Field dependence of magnetocaloric properties of 20% Cr-doped Pr_0.7Ca_0.3MnO₃ perovskite

Abstract

In this study, we have investigated the effect of 20% Cr doping at the Mn site on the magnetic and magnetocaloric properties of Pr_0.7Ca_0.3MnO₃ polycrystalline manganites elaborated using the solid-state reaction. The X-ray powder diffraction shows that compounds crystallize in the orthorhombic system with Pnma space group. The parent compound Pr_0.7Ca_0.3MnO₃ is a well-known charge-ordered (CO) compound. The selective substitution of Cr for Mn in the pristine compound destroys the CO ordering observed in Pr_0.7Ca_0.3MnO₃, and a paramagnetic to ferromagnetic transition takes place. The field dependence of the magnetic entropy change ∆S and of the relative cooling power RCP is reported. The $\left| {\Delta S^{\hbox{max} } } \right|$ are found to be 0.8 and 2.87 J kg⁻¹ K in a magnetic field change of 5 T for x = 0 and 0.2, respectively. The RCP is found to be 253.2 J kg⁻¹ in the Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ at 5 T. With the scaling laws, the experimental ∆S and n collapse onto a universal curve for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample, where an average curve is obtained for ∆S. With the phenomenological universal curve, the magnetic entropy change ∆S curves are well predicted for Cr-substituted sample. Additionally, Banerjee’s criteria, the universal behavior and Landau theory of phase transitions were also studied to assess magnetic ordering in the Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ material.

Introduction

The magnetic cooling based on the magnetocaloric effect (MCE) has been of particular interest due to its high efficiency, compactness and environment-friendly nature compared to the conventional gas refrigeration [1–3]. The major advantages of magnetic refrigeration technology over compressor-based systems are that magnetic refrigeration eliminates toxic gas emission, is a relatively quiet technology (no compressors) and has a higher energy efficiency [4–8]. It offers a green solution to refrigerant fluids such as chlorofluorocarbons, hydrochlorofluorocarbons and ammonia-based compounds. Furthermore, the intrinsic better performance is the reason that this cooling system reduces the electric energy used to run refrigeration units [9–11]. Pr-based manganites have been the subject of extensive study, due to their large magnetocaloric effect (MCE) observed in the wide range of temperature [12–14]. Moreover, these materials present interesting properties such as low cost, good chemical stability, easy preparation and ability to control their magnetic transition temperatures (T _C) close to room temperature by substitutions. This makes these systems particularly promising for magnetic refrigeration [8, 13, 15–17]. The perovskite manganites are strongly correlated systems in which competing interactions give rise to a variety of interesting phenomena including colossal magnetoresistance and magnetocaloric effect [12]. Physical properties in these systems come from the interplay between the charge, orbital, spin and lattice degrees of freedom. These factors can be modified by calcination temperature [18], pressure [19], impurities [20, 21] and particle size [13, 15]. To maintain charge neutrality in doped manganite, the Mn ion exists in two different oxidation states such as Mn³⁺ and Mn⁴⁺. The mixed valence introduces mobile electrons leading to a quasi-metallic conductivity and the ferromagnetic ground state in these compounds. This behavior is usually interpreted by means of double-exchange (DE) interaction [22], Jahn–Teller effect [23] and phase separation [24]. The magnetic coupling between Mn³⁺ and Mn⁴⁺ ions results from the motion of an electron between the two partially filled d orbitals with strong on-site Hund’s coupling [25]. Since the magnetic property is dependent on the strength of double-exchange interaction between Mn³⁺ and Mn⁴⁺ via oxygen, the doping at the A-site is of great importance in modifying the double-exchange strength [26]. Recently, many manganites concerning second-order magnetic transition have been studied in an attempt to achieve a large magnetocaloric effect. The origin of the observed MCE is based on the adiabatic demagnetization: The application of a magnetic field in a ferromagnetic material induces a spin reorientation leading to decrease in the spin entropy. This process is accompanied by a rise of the lattice entropy when the field is applied adiabatically. The strong coupling between the spin and lattice has been shown by the observed lattice changes accompanying magnetic transitions in these manganites; the lattice structural change in the Mn–O bond distances and Mn–O–Mn bond angles with temperature which exhibit variation in the volume can cause an additional change in the magnetism [12, 27]. In Pr_1−xCa_xMnO₃ materials, the so-called CE-type charge/orbital-ordered (CO) state is stabilized. Even so, a FM state can be obtained by applying a magnetic field or by a small quantity of Mn substitution by means of other 3d transition metal elements like Fe³⁺, Co³⁺, Ni³⁺, Ru³⁺, Al³⁺ and Cr³⁺ [14, 28–31]. However, the Cr substitution is particularly interesting as Cr³⁺ is isoelectronic with Mn⁴⁺ and is a non-Jahn–Teller ion [32]. A literature overview highlights that the ionic radius of chrome ion is near to that of manganese ion, and thus, it can easily substitute Mn on B-site in the manganites system. Much investigation was focused on the structure, electrical and magnetic properties performance of Cr-doped manganites. Indeed, Cr doping in the Mn site is an effective way to change the Mn³⁺ –O– Mn⁴⁺ network, because it will affect remarkably the double exchange and, consequently, it will change the final physical properties of the material [28]. Therefore, one might expect to induce a ferromagnetic phase in an otherwise antiferromagnetic, charge-ordered ground state. In fact, a ferromagnetic phase has been observed to coexist with the CO phase in Cr-substituted Pr_0.5Ca_0.5MnO₃ for low Cr concentrations at low temperatures. The results account for the replacement of Mn³⁺ by Cr³⁺, which makes more disorder in the compound, resulting in an increase in the disorder parameter and the fluctuation of the spin. This raison indicates that the chromium substitution at Mn site alters magnetic response in the sample. So, the effect of substitution on the parent material has been the interesting subject for the systematic study of various characteristics like structural and magnetocaloric behavior [33]. In order to continue our investigation in the Pr-based manganites, we report in this paper the field dependence of Cr doping on magnetic and magnetocaloric properties of Pr_0.7Ca_0.3Mn_1−xCr_xO₃.

Experimental

Ceramic materials with the nominal composition of Pr_0.7Ca_0.3MnO₃ (x = 0.0) and (x = 0.2) samples were synthesized from high-purity precursors: Pr₆O₁₁, CaCO₃, Cr₂O₃ and Mn₂O₃ (Aldrich 99.9%; USA) by the conventional solid-state reaction method at high temperature. Stoichiometric ratios of starting chemicals precursors were taken and thoroughly mixed for 35 min using an agate mortar. The obtained powders were then pressed into cylindrical shape of diameter 10 mm using a hydraulic press by applying a pressure of 2 tons cm⁻² and grounded and pre-annealed at 800, 1000, 1100 and 1300 °C for 24 h (heating rate of the order 10 °C per minute was used) for each cycle to ensure a better crystallization according to the following reaction [34]:

$$\frac{0.7}{6}\Pr_{6} {\text{O}}_{11} + 0.3\,{\text{CaCO}}_{3} + (1 - x){\text{MnO}}_{2} + x{\text{Cr}}_{2} {\text{O}}_{3} \to \;\Pr_{0.7} {\text{Ca}}_{0.3} {\text{Mn}}_{1 - x} {\text{Cr}}_{x} {\text{O}}_{3} + \delta \,{\text{CO}}_{2} + \delta '{\text{O}}_{2} .$$

The obtained disk-shaped samples were cooled from high temperature to room temperature following the cooling inertia of the furnace (~8 h). As both samples have been elaborated in air, they are consequently stoichiometric in oxygen [35, 36]. Phase purity, homogeneity and cell dimensions were determined by powder X-ray diffraction at room temperature for both samples. The room temperature X-ray diffraction (XRD) patterns were measured by a PANalytical X’PERT Pro MPD diffractometer, using θ/2θ Bragg–Brentano geometry with diffracted beam monochromatized Cu Kα radiation. The diffraction patterns were collected by steps of 0.017° over the angle range 10°–80°. Structural analyses were carried out by the standard Rielveld method [37, 38] using Fullprof software. Magnetization measurements versus temperature in the range 5–300 K and versus magnetic applied field up to 5 T were taken using a vibrating sample magnetometer. Magnetocaloric results were deduced from the isothermal magnetization measurements.

Results and discussion

X-ray diffraction analysis

The results of the XRD studies at room temperature of Pr_0.7Ca_0.3MnO₃ and Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ materials show that our samples crystallize in the orthorhombic system with the Pnma space group. The atoms (Pr, Ca) are located at 4c (x, 1/4, z), (Mn, Cr) at 4b (1/2, 0, 0), O₁ at 4c (x, 1/4, z) and O₂ at (x, y, z). Figure 1 shows a typical Rietveld refinement for both samples registered at 300 K including the observed and calculated profiles as well as the difference profile. The quality of the refinement is evaluated through the goodness of the fit indicator χ ² (1.21 for x = 0.0 and 1.41 for x = 0.2). A good fit between the observed and the calculated profiles was obtained. The cell parameters for Pr_0.7Ca_0.3MnO₃ are found to be a = 5.459(8) Å, b = 7.674(1) Å and c = 5.430(3) Å. For 20% Cr-doped manganite, they are found to be a = 5.426(4) Å, b = 7.666(8) Å and c = 5.446(4) Å. One may remark that the volume decreases from 227.52 to 226.58 Å³ with the replacement of partial Mn ions by Cr ions. This can be explained by the difference in the ionic radii between the larger Mn³⁺ ions (0.65 Å) which substituted for the smaller Cr⁺³ (0.61 Å) in B sites. It is noticed that for 20% Cr substitution, the average distances of Mn–O drop from 1.970(1) to 1.959(1) Å, and the average angles of Mn–O–Mn jump from 157.55(4)° to 159.89(8)°. However, chromium ions with smaller ionic radius induce local distortions of the 〈Mn–O–Mn〉 angle in the system. The internal stress caused by substituting Mn³⁺ by Cr³⁺ may result in a larger rotation of the MnO₆ octahedra [39]. These modifications can be understood on the basis of structural parameters, B-site average ionic radius ‘<r_B>,’ B-site size variance ‘σ ²_B ’ and tolerance factor ‘t’. Mainly, the remarkable observation from the XRD data was that the principal diffraction peak position of Pr_0.7Ca_0.3MnO₃ shifted toward higher diffraction angle with Cr substitution, which confirms that the unit cell volume decreases with the doping of Cr. Thus, the lattice structural change, which results in a variation of the volume, can cause an additional change in the magnetic properties of the material.

Fig. 1

X-ray powder diffraction patterns and refinement at room temperature for Pr_0.7Ca_0.3Mn_1−xCr_xO₃ (x = 0 and x = 0.2)

Magnetic properties

We plot in the inset of Fig. 2 the low field (µ ₀ H = 0.05 T) temperature dependence of the magnetization M(T) for the pristine compound. The measurements were taken after field cooling (FC) process. A signature of anomaly can be observed in the M(T) curve around 250 K, associated with the charge-ordering state (ordering of electrons and holes of Mn³⁺ and Mn⁴⁺ ions, respectively). This behavior is promoted by the cooperative Jahn–Teller ordering of, for example, orbitals. At T ~ 100 K, the rapid increase in M(T) in this compound is generally attributed to polarization of the Pr moments. Figure 2 shows the field dependence of magnetization measurements versus magnetic applied field at several temperatures. At low temperature, the M(µ ₀ H) of Pr_0.7Ca_0.3MnO₃ increases linearly up to µ ₀ H = 3.2 T and then shows a rapid increased metamagnetic transition due to the progressive conversion of the charge-ordered (CO) state into a ferromagnetic behavior [40]. At high temperature, magnetization curves completely become linear with magnetic field as expected for the paramagnetic state. In our work, the chromium substitution is established to destroy the charge ordering observed in the parent compound and to drive the system toward a ferromagnetic state at low temperatures.

Fig. 2

Isothermal magnetization curves at various temperatures for Pr_0.7Ca_0.3MnO₃. The inset is the temperature dependence of magnetization under 0.05 T for Pr_0.7Ca_0.3MnO₃ compound

To recall the characteristic magnetic behavior of Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ system, the M-T measurement employing the field-cooled (FC) protocol at 0.05 T in the temperature range of 5–300 K is shown in Fig. 3a. The sample exhibits a clear transition from paramagnetic to ferromagnetic state with decreasing temperature. The replacement of Mn³⁺ by 20% Cr³⁺ introduces a diminution of the Mn³⁺ ion concentration which decreases the Mn³⁺/Mn⁴⁺ ratio, changes both the average bond length 〈Mn–O〉 and angle 〈Mn–O–Mn〉 and influences ferromagnetic interaction strengths [41]. This effect has a direct impact on the double-exchange (DE) interactions. Also, it affects other mechanisms such as ferromagnetic interactions and antiferromagnetic super-exchange which are possible to occur between Cr and Mn ions. Here, the main role of the Cr³⁺ ion is to polarize the Mn³⁺ ions. Consequently, these factors lead to a second-order magnetic phase transition in this compound and reveal a strong variation of magnetization around the Curie temperature T _C. The derivative dM/dT of the M(T) curve is used to determine the value of Curie temperature T _C ~ 155 K. The dM/dT curve reveals a strong variation of magnetization around the T _C. It indicates that there is a possible large magnetic entropy change around T _C [42].

Fig. 3

a Temperature dependence of magnetization under 0.05 T for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ compound. The inset is the dM/dT curves. b Magnetization versus magnetic applied field at 20 K for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. The insets determination of the saturation magnetization of the samples at 20 K and isothermal magnetization curves at various temperatures for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. c M ² versus µ ₀ H/M isotherms for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. The inset is the temperature dependence of the spontaneous magnetization M _sp and 1/χ for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃

To understand the nature of the magnetic order fluctuations in our samples, we measured the field dependence of isothermal magnetization at several temperatures. The inset of Fig. 3b shows isothermal magnetization curves for the Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample. This plots show clearly the evolution of our material at low temperatures from the CO state in the undoped compound to the ferromagnetic state in the substituted sample. For Cr-doped system and below T _C, the M(µ ₀ H) curves increase sharply for magnetic field <1 T and tend to saturation. This result confirms well the ferromagnetic behavior at low temperature of our material. It can be expected that the Cr substitution at the Mn sublattices is a disorder effect, which can disrupt the cooperative Jahn–Teller ordering of, for example, orbitals, which in turn can lead to destabilization of the CO state [43, 44]. We plot in Fig. 3b the magnetic isotherm at 20 K for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. The obtained curve was used for evaluation of the characteristic magnetization values by means of procedures described in [45]. Namely, the saturations magnetizations M _sat at 20 K was calculated from M versus 1/µ ₀ H plot at 1/µ ₀ H → 0 (see inset of Fig. 3b). The obtained values of M _sat = 84.15 emu g⁻¹ were of the same order as that of Pr_0.7Ca_0.3Mn_0.95X_0.05O₃(X = Cr, Ni, Co and Fe) [14], Pr_0.5Eu_0.1Sr_0.4MnO₃ [13] and La_0.85Ag_0.15Mn_1−yM_yO₃ [46]. At low temperature, the spontaneous magnetization M _sp(exp) determined by the extrapolation of the M versus µ ₀ H data in the high field to zero field is found to be 72.55 emu g⁻¹ (~2.77μ_B/Mn) (see Fig. 3b). The magnitude of the M _sp(exp) is smaller than the theoretical value of 3.5 μ_B/Mn calculated for full spin alignment, leading to small canted spins. This result confirms the existence of the antiferromagnetic (AFM) interactions and signifying that the magnetic moment of the rare-earth ions are polarized and pinned by the Mn sublattice [47, 48]. The order of the transition in Cr-doped manganite is clarified by Banerjee criterion [49]. This criterion suggests a magnetic transition of first order with negative slopes and a second-order transition with positive slopes. Figure 3c shows the Arrott plots (M ² vs μ ₀ H/M) for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample. The M ² versus μ ₀ H/M curves exhibit a positive slope without inflection points, indicating that the ferromagnetic–paramagnetic transition is of second order [50]. It can be mentioned that Arrott plot assumes the critical exponents following the mean field theory (β = 0.5 and γ = 1). Hence, linear behavior of isotherms in high field indicates the presence of mean field interactions. The advantages of this plot are (1) T _C can be determined accurately since the isotherm at T _C will pass through the origin, (2) it gives spontaneous magnetization M _sp(T) as intercept on positive M ² axis, and (3) it directly gives the inverse of the magnetic susceptibility χ ⁻¹ versus T as an intercept on µ ₀ H/M axis. The T _C value deduced from the Arrott curves is close to this determined from M(T) curve. The inset of Fig. 3c shows the temperature dependence of the spontaneous magnetization M _sp and the inverse of the magnetic susceptibility χ ⁻¹ evolution versus temperature for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. The M _sp(T) curve confirms the FM behavior of chromium-substituted material at low temperatures. The critical exponents β and γ are defined by:

$$M_{\text{sp}} (T) = M_{\text{sp}} (0)\left[ {1 - \frac{T}{{T_{\text{C}} }}} \right]^{\beta } \, {\text{and}}\,\chi^{ - 1} (T) = \chi^{ - 1} (0)\left[ {\frac{T}{{T_{\text{C}} }} - 1} \right]^{\gamma }$$

(1)

The estimated values of the critical exponents for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample associated with the spontaneous magnetization and the inverse of the magnetic susceptibility are found to be β = 0.365(9) and γ = 1.125(1). Fundamentally, β keeps close to the value expected for the 3D Heisenberg ferromagnets with ferromagnetic short-range interactions, while γ is near the critical value of the mean field theory. Basically, the determined value of δ will be close to that calculated from the Widom relation δ = 1 + γ/β = 4.074(8). These values of critical exponents are between the predicted values for the three-dimensional Heisenberg model (β = 0.37, γ = 1.39, δ = 4.7) and for the mean field model (β = 0.5, γ = 1, δ = 3) as observed in other manganites [51, 52].

Magnetocaloric effect

The isothermal magnetization curves are used to determine the magnetic entropy change ΔS of our samples under a magnetic applied field change Δµ ₀ H according to the classical thermodynamic theory based on Maxwell’s relations using the following expression:

$$\Delta S(T,\mu_{0} H) = \sum {\left( {\frac{{M_{\text{i}} - M_{{{\text{i}} + 1}} }}{{T_{{{\text{i}} + 1}} - T_{\text{i}} }}} \right)} \, \Delta \mu_{0} H_{\text{i}}$$

(2)

In this equation, M _i and M _i+1 are the experimental values of magnetization measured at temperatures T _i and T _i+1, respectively, under magnetic applied field µ ₀ H _i. Experimentally, the ΔS is often evaluated by some numerical approximation methods. One is to use directly the measurements of M–T curve under different applied magnetic fields. Another method is estimation through either the adiabatic change in temperature (ΔT _ad) by the application of a magnetic field or isothermal magnetization measurements at several temperatures. In this work, we have used the third method based on magnetization measurements versus a magnetic applied field. Figure 4a depicts the magnetic entropy change as a function of temperature at several magnetic fields for the undoped compound. Pr_0.7Ca_0.3MnO₃ material exhibits a positive entropy change due to the dominance of the antiferromagnetic interactions with a maximum value of 0.8 J kg⁻¹ K upon a magnetic field change of 5 T. The substitution of Mn³⁺ by 20% Cr³⁺ introduces a diminution of the Mn³⁺ ion concentration which changes both the average bond length 〈Mn–O〉 and angle 〈Mn–O–Mn〉 and influences MCE properties. Figure 4b shows the magnetic entropy change ΔS evolution as a function of temperature of Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample. The ΔS exhibit a broad negative peak around T _C (indicating a ferromagnetic behavior) for the substituted material. For Δµ ₀ H = 2 T and 5 T, the maximum magnetic entropy change |ΔS ^max| is found to be 1.5 and 2.87 Jkg⁻¹ K⁻¹, respectively. In particular, the magnetic entropy change in the Cr-doped sample demonstrates a significant broad variation with temperature around T _C. Such variation covering a broad temperature range is beneficial for magnetic cooling applications [53]. This is a very interesting result, because we are able to change MCE and width of peak temperature range of the material by substituting manganese ion and may select optimization, which is desirable for Ericsson-cycle magnetic refrigeration and heat exchange technology [2]. In the magnetic cooling technology, it is important that the magnetocaloric effect (MCE) extends over a large temperature range. We can then consider the magnitude of ΔS and its full width at half maximum, known as the relative cooling power (RCP) [12]. The RCP is evaluated as ${\text{RCP}} = \left| {\Delta S_{\text{M}}^{\hbox{max} } \, (T,H)} \right| \times \delta T_{\text{FWHM}}$. The obtained value of RCP for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ is found to be 253.2 J kg⁻¹ at 5 T. In Table 1, we have compared the entropy changes in our sample with other B-site multi-element doping effect like Cr, Ni and Fe in several manganites system. The obtained values are comparable with those mentioned in Table 1. All these studies indicate that the large MCE in the perovskite manganites can originate from the spin-lattice coupling related to the magnetic ordering process. The lattice structural change in B–O bond distances and B–O–B, bond angles with temperature, which results in a variation of the volume, can cause an additional change in the magnetic properties of manganite.

Fig. 4

a Magnetic field dependence of magnetic entropy change for Pr_0.7Ca_0.3MnO₃. b Experimental (open symbols) magnetic entropy change and predictions using the universal curve (continuous lines) for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. c The collapse of the experimental data onto the universal curve for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃

Table 1 T _C, ΔS _max and RCP values for the present samples under a magnetic applied field of 5 T and for some previous works

Currently, a phenomenological construction of universal curve for ∆S(T) has been proposed as a method to compare the properties of several magnetic materials and to make extrapolations to fields and/or temperatures outside the available experimental range [61]. It is interesting to note that the collapse of these curves is observed not only in the near vicinity of the transition but in a wide temperature range. This raises the question as to whether the collapse of the ∆S(T) curves is a manifestation of a universal behavior or not. A study on first-order phase transitions should shed light on the subject: A breakdown of the universal curve is expected for first-order phase transitions if the underlying cause is universality associated with critical phenomena and intrinsic to second-order phase transitions. Otherwise, ∆S(T) curves may collapse in the same way for first-order as for second-order transitions. The magnetocaloric universal behavior of Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ is studied as a verification of Banerjee criterion [49]. This new criterion based on the entropy changes can be helpful in distinguishing the order of magnetic phase transitions. For ferromagnets undergoing second-order phase, the rescaled magnetic ΔS curves follow a universal behavior. While the scaled ΔS curves do not collapse as a single curve, the ferromagnets undergo a first-order phase transition. The phenomenological universal curve can be constructed by normalizing and rescaling the temperature axis below and above T _C as defined in Eq. (3) with an imposed constraint that the position of two additional reference points in the curve corresponding to θ ± 1. This procedure is used to compare the properties of several materials and to make extrapolations to temperatures and/or fields outside the accessible experimental range. A way of doing this would be to normalize all ∆S curves with their peak entropy change, respectively, as ∆S′ = ∆S/2, and the temperature axis is rescaled by a new term denoted by θ and is given as below [7, 62]:

$$\theta = \left\{ {\begin{array}{*{20}c} { - \left( {T - T_{\text{C}} } \right)/\left( {T_{{{\text{r}}1}} - T_{\text{C}} } \right);} & {T \le T_{\text{C}} } \\ {\left( {T - T_{\text{C}} } \right)/\left( {T_{{{\text{r}}2}} - T_{\text{C}} } \right);} & {T > T_{\text{C}} } \\ \end{array} } \right.$$

(3)

where T _r1 and T _r2 are the temperatures of the two reference points of each curve that correspond to ∆S/2. The universal curves of Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ at several magnetic fields are constructed in Fig. 4c. It is seen that all curves collapse onto a single master curve revealing second-order phase transition in this compound which confirms the result obtained by Banerjee criterion. The average curve is obtained from the universal scaling, as shown as solid curve in Fig. 4c which gives a smoother depiction of the curve. The mentioned curve, once the temperature axis is backtransformed from the reduced temperature to the unnormalized one, allows making extrapolations to lower/higher temperatures for the high field data and obtaining a finer description of the peak for the low field curves [63, 64]. The combination of all data in the average universal curve and the inverse transformation of the experimental ΔS(T) axes permits to provide a prediction about the behavior of the curve. The good agreement between predicted and experimental points is remarkable. The predicted curves are given as red solid curves for comparison. A comparison between both curves is presented in Fig. 4b for Cr-substituted material.

The change in specific heat associated with a magnetic field variation from zero to μ ₀ H can be calculated from the magnetic contribution to the entropy change induced in the materials, ∆C _P, by the following expression:

$$- \Delta C_{\text{P}} = T\frac{{\partial \Delta S_{\text{M}} }}{\partial T}$$

(4)

Through this relation, ΔC _P of Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ material versus temperature at several magnetic fields is depicted in Fig. 5. As the figure shows anomalies are observed in all curves around T _C, which are due to the magnetic phase transition [20, 65, 66]. The magnitude of ΔC _P varies abruptly from (+) to (−) near the Curie temperature and speedily decreases with decreasing temperature. The sum of the two parts is the magnetic contribution to the total specific heat which affects the cooling or heating power of the magnetic refrigerator [67]. Specific heat presents the advantage of delivering values necessary for further refrigerator design, should the material in question be selected [68].

Fig. 5

Heat capacity changes −∆C _p as function of temperature for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃

It has also been demonstrated that the field dependence of the maximum magnetic entropy change shows a power law ΔS ^max ≈ a(μ ₀ H)ⁿ for some materials [53, 61]. In order to check the validity of n value, the experimental data of field dependence of −∆S at T _C for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample are plotted in Fig. 6a. The fitting value of n is about 0.781(9) which is significantly higher than 2/3, as predicted by the mean field model. The field dependence of the RCP for the Cr-doped manganite is analyzed. In addition, the magnetic field dependence of the RCP is shown in Fig. 6b. It can be expressed as a power law by taking account of the field dependence of entropy change ∆S and reference temperature into consideration, i.e., RCP ≈ A(μ ₀ H)^N where N is the critical exponent of the magnetic transition. RCP should scale with field as a power law with an exponent N. The N value calculated from experimental data is N = 1.138(7) for our material. This N result is compatible with values previously reported for (Na, K)-doped Pr_0.6Sr_0.4MnO₃ [69, 70] and those observed for Gd alloys (N = 1.16) by Law et al. [71]. Refrigerants with a wide working temperature span and high RCP are in fact very beneficial to magnetic cooling applications [5]. The polycrystalline Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ manganite is particularly pronounced with wide operative temperature range, as well as inexpensiveness and easy fabrication, and therefore could be considered as a potential magnetic cooling [16, 72].

Fig. 6

Magnetic field dependence of: a magnetic entropy change ∆S _max and b RCP for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃

Moreover, the n exponent can be locally calculated from the logarithmic derivative of the ΔS with magnetic field as [71]:

$$n\left( {T,\mu_{0} H} \right) = \frac{{{\text{d}}\ln (\left| {\Delta S} \right|)}}{{{\text{d}}\ln (\mu_{0} H)}}$$

(5)

given n = 1 in the ferromagnetic range for T ≪ T _C, n = 2 in the paramagnetic range for T ≫ T _C and $n(T_{\text{C}} ) = 1 + \left[ {\frac{\beta - 1}{\beta + \gamma }} \right]$ at T = T _C. According to the mean field approach, n = 2/3. However, current experimental result reveals deviation from n = 2/3 in the case of some ferromagnetic materials [61]. Figure 7a depicts the temperature dependences of the local exponent n for several values of µ ₀ H describing the field dependence of the magnetic entropy change calculated from Eq. (5) when fields of different values are applied. It has to be noted that n evolves with field in the entire studied temperature range. However, n(T, µ ₀ H) plots show the minima around the Curie temperature. The minimum value of n reaches 1 T ≪ T _C and 2 T ≫ T _C. In fact, in the present case the value of n at T _C is greater than the predicted value of 2/3 in the mean field approach due to the local inhomogeneities or super-paramagnetic clusters in the vicinity of a transition temperature T _C existed in the Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ specimen [73]. A similar behavior has been observed in La_0.7Ca_0.3Mn_0.91Ni_0.09O₃ system [74].

Fig. 7

a Local exponent n at several magnetic fields for Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ compound. b Local exponent n as a function of the rescaled temperature θ for all magnetic fields

The n(T, µ ₀ H) curves also collapse when plotted against the same rescaled temperature axis for which the normalized values of −∆S(T) collapse onto the universal curve. The procedure consists of identifying the reference temperatures as those which have a certain value of n. This value has been arbitrarily selected as n(Tr) = 1.5. The transformed curves of different fields are plotted in Fig. 7b. This figure shows the rescaled temperature θ dependence of the exponent n for the Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ compound. For several magnetic fields, all the data points collapse into a single master curve revealing universal behavior in Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. In general, deviations from the collapse might indicate either the influence of the demagnetizing field associated with the shape of the sample [75] or to the presence of additional magnetic phases [76].

The observed ∆S(T) in La-based manganites have been fitted with models based on Weiss molecular mean field theory or Landau theory of phase transition. It was found that the Landau’s theory fits the ∆S(T) data very closely. Hence, we have attempted to fit the ∆S(T) data of Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample to the Landau theory using the Gibbs free energy following:

$$G(M,T) = G_{0} + \frac{1}{2}A \times M^{2} + \frac{1}{4}B \times M^{4} + \frac{1}{6}C \times M^{6} - M\mu_{0} H$$

(6)

where A(T), B(T) and C(T) are temperature dependent usually known as Landau coefficients.

In order to retain the state of energy minimization in Eq. (6), the magnetic equation of the state is given as

$$\frac{{\mu_{0} H}}{M} = A(T) + B(T) \times M^{2} + C(T) \times M^{4}$$

(7)

These Landau coefficients A, B and C are identified from polynomial fits of the Arrott plots to Eq. (7) which are displayed in Fig. 8. According to Inoue–Shimizu model [77], A(T) is the inverse of the magnetic susceptibility, with a minimum at T _C. The sign of B can be used to distinguish the magnetic transitions order, as it indicates the first-order transition (B < 0) or the second-order transition (B > 0). As shown in Fig. 8, B(T) has a positive value when A(T) reaches the minimum, denoting a second-order transition in Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. The corresponding magnetic entropy can be obtained by differentiating Gibbs free energy with respect to temperature as:

$$\begin{aligned} \Delta S(T,\mu_{0} H) & = S(T,\;0) - S(T,\mu_{0} H) \\ & = - \frac{1}{2}A'(T) \times (M_{0}^{2} - M^{2} ) - \frac{1}{4}B'(T) \times (M_{0}^{4} - M^{4} ) \\ & \quad - \frac{1}{6}C'(T) \times (M_{0}^{6} - M^{6} ) \\ \end{aligned}$$

(8)

Fig. 8

Temperature dependence of the Landau coefficients A(T), B(T) and C(T) for the Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ sample [the units for A(T), B(T) and C(T) are T ² kg/J, T ⁴ kg³/J³ and T ⁶ kg⁵/J⁵, respectively] and experimental and theoretical magnetic entropy changes ΔS for the Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃ at magnetic fields of 1–5

Thus, ∆S(T) can be calculated using Eq. (8) where A′(T), B′(T) and C′(T) are the temperature derivatives of Landau coefficients. The value of M ₀ can be obtained by extrapolating the magnetization at µ ₀ H = 0. As shown in Fig. 8, the red lines represent the calculated magnetic entropy change by the Landau theory and the symbols represent the experimental data at µ ₀ H = 1,2,3,4 and 5 T, respectively. The plot exactly matches the experimental data above Curie temperature. However, this mean field approximation cannot interpret the results below T _C. The obtained results are similar to that observed in hole-doped manganites where Landau analysis nicely corroborated the experimental data for polycrystalline compound [78]. This agreement considering the fact that the present model does not take into account the influence of the Jahn–Teller effect and exchange interactions on the magnetic properties of manganites. The analysis clearly reveals the importance of magnetoelastic coupling and electron interaction in understanding the magnetocaloric properties of lanthanum manganites [11, 17, 78].

Conclusions

In summary, the polycrystalline compounds were prepared by the conventional solid-state reaction method. X-ray results indicated that all the samples are single phase and crystallize in the orthorhombic system with Pnma space group. Magnetocaloric effect and transition order were investigated by using magnetization measurements. The 20% Cr substitution drives the system from charge-ordered state to ferromagnetic one. Our results indicate that chromium substitution in charge-ordered manganites is one of the possible ways to enhance magnetocaloric effect in these complex oxides and it is certainly better than doping at Mn site in ferromagnetic manganites. Banerjee criterion, the universal behavior and Landau analysis reveal a second-order transition in Pr_0.7Ca_0.3Mn_0.8Cr_0.2O₃. Comparison between the calculated and experimental values of entropy changes indicates a potential transition of magnetic interaction in the vicinity of T _C. The Cr-substituted material possesses some appropriate properties for a good candidate as magnetic cooling in a low temperature range.