetic entropy changes of the two samples were 2.76 ± 0.002 J·kg⁻¹·K⁻¹ and 2.81 ± 0.002 J· kg⁻¹· K⁻¹, respectively, and the temperature corresponding to the maximum absolute values of the magnetic entropy change was approximately 130 K. The magnetic refrigerant capacities (RC) of the two La_0.9-xEu_xSr_0.1MnO₃ samples were 436.01 ± 0.002 J· kg⁻¹ and 438.09 ± 0.002 J· kg⁻¹, respectively. Our results revealed that the changes in magnetic entropy and magnetic refrigerating capacity of the doped sample increased compared to the parent sample, suggesting that the double-doped manganese oxide La_0.9-xEu_xSr_0.1MnO₃ samples can be used as magnetic refrigerants in the medium temperature range of 80 to 250 K.

1 Introduction

Magneto-refrigeration technology based on magnetocaloric effects (MCE) is important for energy saving, is environmentally friendly, and can replace traditionally used refrigeration technologies [1,2,3]. Previous studies show that the rare earth element Gd has a large MCE. However, the use of Gd as a magnetic refrigerant is limited due to its high price [4,5,6]. Doped manganese oxides have attracted some attention in the field of magnetic refrigeration research. Studies reveal that the double-doped manganese oxide with the formula R_1-xA_xMnO₃, where R is a trivalent rare-earth cation, and A is a divalent alkaline earth cation, has a large MCE.

Furthermore, double-doped manganese oxides showed advantages compared to Gd and other magnetic refrigerating materials. The double-doped manganese oxides are low-cost and convenient to prepare, while they exhibit high chemical stability, high resistance, and favorable eddy current reduction [3, 5].

The transport and magnetic properties of manganese oxide can be explained by Zener double exchange (DE) [3, 7,8,9]. It is well known that the DE is very sensitive to the Mn–O bond length and the Mn–O–Mn bond angle and is a function of the average radius of the ions at A or Mn sites. If the ion at the A site is substituted with a rare earth element with a different ionic radius, internal stress changes in the crystal lattice lead to the deformation of the MnO₆ octahedron. This results in a decrease in the Mn–O–Mn bond angle [1,2,3, 9]. A difference of the Neel temperature was achieved with varying degrees of the average radius of the cations, which is important for equivalent ion valence substitutions [1, 8,9,10] except for the change of the Mn³⁺/Mn⁴⁺ ratio. Thus, changing the average anion radius < r_A > via doping is beneficial for the adjustment of the ferromagnetic–paramagnetic transition temperature and the regulation of MCE [8, 11]. The ferromagnetic–paramagnetic phase transition temperature of the Sr-doped manganese oxides was higher [1, 8].

Numerous efforts were made to dope the A site. In our previous study, the magnetic and magnetocaloric properties of La_0.9-xEu_xSr_0.1MnO₃ (x = 0.05) were investigated. Compared to the undoped La_0.9Sr_0.1MnO₃ sample, the magnetic entropy change in a 7-T magnetic field and the RCP decreased to 2.53 J· K⁻¹ · kg⁻¹ and 354.93 J· kg⁻¹. The doping levels were further decreased to x = 0.025in order to improve the properties of doped La0.9-xEuxSr0.1MnO3, In this study, the MCE and critical field behavior of the La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples were investigated after adjusting the average radius of the cation at the site A.

2 Material and Methods

The polycrystalline La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples were prepared using a traditional high-temperature solid-phase reaction. High purity (≥ 99.9 wt%) La₂O₃, Eu₂O₃, SrCO₃, and MnCO₃ were strictly weighed following the stoichiometry ratio and then ground with an agate mortar. The mixture was sintered twice for 12 h at atmospheric pressure and a temperature of 1000 °C and 1100 °C, respectively, and was ground between two sintering steps. The sample was ground for a final round at the end and pressed into a circular pellet with a diameter of 10 mm and a thickness of ~ 2 mm using a pellet press set. The pellets were sintered at 1250 °C for 24 h at atmospheric pressure in a tube sintering furnace, and bright, high-quality samples with smooth and crack-free surfaces were obtained. An X-ray diffractometer (XRD) was used to examine the crystalline structure and quality of the samples. The magnetic properties of the samples were obtained using the multifunction physical property measurement system (US Quantum Design, PPMS) with a vibrating sample magnetometer (VSM) module.

3 Results and Discussion

The sample quality and crystal structure of the polycrystalline La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples were measured using an X-ray diffractometer at room temperature. As shown in Fig. 1, the XRD patterns revealed that all samples are in the space group of Pbnm and have an orthogonal crystal structure with good unidirectionality. The Eu³⁺ doping did not affect the spatial structure compared to its parent compound. The XRD results showed the presence of extremely small amounts of Eu₂O₃ in the LSMO matrix. However, Eu₂O₃ impurities did not affect the changes of the adiabatic temperature (∆T_ad) since the lattice-specific heat dominates. Table 1 shows the lattice parameters and c/a of the La_0.9-xEu_xSr_0.1MnO₃samples with x = 0.000 and 0.025. The results show that the doping decreased the lattice parameters a and c and increased b. The slight change in the lattice parameters with doping was attributed to the radius difference between La²⁺ and Eu³⁺ [2, 12,13,14] since the La²⁺ radius (0.117 nm) was larger than the Eu³⁺ radius (0.0947 nm), which resulted in a shrinkage of the unit cell volume after Eu³⁺ doping.

Fig. 1

XRD patterns of polycrystalline La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 at room temperature

Table 1 Crystal parameters and c/a of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025

The distortion of the crystalline structure was measured using the tolerance factor t [15], which was calculated as

$$t=\left({r}_{A}+{r}_{O}\right)/{2}^{1/2}\left({r}_{Mn}+{r}_{O}\right)$$

(1)

where ${r}_{A}$, ${r}_{O}$, ${r}_{Mn}$ are the radii of the rare earth ion, oxygen ion, and manganese ion, respectively. The calculated results show that the tolerance factors of the polycrystalline La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 were t_0.000 = 0.876 and t_0.025 = 0.875, respectively. Distortion occurs when t = 0.75 ~ 0.9, at which the symmetry decreases, but the orthogonal crystal structure is stable [16]. After the Eu³⁺ doping, the tolerance factor decreased slightly, and the degree of the structure distortion of the sample increased.

The magnetization was a function of temperature (M-T curve) of the La_0.9-xEu_xSr_0.1MnO₃ samples (x = 0.000, 0.025) at a magnetic field of 0.05 T, as shown in Fig. 2. The zero field cooling (ZFC) and zone cooling (FC) curves of each sample were shown in Fig. 2. The inset presents the dM/dT versus T curve. The results reveal that at high temperatures, these two La_0.9-xEu_xSr_0.1MnO₃ samples were paramagnetic. The magnetic property of the samples changed from paramagnetic to ferromagnetic with increasing temperatures. At Curie temperatures, Tc1 ≈ 130 K and Tc2 ≈ 120 K, the magnetization increased dramatically, and the samples had ferromagnetic properties [17,18,19,20,21]. After Eu³⁺ doping, the decrease of the average ionic radius increased the lattice distortion, decreased the Mn–O-Mn bond angle, and reduced the ferromagnetic interaction between Mn³⁺ and Mn⁴⁺ through oxygen, resulting in a decrease of the Curie temperature and magnetization [1, 22, 23]. As the temperature in the lower temperature region further decreased, the ZFC and FC curves of the samples deviated from each other. This could be attributed to the competition between the ferromagnetic and antiferromagnetic states of the molecules. The deviations of the ZFC and FC curves of the samples from each other indicate a cluster spin-glass behavior [2].

Fig. 2

Magnetization versus temperature (M-T) curve of La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples at a magnetic field of 0.05 T. The inset is dM/dT versus T curve

At a magnetic field of 0.05 T, the magnetic susceptibility times temperature (χT) versus T curves of the two polycrystalline La_0.9-xEu_xSr_0.1MnO₃ samples (with x = 0.000 and 0.025) are shown in Fig. 3. The χT decreased as the temperature increased and reached the maximum near 105 K before it rapidly decreased. Thus, the Neel temperature obtained is T_N1 ≈ 105 K. The samples are antiferromagnetic below the Neel temperature, which is consistent with our previous results. The Neel temperature for the Eu³⁺-doped samples was slightly lower than the undoped sample.

Fig. 3

Magnetic susceptibility times temperature (χT) versus T curves of the polycrystalline La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 at a magnetic field of 0.05 T

The inverse of the magnetic susceptibility (χ⁻¹) versus T and the Curie–Weiss fitting curves of La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples was calculated at a magnetic field of 0.05 T (Fig. 4). The Curie–Weiss law for fitting was calculated as follows [24,25,26,27].

$$\chi \left(T\right)=\mathrm{C}/\left(T-\theta \right)$$

where C is the Curie constant and θ is the Curie–Weiss temperature. The fitting results revealed that the Curie’s temperature of the two La_0.9-xEu_xSr_0.1MnO₃ samples (with x = 0.000 and 0.025) were θ 1 ≈ 200 K and θ 2 ≈ 190 K, respectively. In the temperature regions of 130–285 K and 120–290 K, the χ⁻¹-T curves of both samples deviated from the Curie–Weiss law as depicted in Fig. 4. This is ascribed to the antiferromagnetic competition within the system, resulting in the anti-parallel arrangement of the ferromagnetic clusters. This suggests that there are preformed clusters in the polycrystalline samples [2]. Combined with our other results from Fig. 2, we discovered that an increase in temperature within the temperature regions of 15–105 K and 15–96 K resulted in antiferromagnetic phase characteristics of both samples. Within the temperature regions of 105–130 K and 96–120 K, the samples were ferromagnetic. At 130–285 K and 120–290 K, the samples presented as preformed clusters. The La_0.9-xEu_xSr_0.1MnO₃ samples (with x = 0.000 and 0.025) presented as pure paramagnetic materials above the temperature of 285 K and 290 K, respectively.

Fig. 4

Inverse of magnetic susceptibility (χ⁻¹) versus T curves of La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 at a magnetic field of 0.05 T

The experimental effective paramagnetic moment was obtained by [3, 16]

$${\mu }_{eff}^{exp}={(3{k}_{B}C/{N}_{Av})}^{1/2}$$

(2)

where the Avogadro constant N_Av = 6.023 × 10²³ mol⁻¹ and the Boltzmann constant k_B = 1.38016 × 10⁻²³ J· K⁻¹.

The theoretical effective magnetic moment was determined by [2, 4, 28]

$${\mu }_{{e}_{ff}}^{the}={\left[0.1{\mu }_{eff}^{2}\left(M{n}^{4+}\right)+x{\mu }_{eff}^{2}\left({Eu}^{3+}\right)+(0.9-x){\mu }_{eff}^{2}\left(M{n}^{3+}\right)\right]}^{1/2}$$

(3)

Table 2 shows the Curie constant C, Curie temperature θ, experimental effective paramagnetic moment, and the theoretical effective magnetic moment of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 at a magnetic field of 0.05 T. The difference between the experimental and theoretical effective paramagnetic moments resulted from the existence of ferromagnetopolaron in the paramagnetic materials.

Table 2 Curie temperature T_C, Curie–Weiss constant C, paramagnetic Curie temperature θ, and the experimental and theoretical magnetic, paramagnetic moments of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 at a magnetic field of 0.05 T

A series of isothermal magnetization measurements were carried out near the Curie temperature to investigate the magnetocaloric effect of the La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples. Figure 5a, b illustrate the isothermal magnetization curves (M-H curves) of the two samples. At the same temperature, the magnetization of the samples increased as the external magnetic field increased. Under the same magnetic field, the system was subjected to more intense thermal perturbation with increasing temperatures, leading to the magnetization of the samples decrease [2, 29,30,31]. Furthermore, the M-H curve of the sample tended to be linear at high temperatures, indicating that the paramagnetic state of the sample was most prevalent in the high-temperature region. In contrast, the magnetization of the two samples rapidly increased and tended to be saturated at low temperatures, indicating a predominantly ferromagnetic phase.

Fig. 5

a and b Magnetization curves (M-H curves) of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025, respectively, at varied temperatures. c and d Isothermal Arrott curves of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025, respectively, at varied temperatures

Figure 5c, d illustrate the isothermal Arrott curves (H/M-M² curves) of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025. No obvious S shape and negative slope were observed for the curves. Following the Banerjee criterion [32,33,34,35], the paramagnetic-ferromagnetic transition of the two samples near T_c was a second-order phase transition.

To study the second-order phase transition of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025, the mean field, three-dimensional Heisenberg, three-dimensional Ising, and tri-critical models were used to fit the magnetization curves of the samples. Figure 6a show the fitting results of the magnetization of the La_0.9-xEu_xSr_0.1MnO₃ sample with x = 0.000 using the mean field, three-dimensional Heisenberg, three-dimensional Ising, and tri-critical models, respectively, at temperatures ranging from 120 to 140 K with an incremental increase of 2 K. The fitting results revealed that for the sample with a doping amount of 0.000 (x = 0.000), the fitting curves of the mean field and three-dimensional Heisenberg and Ising models were parallel to the experimental curve spanning a large range of temperatures and H·M⁻¹. The tri-critical model fittings showed a divergence from the experimental magnetization curve. The mean field model had the best fitting for the experimental magnetization curve of the sample. Figure 6b show the fitting results of the magnetization of the second La_0.9-xEu_xSr_0.1MnO₃ sample with x = 0.025 using again the mean field, three-dimensional Heisenberg, and Ising, and tri-critical models, respectively, at temperatures ranging between 112 and 152 K with an incremental increase of 4 K. The results from Fig. 6b illustrated that for the La_0.9-xEu_xSr_0.1MnO₃ sample with x = 0.025, the experimental magnetization curve fits the three-dimensional Ising model the best compared to other models.

Fig. 6

a The fitting results of the magnetization of the La_0.9-xEu_xSr_0.1MnO₃ sample with x = 0.000 using the mean field, three-dimensional Heisenberg, three-dimensional Ising, and tri-critical models, respectively, at temperatures from 120 to 140 K with an incremental increase of 2 K. b The fitting results of the magnetization of the La_0.9-xEu_xSr_0.1MnO₃ sample with x = 0.025 using the mean field, three-dimensional Heisenberg, three-dimensional Ising, and tri-critical models, respectively, at temperatures from 112 to 152 K at an increment step of 4 K

To further examine which model fits the La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples best, the critical exponents β, γ, and δ [1, 36] of the samples were obtained using the Kouvel-Fisher (K-F) method. β is the critical exponent associated with the critical behavior of spontaneous magnetization; γ is the critical exponent related to the critical behavior of the magnetic susceptibility with no applied magnetic field; δ is the critical exponent of the state function. The Kouvel-Fisher equations are as follows:

$${M}_{s}\left(T\right){\left[\mathrm{d}{M}_{s}\left(T\right)/\mathrm{d}T\right]}^{-1}=\left(T-{T}_{c}\right)/\beta$$

(4)

$${\chi }_{0}^{-1}\left(T\right){\left[\mathrm{d}{\chi }_{0}^{-1}\left(T\right)/\mathrm{d}T\right]}^{-1}=\left(T-{T}_{c}\right)/\gamma$$

(5)

$$\delta =1+\left(\gamma /\beta \right)$$

(6)

where M_S is the spontaneous magnetization, χ₀⁻¹ is the susceptibility at a zero magnetic field. The Arrott plots in the temperature ranges of T ≤ Tc and T > Tc were linearly fitted, and the intercepts obtained from the fitting curves are M_S² and χ₀⁻¹, respectively. Figure 7A(a), B(a) illustrates the spontaneous magnetization versus temperature (Ms-T) plots below the Curie temperature, and Figs. 7A(b), B(b) shows the Ms(T)·(dMs/dT)⁻¹ versus T curves. Figures 7A(c) to (d), B(c) to (d) show the inverse of the magnetic susceptibility at a zero magnetic field versus temperature and χ₀⁻¹(dχ₀⁻¹/dT) versus T plots with the temperature that is higher than the Curie temperature. The 1/β value represents the slope of the fitted Ms(T)·(dMs/dT)⁻¹ versus T curve, as shown in Eq. (4). Based on the slope, we determined that the β1 of the polycrystalline La_0.9-xEu_xSr_0.1MnO₃ sample with the doping amount x = 0.000 was 0.52 ± 0.02. The 1/γ was expressed as the slope of the (d)χ₀⁻¹(dχ₀⁻¹/dT) versus T fitting curve, as shown in Eq. (5), and was γ1 = 1.02 ± 0.02. Based on Eq. (6), we obtained δ1 = 2.96 ± 0.01. Comparing the results with the data from the models, the results of the sample fit best with the mean field model. Furthermore, the critical exponents for the Eu³⁺-doped sample, β2 = 0.368 ± 0.02, γ2 = 1.303 ± 0.02, and δ2 = 4.54 ± 0.01, fit best with the 3D- Heisenberg model, which is consistent with the previous results.

Fig. 7

Spontaneous magnetization (Ms) of the La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000 (A), 0.025 (B)) sample as a function of temperature plots when the temperature is smaller than the Curie temperature (a) Ms versus T plot (b) Ms(T)·(dMs/dT)⁻¹ versus T plot, and the reciprocal zero-field magnetic susceptibility as a function of temperature plots (c) χ₀⁻¹ versus T plot and (d) χ₀⁻¹ (dχ₀⁻¹/dT) versus T plot

The magnetic equation of state is a relationship between the variables M(H,ɛ), H, and T. From the scaling hypothesis, the equations is as follow:

$$M\left(H,\varepsilon \right)={\varepsilon }^{\beta }f\pm \left(H/{\varepsilon }^{\beta +\gamma }\right)$$

(7)

where f stands for regular analytic function, f + for T > T_C, and f- for T < T_C. ɛ is the reduced temperature, (T-T_C)/T_C. The scaled M/|ɛ|^β is plotted as a function of the scaled H/|ɛ|^β+γ. Two individual branches appeared in the vicinity of T_C, confirming that the critical parameters determined are in good conformity to the scaling hypothesis. The two different curves in ln (M/|ɛ|^β) as a function of ln(H/|ɛ|^β+γ) strongly support the accuracy of obtained values of the critical exponents and T_C as shown in the inset of Fig. 8.

Fig. 8

Scaling plots indicate two universal curves below and above T_C for two compounds. Inset represents the same plots on logarithmic scale

Critical exponents affect how magnetic caloric reactions respond. According to Oesterreicher et al. [1], the equation for field dependence of change in magnetic entropy ∆S_M for the second-order magnetic phase transition is

$$\left|\Delta {S}_{M}^{\mathrm{m}\mathrm{a}\mathrm{x}}\right|=a{\left(H\right)}^{n}$$

(8)

where a is a constant and n depends upon the magnetic state of the compound. n can be calculated by the logarithmic derivative of the ∆S_M to the magnetic field:

$$n=\frac{d\mathrm{l}\mathrm{n}\Delta {S}_{M}}{d\mathrm{l}\mathrm{n}H}$$

(9)

n reaches about “2” in the paramagnetic region (T > T_C) and about “1” in ferromagnetic region (for T < T_C). n is field independent at T_C and can be found by

$$n=1+\frac{\beta -1}{\beta +\gamma }=1+\frac{1}{\delta }(1-\frac{1}{\beta })$$

(10)

Figure 9 uses Eq. (9) to demonstrate how temperature affects the exponent n (T, H) at different fields. The figure shows that n changes with the field over the temperature range, and its values are very similar above the Tc. n approaching “2″ for temperatures greater than Tc is a consequence of the Curie–Weiss law, suggesting that the magnetization has a linear field dependence in the higher temperature range. At T < T_C, n is approximately ‘‘1.” The n at Curie temperature (T_C) can be calculated with critical exponents β and γ with Eq. (10). Mean field theory and 3D-Heisenberg theory explains that a minimum value of n at T = T_C is projected to be around 0.67 and 0.637 [37]. In our compounds, n values near the temperature T_C were measured to be 0.68 ± 0.001 and 0.623 ± 0.002 for x = 0.000 and 0.025, respectively. Our simulated values of n are very close to the theoretical values of n near Curie temperature, indicating that the critical model we built is reasonable. Before doping, the model is an average field model, and the interaction of the magnet falls into the magnetic field fluctuation effect. After doping, the model changed to the Heisenberg model, and the apparent critical fluctuation near Curie temperature (Tc) led to the critical indices β < 0.5 and γ > 1, indicating that the doping resulted in double exchange, which led to the increase in the magnetic field and the magnetic entropy change.

Fig. 9

Temperature dependence of the exponent (n) of the change in magnetic entropy with different applied magnetic fields from 1 to 7 T

The isothermal magnetic entropy ∆S_M was generally calculated using Maxwell’s thermodynamic equation. Then the magnetocaloric effects of the magnetic material were estimated as described previously [37,38,39]. The equations involved were as follows

$$\Delta {S}_{M}\left(T,H\right)={S}_{M}\left(T,H\right)-{S}_{M}\left(T,0\right)={\int }_{0}^{H}(\partial S/\partial H)dH$$

(11)

Substituting ${(\partial S/\partial H)}_{T}={(\partial M/\partial T)}_{S}$ into Eq. (11), the following equation was obtained:

$$\Delta {S}_{M}\left(T,H\right)={S}_{M}\left(T,H\right)-{S}_{M}\left(T,0\right)={\int }_{0}^{H}\left[\partial M(T,H)/\partial T)\right]dH$$

(12)

The magnetic entropy change of the polycrystalline sample La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) under different applied magnetic fields was obtained using Eq. (12). It is necessary to approximate Eq. (12) numerically because of discrete experimental data. This change of magnetic entropy of the polycrystalline samples was expressed as a function of temperature, as shown in Fig. 10. The results show that the magnetic entropy change increased as the applied magnetic field increased. The increase in the magnetic entropy change was attributed to the increased magnetic order of the samples and the applied magnetic field, which led to an increase in magnetization near T_c [1]. For the sample with x = 0.000, a maximum magnetic entropy change was observed around 130 K. When the applied magnetic field reached 7 T, the maximum magnetic entropy was 2.76 ± 0.002 J· kg⁻¹· K⁻¹. For the second sample with x = 0.025, the maximum magnetic entropy change was obtained near 125 K. The maximum magnetic entropy change was 2.81 ± 0.002 J· kg⁻¹· K⁻¹ when the applied magnetic field reached 7 T. Small errors are introduced with the experimental magnetization data and the numerical approximation of Eq. (12). The increase in magnetic entropy after doping Eu³⁺ resulted from the decrease in the average ionic radius < r_A > of site A. Therefore, the lattice disorder increased, and the Mn–O-Mn bond angle was inhibited [3, 9, 40]. Furthermore, the spin–lattice coupling played an important role in the change of magnetic entropy in the magnetic ordering, and the coupling strength changes as the average ionic radius of site A changed. Eu³⁺ doping reduced the < Mn–O–Mn > bond angle and increased the distortion of MnO₆ octahedron, which in turn weakened the interaction and affected the parallel arrangement of Mn³⁺/Mn⁴⁺ spins. Therefore, there was a trend that the enhanced distortion tends to increase the spin–lattice coupling, which in turn increases the change in magnetic entropy [1, 8, 41].

Fig. 10

The negative magnetic entropy change under the same applied magnetic field versus temperature (− ∆S-T) plots for La_0.9-xEu_xSr_0.1MnO₃ samples with a x = 0.000 and b x = 0.025

Good magnetic refrigerating materials have a large ∆S_M and a high cooling efficiency (RC). RC is another critical parameter for magnetic refrigerating materials used in the field of magnetic refrigeration technology and was used to determine the performance of magnetic refrigerating materials [42, 43]. The definition of RC is as follows: RC =|− ∆S_M|·∆T_FWHM where ΔT_FWHM is the full width at half maximum of the magnetic entropy of the magnetic entropy versus temperature plot. The RC of the La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 at an applied magnetic field of 7 T are 436.01 ± 0.002 J· kg⁻¹ and 438.09 ± 0.002 J· kg⁻¹, respectively. The comparison of the magnetocaloric parameters of La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples and similar magnetic refrigeration materials in a 5-T magnetic field is shown in Table 3. The magnetic entropy and RC of Gd are 9 J· kg⁻¹· K⁻¹ and 405 J· kg⁻¹ in a 5-T magnetic field, respectively. The results show that the La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples achieved magnetic refrigeration in the middle of the temperature curve and can, therefore, be used as good magnetic refrigerating molecules.

Table 3 Comparison of the magnetocaloric parameters of La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples and similar magnetic refrigerant materials under a magnetic field of 5 T

The ∆Tad is an important parameter for the magnetocaloric effect (MCE) of the magnetic materials. The relationship between the ∆Tad and the isothermal magnetic entropy change is represented by the following equation [44]:

$$\Delta {T}_{ad}{\left(T\right)}_{\Delta H}=-{\int }_{{H}_{1}}^{{H}_{2}}\frac{T}{{C}_{H,P}}\frac{\partial M\left(H,T\right)}{\partial T}dH=\frac{T}{{C}_{H,P}}\Delta {S}_{M}(T)$$

(13)

where $\Delta {S}_{M}(T)$ reveals the temperature-dependent magnetic entropy change; ${C}_{H,P}$ stands for the specific heat of the material in an applied magnetic field H at a constant pressure; the specific heat without an applied magnetic field is given by the following equation [45]:

$$C_{p,m}=2.82+0.874T-\frac{6.93\times10^4}{T^2}+398\times10^{-7}-3.31\times10^{-2}T^{3/2}\;\;\;\;\left(77\leq T\leq298\right)$$

(14)

The maximum adiabatic temperatures of both samples at an applied field of 7 T were obtained using Eqs. (13) and (14) and are $\Delta {T}_{ad}^{\mathrm{m}\mathrm{a}\mathrm{x}}\left({T}_{c1}\right)=5.67\pm 0.001K$ and $\Delta {T}_{ad}^{\mathrm{m}\mathrm{a}\mathrm{x}}\left({T}_{c2}\right)=4.89\pm 0.002K$, respectively. Figure 11 shows the maximum adiabatic temperature curves of both samples at the varied applied field. The trend of the change in the maximum adiabatic temperature ($\Delta {T}_{ad}^{\mathrm{m}\mathrm{a}\mathrm{x}}$) is similar to the trend of the maximum magnetic entropy change. Both the maximum adiabatic temperature and the maximum magnetic entropy change increased as the applied field increased.

Fig. 11

The field dependence of the maximum values of adiabatic temperature change $\Delta {T}_{ad}^{\mathrm{m}\mathrm{a}\mathrm{x}}$

Based upon the rescaled magnetic entropy change curves, Franco et al. [46] hypothesized that the Fe-doped Gd₅Ge₂Si₂ underwent a second-order magnetic phase transition if the modified curve construction recollapsed the magnetic entropy change onto a single curve in a temperature range near the peak of the magnetic entropy change at different applied fields. The samples had a first-order phase transition if the magnetic entropy change did not collapse to a single curve in a range of T < T_C^3D.

Figure 12 shows a rescaled magnetic entropy change plot of the La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) samples. The y-axis is the ratio of the magnetic entropy change (ΔS) to the maximum magnetic entropy change (ΔS_M) at the applied fields. The x-axis is θ, which is defined as

$$\theta=\left\{\begin{array}{c}-\frac{T-T_c}{T_1-T_c}\;\;T\leq T_c\\\frac{T-T_c}{T_2-T_c}\;\;\;T\geq T_c\end{array}\right.$$

(15)

where T₁ and T₂ stand for the two reference temperatures corresponding to ΔS_M/2.

Fig. 12

Behavior of the master curve as a function of the rescaled temperature for different magnetic fields

Figure 12 illustrates that the rescaled magnetic entropy change collapses to a single curve, indicating the occurrence of a second-order phase transition, which is in good agreement with the previous results.

4 Conclusions

The effects of Eu³⁺ and Sr²⁺ cationic double-doping on the preformed clusters, critical behavior, and magnetic entropy change of two LaMnO3 samples were investigated. Our results show that the La_0.9-xEu_xSr_0.1MnO₃ (x = 0.000, 0.025) polycrystalline samples we prepared through a traditional solid-state reaction were unidirectional orthogonal crystal structure with the space group of Pbnm. As the temperature increased from 15 to 380 K, the two La_0.9-xEu_xSr_0.1MnO₃ samples sequentially presented the characteristics of cluster spin glass, ferromagnetic phase, preformed cluster, and paramagnetic phase. The Curie temperatures of the two samples were 130 K and 120 K, respectively. The phase transition of the two samples was the second-order phase transition, and the critical indexes of the samples were analyzed. The critical index of the sample with x = 0.000 was best fitted to the mean field model, and the critical index of the sample with x = 0.025 fit best with the 3D- Heisenberg model. At an applied magnetic field of 7 T and a temperature of 130 K, the maximum magnetic entropy changes of the two polycrystalline La_0.9-xEu_xSr_0.1MnO₃ samples (with x = 0.000 and x = 0.025) were 2.76 ± 0.002 J· kg⁻¹· K⁻¹ and 2.81 ± 0.002 J· kg⁻¹· K⁻¹, respectively, and the magnetic refrigerating capacities were 436.01 ± 0.002 J· kg⁻¹ and 438.09 ± 0.002 J· kg⁻¹, respectively. Our results show that the magnetic entropy change and magnetic refrigerating capacities of the samples increased after doping. The results suggest that La_0.9-xEu_xSr_0.1MnO₃ samples with x = 0.000 and 0.025 can be used as magnetic refrigerants in the medium temperature range of 80–250 K.