1 Introduction

Manganite-based perovskites are attractive functional materials, because they possess a wide range of reported inclusion physical properties, such as the magnetocaloric effect and thermoelectric properties [1,2,3,4,5,6]. These physical properties make these materials intended for several applications [7,8,9,10,11,12,13,14]. Experimentally, the magnetic and structural properties of Pr0.7Ba0.2Ca0.1MnO3 were investigated [15]. A study was conducted to investigate the impact of compaction pressure on the enhancement of ferromagnetic properties in La0.7Ba0.3MnO3 [16]. Pr0.75Sr0.25MnO3 exhibited a semiconductor–metal transition [17]. Pr0.75Sr0.25MnO3 has a half-metallic character with a huge band gap of 2.8 eV in the minority band [18]. This property of the half metals make them potential candidates for application in spintronic devises and magnetic sensors.

In the present work, we have prepared Pr0.7Ba0.3MnO3 using solid-state reaction method (more detailed is given in Refs. [19, 20]. The structural and morphological of the prepared samples were investigated. Furthermore, the electronic and thermoelectric properties of Pr0.75Ba0.25MnO3 are investigated using DFT calculation [21,22,23,24,25,26]. In addition, The Monte Carlo simulations (MCSs) calculations are used to shed light on the magnetic and magnetocaloric properties of Pr0.75Ba0.25MnO3. The T dependence of the magnetization is given. The thermal specific heat, variation of heat specific, magnetic entropy changes, relative cooling power, and magnetic hysteresis cycle are obtained.

2 Model and simulations method

The Hamiltonian of this system

$$H = - J_{1} \sum\limits_{ < i,j > } {S_{i} S_{j} } - J_{2} \sum\limits_{ < < i,k > > } {S_{i} S_{k} } - J_{3} \sum\limits_{ < < < i,l > > > } {S_{i} S_{l} } - J_{4} \sum\limits_{ < < < < i,m > > > > } {S_{i} S_{m} } - h\sum\limits_{i} {S_{i} } .$$
(1)

The J1, J2, J3 and J4 are the first, second, third, and fourth exchange interactions between Mn–Mn in Pr0.75Ba0.25MnO3 with S(Mn4+) = 3/2.

The obtained results by DFT were used to calculate the exchange interaction between the magnetic atoms, J1 = 15.3, J2 = 12.3, J3 = 12.1, J4 = 9.5 K. Magnetocaloric effect Pr0.7Ba0.3MnO3 has been studied.

The magnetic and magnetocaloric properties of Pr0.75Ba0.25MnO3 were investigated using Monte Carlo simulations (MCSs) in conjunction with the Metropolis algorithm. Equation (1) was employed for this purpose.

The magnetization of Mn4+ in this perovskite

$$M = \left\langle {\frac{1}{N}\sum\limits_{i} {S_{i} } } \right\rangle$$
(2)

with N = 2465 spins.

The specific heat of this perovskite

$$C_{P} = \frac{\beta }{{N^{2} }}\left( {\left\langle {E^{2} } \right\rangle - \left\langle E \right\rangle^{2} } \right),$$
(3)

where \(\beta = \frac{1}{{k_{B} T}}\).

The magnetic entropy changes

$$\Delta S\left( {T,h} \right) = \int\limits_{0}^{{h_{\max } }} {\left( {\frac{\partial M}{{\partial T}}} \right)_{{h_{i} }} } dh.$$
(4)

The RCP

$${\text{RCP}} = \int\limits_{{T_{1} }}^{{T_{2} }} {\Delta S} \left( T \right)dT.$$
(5)

3 Crystallographic structure and computational details

The structure de Pr0.7 Ba 0.3MnO3 dans le plan (b, c) and in [111] plane is presented in Figs. 1 and 2, respectively. The lattice parameters of our system are given in Table 1.

Fig. 1
figure 1

Structure of Pr0.75Ba0.25MnO3 (b, c) plane

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Fig. 2
figure 2

Structure of Pr0.7Ba0.3MnO3 in [111] plan

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Table 1 Lattice parameters of Pr0.7Ba0.3MO3 given by Refs. [19, 20]
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In Table 2, we have presented the crystallographic parameters Pr0.7 Ba0.3MnO3, and n is percentage occupancy.

Table 2 Crystallographic parameters Pr0.7Ba0.3MnO3, and n is percentage occupancy [20]
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The numbers in parentheses represent the error and the results in braces are for the Pr0.7 Ba 0.3MnO3 compound.

In this work, DFT calculations were perfumed using approximate XC functional GGA executed via operating the PBE method as implemented in the Wien2k package [27,28,29,30,31,32]. In the electronic properties part, we take a concentration of 0.25 instead of 0.3. To avoid supercells, which give us too many atoms in the structure and which can spoil the convergence of the calculation. A cut-off parameter is RMT × Kmax = 8 and the Fourier expansion parameter Gmax = 12.0. The transport properties of Pr0.75Ba0.25MnO3 are obtained using the semi-classical Boltzmann theory as implemented in the BoltzTraP code [33]. Furthermore, in the BoltzTraP code, the semi-classical transport equations of Bloch–Boltzmann are solved in the constant relaxation time approximation [33]. The transport coefficients using the constant relaxation time approximation are τ = 10−14 s [34].

4 Results and discussion

The spin-polarized total and partial electronic density of states (DOS) for Pr0.75Ba0.25MnO3 are depicted in Figs. 3 and 4, respectively. The asymmetrical nature observed between the spin-up and spin-down DOS confirms the magnetic properties of the material. The magnetization is attributed to the transition metal Mn and the rare-earth element Pr. Specifically, the spin moment values are determined to be 1.89023 μB for Pr and 3.37955 μB for Mn. At the Fermi level (EF), the majority band exhibits conducting behavior, while the minority band displays insulating characteristics. This configuration allows for 100% spin polarization and exhibits a half-metallicity feature, with a band gap of 2.274 eV in the minority band. This half-metallicity property positions Pr0.75Ba0.25MnO3 as a promising candidate for applications in spintronic devices and magnetic sensors.

Fig. 3
figure 3

Total DOS of Pr0.75Ba0.25MnO3 Pr, Ba, Mn, and O atoms calculated using GGA approximation

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Fig. 4
figure 4

Total PDOS of Pr0.75Ba0.25MnO3 calculated using GGA approximation

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The partial DOS shows that Mn-3d and O-2p have an important electron density contribution in region [– 6, – 0.25 eV]. In the energy range of [– 0.25, + 2 eV], which encompasses the conduction band, the orbital hybridization is primarily influenced by the spin-up states of Pr-4f, Mn-3d, and O-2p orbitals. Within this range, an exchange splitting can be observed between the spin-down and spin-up partial states of Pr-4f and Mn-3d orbitals. This exchange splitting plays a significant role in contributing to the majority portion of the total spin magnetic moments of the unit cell in Pr0.75Ba0.25MnO3 compound.

Figure 5 illustrates the relationship between magnetization and temperature (magnetization vs. temperature curve) and magnetic susceptibility vs. temperature for Pr0.75Ba0.25MnO3. The data reveal a transition from a ferromagnetic phase to a paramagnetic phase, which occurs at a critical temperature (TC) of 156 K. Our value is near to that given by Ref. [20] (TC = 164 K), and this difference may be due to the atoms that we did not consider in our Pr, Br, and O calculation, because they are not magnetic. The value of saturation magnetization is equal 16 emu/g. This value is near to that obtained by the experiment results [20].

Fig. 5
figure 5

The thermal magnetization and magnetic susceptibility of Pr0.75Ba0.25MnO3

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The magnetic entropy changes are given in Fig. 6. The value of ΔSmax is situated at the Tc = 156 K. For h = 5.0 T, the value of ΔSmax is 12 and 2.46 J/kg.K for 0.5 T. ΔS vs. temperature was given in Ref. [35]. In previous work [15] for Pr0.7Ba0.2Ca0.1MnO3, they found that 2.2 J/kg.K for 5 T.

Fig. 6
figure 6

The thermal entropy magnetic change of Pr0.75Ba0.25MnO3 for several magnetic fields h

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The variation of specific heat CP vs. temperature is shown in Fig. 7. The maximum of CP is situated at 156 K. As the magnetic field values increase, there is a decrease in the maximum value of the specific heat capacity (CP). The large anomaly in the heat capacity of DyAl2 was observed due mainly to spin reorientation from [100] to [111][36].

Fig. 7
figure 7

The thermal specific heat of Pr0.75Ba0.25MnO3

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The CP changes exhibit anomaly near to TC due to the ferromagnetic–paramagnetic transition. The thermal specific heat changes strongly changes from the negative value to the positive one as temperature increases (Fig. 8).

Fig. 8
figure 8

The thermal specific heat changes and magnetic susceptibility of Pr0.75Ba0.25MnO3

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The relative cooling power vs. magnetic field is given in Fig. 9. The value of RCP obtained by Ref. [15] is 261 J/kg for 5 T and in this work for Pr0.7Ba0.3MnO3 is 126 J/kg for 5 T. The RCP vs. temperature is illustrated in Fig. 10. The RCP exhibits an anomalous around the TC = 156 K. The relative cooling power (RCP) increases with increasing temperature until it reaches a saturation point for each value of the magnetic field (h). Regarding the effect of the magnetic field (h), RCP increases as the value of h increases, as shown in Fig. 8. The magnetization versus magnetic field (h) is depicted in Fig. 11 for three different temperatures: 150 K, 130 K, and 170 K. It can be observed that the remanent magnetization and coercive field decrease with increasing temperature. Pr0.75Ba0.25MnO3 demonstrates a superparamagnetic behavior around the temperature TC.

Fig. 9
figure 9

The RCP vs. h(T) of Pr0.75Ba0.25MnO3 for TC = 0.015 K and Th = 350 K

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Fig. 10
figure 10

The RCP vs. T(K) of Pr0.75Ba0.25MnO3 for several magnetic field

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Fig. 11
figure 11

The M(emu/g) vs. h(T) of Pr0.75Ba0.25MnO3 for several temperatures

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To provide a more accurate response, I would need the complete relation for calculating the Seebeck coefficient. Could you please provide the full relation or equation for calculating the Seebeck coefficient

$$S=T\frac{8{\pi }^{2}{k}_{B}^{2}{m}^{*}}{3e{h}^{2}}{\left(\frac{\pi }{3n}\right)}^{2/3};$$
(6)

so, n is charge carrier concentration, m* is the effective mass, kB is Boltzmann constant, \(e = \, 1.67 \times 10^{ - 19} C\), and \(h = 6.626 \times 10^{ - 26} \;{\text{k.g.m}}^{2} \sec^{ - 1}\) ℎ.

Figure 12 illustrates the variation of the Seebeck coefficient (S) for Pr0.75Ba0.25MnO3. The data reveal that the Seebeck coefficient increases as the temperature increases. At 800 K, it reaches a value of 8 × 10–15 V/K. The positive value of the Seebeck coefficient for Pr0.75Ba0.25MnO3 indicates that the dominant charge carriers are holes. This positive Seebeck coefficient aligns with the p-type behavior exhibited by the material.

Fig. 12
figure 12

Variation of Seebeck coefficient with temperature for Pr0.75Ba0.25MnO3

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The electrical conductivity (σ) of Pr0.75Ba0.25MnO3 is depicted in Fig. 13. It is generally observed that the electrical conductivity is directly proportional to the charge carrier concentration (n) and the mobility (μ). At 800 K, Pr0.75Ba0.25MnO3 exhibits a high electrical conductivity with a value of \(9.35.10^{18} \Omega^{ - 1} .m^{ - 1} .s^{ - 1}\). On the other hand, the thermal conductivity (κ) reflects a material's ability to conduct and transmit heat. Figure 14 displays the variation of thermal conductivity with temperature. As depicted, the thermal conductivity increases as the temperature rises. The maximum value of thermal conductivity observed in the figure is \(2.75.10^{14} W.K^{ - 1} .m^{ - 1} .s^{ - 1}\).

Fig. 13
figure 13

Variation of electrical conductivity vs. T(K) for Pr0.75Ba0.25MnO3

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Fig. 14
figure 14

Variation of electronic thermal conductivity κe/τ vs. T(K) for Pr0.75Ba0.25MnO3

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The figure of merit ZT is calculated by the following relation:

$$\mathrm{ZT}=T\frac{{S}^{2}\sigma }{\kappa },$$
(7)

with k = kl + ke, kl defines the lattice thermal conductivity and ke denotes electronic thermal energy. The lattice component of thermal conductivity kl caused by the phonon scattering is not considered in our calculations. As an estimate of Pr0.75Ba0.25MnO3 efficiency.

In Fig. 15, the variation of the ZT coefficient for Pr0.75Ba0.25MnO3 is displayed. The plot demonstrates that the ZT value increases as the temperature rises. The highest ZT value obtained is 0.19 at 800 K. Given that the thermal conductivity due to phonons is included in the calculations, this value of 0.19 is considered reasonable. It suggests that the thermal conductivity contribution from the lattice component (kl) is significant.

Fig. 15
figure 15

Variation of figure of merit vs. T(K) for Pr0.75Ba0.25MnO3

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5 Conclusions

In our study, we employed a combination of experimental techniques, density functional theory (DFT), and Monte Carlo simulations (MCSs) to investigate the electronic, thermoelectric, magnetic properties, and magnetocaloric effect of the Pr0.75Ba0.25MnO3 perovskite system. One of the key findings is that the Pr0.75Ba0.25MnO3 perovskite exhibits a half-metallic character, with a band gap of 2.274 eV in the minority band. This indicates that it can selectively conduct one spin direction while behaving as an insulator for the opposite spin direction. We also determined the transition temperature, which was found to be in close agreement with experimental results, validating the accuracy of our approach. By analyzing the magnetization as a function of temperature, we were able to deduce the magnetic entropy change (∆S). It was observed that ∆Smax increases with the applied magnetic field (h). This suggests that the ferromagnetic order in the samples exhibits long-range characteristics. Furthermore, we observed a significant magnetocaloric effect at a magnetic field of 5 T, accompanied by large values of ∆Smax and relative cooling power (RCP). These findings indicate that the Pr0.75Ba0.25MnO3 perovskite system holds promise for magnetic refrigeration applications. In terms of thermoelectric properties, our theoretical investigation revealed that Pr0.75Ba0.25MnO3 exhibits p-type behavior, with holes being the dominant charge carriers. This characterization of the material's electronic behavior provides valuable insights into its thermoelectric performance. Overall, the combination of the observed large magnetocaloric effect, relatively high RCP, and high magnetization makes the Pr0.75Ba0.25MnO3 perovskite system a promising candidate for applications in magnetic refrigeration and spintronics.