Evolution of Griffiths Phase and Critical Behaviour of La1-xPbxMnO3+-y Solid Solutions

Polycrystalline La1-xPbxMnO3+-y (x = 0.3, 0.35, 0.4) solid solutions were prepared by solid state reaction method and their magnetic properties have been investigated. Rietveld refinement of X-ray powder diffraction patterns showed that all samples are single phase and crystallized with the rhombohedral structure in the R-3c space group. A second order paramagnetic to ferromagnetic phase transition was observed for all materials. The Griffiths phase (GP), identified from the temperature dependence of the inverse susceptibility, was suppressed by increasing magnetic field and showed a significant dependence on A-site chemical substitution. The critical behaviour of the compounds was investigated near to their Curie temperatures, using intrinsic magnetic field data. The critical exponents (\b{eta}, {\gamma} and {\delta}) are close to the mean-field approximation values for all three compounds. The observed mean-field like behaviour is a consequence of the GP and the formation of ferromagnetic clusters. Long-range ferromagnetic order is established as the result of long-range interactions between ferromagnetic clusters. The magnetocaloric effect was studied in terms of the isothermal entropy change. Our study shows that the material with the lowest chemical substitution (x = 0.3) has the highest potential (among the three compounds) as magnetic refrigerant, owing to its higher relative cooling power (258 J/kg at 5 T field) and a magnetic phase transition near room temperature.


INTRODUCTION
The perovskite manganite oxides have been studied extensively for their colossal magnetoresistance, 1 Jahn-Teller (J-T) distortions, 2 metal-insulator transitions, 3 and most importantly the strong coupling between lattice, charge, orbit and spin degrees of freedom. 4 Moreover, owing to their chemical stability, low-cost synthesis, zero field-hysteresis and large value of the isothermal entropy change near room temperature, 5 the possible application in energy-efficient and environment-friendly magneticrefrigeration 6-10 has drawn attention. The general formula of a manganite perovskite is AMnO3, where A is a trivalent atom. Replacing the A site atom with a divalent atom with smaller or larger atomic radius, known as hole doping, 1 results in a change of unit cell volume which is often referred to as a change in chemical pressure inside the material. An increment of the unit cell volume corresponds to a decrease of the chemical pressure. 11 The chemical substitution process causes the replacement of some of the Mn 3+ ions with Mn 4+ ions to maintain charge equilibrium. This creates the Mn 3+ -O 2--Mn 4+ doubleexchange interaction in the compound, which is responsible for the low temperature ferromagnetic phase of the material and hence determines the Curie temperature ( ) of the material. Thus, varying the amount of divalent cation in the A site, enables to tune the phase transition temperature of the material.
Phase inhomogeneity and quenched disorder in hole-doped manganites often leads to Griffiths phase (GP) formation. In manganites, quenched disorder can arise from several sources, such as, bending of the Mn-O-Mn bond angle, size variation of cations by chemical substitution, J-T distortions, etc. 12 In 1969, R. Griffiths observed a nonanalytic behaviour of the magnetization above in a randomly diluted Ising ferromagnet, caused by the formation of ferromagnetic clusters above TC. 13 In a GP material, some lattice sites are either vacant or occupied with spin-zero atoms. 12 16. 15 According to the theoretical model of Bray 16 and Bray-Moore 17 , the ferromagnetic clusters in the GP grow in size with decreasing temperature in a way that it creates an effective magnetic long-range order in the material. 18 One approach to understand the magnetic ordering, is to investigate the critical scaling behaviour and to evaluate the critical exponents (β, γ and δ) as mentioned by Arrott-Naokes 19 and Kouvel-Fished. 20  .82) and tricritical mean-field (β = 0.25, γ = 1, δ = 5). 21 Previously, the magnetic ordering of La0.7Pb0.3MnO3 has been explained in the framework of the long-range, mean-field like interactions at lower field and the observed critical exponents at higher field were reported to be more close to the 3D short-range interaction model. 22,23 Moreover, independent of magnetic field, a short-range, 3D-Heisenberg like critical behaviour was observed for La0.9Pb0.1MnO3. 24 While performing the critical analysis it is important to perform the analysis in the critical region (~TC ± 0.01TC) as described by Arrott and Noakes 19 as well as to compensate for the demagnetization effect and to use the intrinsic magnetic field in the analysis. 25 There are a large number of publications 26-37 on manganite materials reporting on critical scaling analysis without considering the two above mentioned criterions, which raises questions on the reliability of such analysis. Some other works 22 compound. The change of magnetic properties and evolution of the GP as a function of chemical substitution have been investigated. With critical scaling analysis (using intrinsic magnetic field in the analysis), the type of critical behaviour in the LPMO samples have been determined and compared with theoretical models. The magnetocaloric effect of the LPMO samples has also been studied. To the best of our knowledge, this is the first report of the evolution of the GP in the Pb-substituted LaMnO3 system.

II. EXPERIMENTAL DETAILS
Polycrystalline LPMO samples were synthesized by using the solid-state reaction route. Stoichiometric amounts of La2O3, PbO and MnCO3 powders were mixed together and calcinated at 1473 K for 24 h in Pt crucible with intermediate heating and grinding. All high-temperature treatments were performed at ambient atmosphere with a programmed heating and cooling rate of 50 K/hour. The samples were characterized using X-ray powder diffraction (XRPD) at 295 K by using Cu-Kβ radiation (Bruker D8 Advance diffractometer) with a step size of 0.013° (counting time was 15 s per step). The oxidation states of the samples were determined by using X-ray photoelectron spectroscopy (XPS). A "PHI Quantera II" system with an Al-Kα X-ray source and a hemispherical electron energy analyzer with pass energy of 26.00 eV were used to collect the XPS spectra. The magnetic measurements were performed using a superconducting quantum interference device (SQUID) based Quantum Design magnetometer (MPMS) in the temperature range from 390 K to 5 K with a maximum field of 5 T.

A. Structural and chemical properties
The single-phase formation of the LPMO samples has been confirmed by the Rietveld refinement of the XRPD patterns (Figure 1 (a)). Three structural models; orthorhombic (Pnma), rhombohedral (R-3c) and monoclinic (P21/n), were tested on the samples by using the Fullprof program 45    The ionic state of Mn plays an important role in the lattice distortion. To check the manganese ionic state, the Mn-2p peak has been analysed by XPS (Figure 2 Figure 3 and values shown in Table 2 obtained from critical scaling analysis), which indicates that for LPMO, the double-exchange model is not sufficient to describe its magnetic behaviour.
To understand the magnetic behaviour in more detail, the variation of the inverse DC-susceptibility with temperature was studied (Figure 4(a)). For a pure PM-FM transition, in the paramagnetic region, the magnetic susceptibility ( ) follows the Curie-Weiss law, Where, is the Curie constant, is temperature and is the Curie-Weiss temperature. For LPMO, deviations from the expected linear dependence between 1/ and are observed. A deviation from a linear dependence can also be expected for ferrimagnets. But for ferrimagnets, the tangent in the paramagnetic region at temperatures >> of the 1/ vs. curve should have negative intercept with the temperature axis, 50 which is not observed for LPMO.
Based on the original work of Griffiths 13 and its following works, 15,51-54 the magnetic behaviour of LPMO resembles that of a GP material. According to this model the magnetic system at temperatures > can be described as a disordered system with randomly distributed FM clusters embedded in a PM background. 54 This GP can be characterized by where is a constant and is the transition temperature associated with the GP. 53,55 There is also another characteristic temperature ( ), above which the pure FM behaviour (following the Curie-Weiss law) is observed. 15 The GP behaviour is observed in the temperature regime ≤ ≤ . 55 The value of is a measure of the deviation from a "pure" ferromagnetic behaviour. Therefore, a higher value of corresponds to a more disordered state. The A-site chemical substitution and field dependence of is shown in Figure 4(b). The most disordered state is observed for the = 0.3 sample and with increasing A-site chemical substitution the effect of disorder is suppressed. Also, with increasing magnetic field, the GP singularity is suppressed.

C. Scaling analysis
The critical scaling analysis has been performed using the intrinsic magnetic-field (Hi) to avoid the effect of the demagnetization field. 25 The magnetic transition was analysed using Arrott-plots 56 Where, = − . The exponents can according to Widom scaling be related as, 21,59 = 1 + , (7) The value of TC is extracted from the Arrott plots as described by D. Kim et al. 25 and are listed in Table   2. The FM-PM transition is expected at TC, but the GP does not allow the material to leave its FM-state until TG. As the GP-region is near the critical region, it limits the use of Equations (4) and (5). To avoid any effect of the GP in the critical analysis, instead of using Equations (4) and (5) the static scaling hypothesis has been used, 25 according to which, where, + is for > 0 and − is for < 0 . Using Equation (7), the equation of state can be expressed as, Using Equation (6), the value of has been extracted for the different samples and then using Equation (9) the value of has been found from the best data-collapse, as is shown in Figure 6(a). The values of the extracted critical exponents are listed in Table 2.   The extracted values of , and , resemble the values valid for the mean-field model. In order to further establish which of the four models (mean-field, 3D-Heisenberg, 3D-Ising and tricritical meanfield) best fits the results of the LPMO samples, modified Arrott plots are shown in Figure 5 for the =0.4 sample (the behaviour of the other samples is similar). For the best fitted model, the critical isotherm at = should pass through the origin and all other isotherms near should be parallel to the critical isotherm. 25 To confirm this parallel-isotherm feature near the TC, the slopes for the =0.4 sample with respect to the critical isotherms are presented in Figure 6(b) for the different models. From this relative slope plot (the behaviour of the three samples is similar), the observed best model is the mean-field model for the LPMO samples.

D. Magnetocaloric Effect (MCE) Analysis
To investigate the MCE indirectly, 6   Another important parameter for the application of the MCE is the relative cooling power (RCP), which decides the temperature span in which the magnetic refrigerator can work. The RCP is defined as, 10 where −∆ is the maximum value of the isothermal entropy change and ∆ is the full width at half maximum of the isothermal entropy change with respect to temperature. showed a GP-singularity at low magnetic field and this singularity was suppressed by increasing external magnetic field and increasing A-site chemical substitution. From the Arrott-plots 56 and the Banerjee-criterion 57 a second order magnetic phase transition was confirmed for all three compounds.
Using the Arrott-Naokes 19 , Kouvel-Fished 20 and Widom 21,59 rules the values of universal critical exponents ( , and ) have been calculated and they are very close to the mean-field model values.
The observed mean-field like long-range interaction is attributed to the formation of ferromagnetic clusters 22 in the presence of the GP. Long-range ferromagnetic order is established as the result of the interaction between ferromagnetic clusters. Using a renormalization group approach, Fisher et al., 61 derived the critical exponents for a system where the decay of the long-range interaction was described as ( ) ∝ 1 + , where is the dimension of the system and is an exponent describing the decay of the interaction potential. The transition from mean-field to short-range critical behaviour was discussed in terms of the variable = 2 − and it was found that mean-field critical exponents were obtained for < 0. From our critical scaling analysis, it can therefore be concluded that the interaction potential describing the interaction between ferromagnetic clusters in the LPMO samples is described by an exponent < 1.5. The magnetocaloric analysis showed a small change in the maximum value of ∆S M (isothermal entropy change) with A-site chemical substitution.