Magnetocaloric study, critical behavior and spontaneous magnetization estimation in La0.6Ca0.3Sr0.1MnO3 perovskite

A detailed study of structural, magnetic and magnetocaloric properties of the polycrystalline manganite La0.6Ca0.3Sr0.1MnO3 is presented. The Rietveld refinement of X-ray diffraction pattern reveals that our sample is indexed in the orthorhombic structure with Pbnm space group. Magnetic measurements display a second order paramagnetic (PM)/ferromagnetic (FM) phase transition at Curie temperature Tc = 304 K. The magnetic entropy change (ΔSM) is calculated using two different methods: Maxwell relations and Landau theory. An acceptable agreement between both data is noted, indicating the importance of magnetoelastic coupling and electron interaction in magnetocaloric effect (MCE) properties of La0.6Ca0.3Sr0.1MnO3. The maximum magnetic entropy change (−ΔSmaxM) and the relative cooling power (RCP) are found to be respectively 5.26 J kg−1 K−1 and 262.53 J kg−1 for μ0H = 5 T, making of this material a promising candidate for magnetic refrigeration application. The magnetic entropy curves are found to follow the universal law, confirming the existence of a second order PM/FM phase transition at Tc which is in excellent agreement with that already deduced from Banerjee criterion. The critical exponents are extracted from the field dependence of the magnetic entropy change. Their values are close to the 3D-Ising class. Scaling laws are obeyed, implying their reliability. The spontaneous magnetization values determined using the magnetic entropy change (ΔSMvs. M2) are in good agreement with those obtained from the classical extrapolation of Arrott curves (μ0H/M vs. M2). The magnetic entropy change can be effectively used in studying the critical behavior and the spontaneous magnetization in manganites system.


Introduction
In the last few decades, the study of the Magnetocaloric Effect (MCE) has attracted the attention and whetted the interest of scientic and engineering communities, not only for its potential applications near room temperature but also for other energy conversion matters 1 as well as certain environmental protection issues. The MCE can be dened as an intrinsic property of magnetic materials. It is characterized by the temperature change (DT ad ) in an adiabatic process and by the entropy change (DS iso ) in an isothermal process originating uniquely from the application and removal of an external magnetic eld in the presence of such ferromagnetic materials as gadolinium which was rstly proposed by G. V. Brown in 1976. 2 The building of a magnetic refrigeration device near room temperature based on the MCE provides tremendous economic, ecological and energetic benets compared to the rest of existing refrigeration machines which are based on conventional gas compression/expansion technique. [3][4][5] Indeed, since the driving force of magnetic refrigerators arises from the variation of the applied magnetic eld, the number of energy consuming elements involved in the refrigeration process is drastically reduced resulting in an enhancement of the cooling efficiency. Moreover, these devices are very environmentally friendly. They do not use any toxic gaseous substances which are normally responsible for damaging our living environment. 6 It is worthy highlighting that Brown's idea has opened the door to a completely innovative technology which is now under development with a notably huge amount of working prototypes. 7 The research on magnetocaloric materials presenting optimal magnetocaloric properties 8-10 was obviously taken further towards the end of the 90's when giant MCE was discovered in Gd 5 Si 2 Ge 2 . 11 Immediately, hundreds of other materials with extraordinary MCE were found [12][13][14][15] and still today dozens of new materials with giant MCE are described every year. Consequently, several magnetic materials which belong to various chemical families have been fully characterized 16 with deeper investigation on the most intimate details of the structural and magnetic properties. Recently, large values of MCE have been observed in the perovskite based-manganite of (R 1 À x M x )MnO 3 formula (where R is a trivalent rare earth ion and M is a divalent alkali earth ion). 17,18 With small thermal and magnetic hysteresis, large magnetic entropy change, and relatively low cost, 19 perovskite manganese oxides have been the subject of continuous research for many years as advantageous materials for refrigeration. This interest arises not only from its dynamic ability for uses in device applications [20][21][22] but also from its impressive physical properties. [23][24][25][26][27] There are numerous sound arguments conrming the fact that perovskite based-manganite compounds will perform a crucial role in the incoming technologies of the near future. 28 Owing to the large amount of known magnetocaloric materials, it was necessary to develop strategies which enable us to compare them accurately, apart from their nature, processing or composition. Nowadays, signicant advances have been carried out allowing a deeper insight to better explore the matter. Phenomenological theories are the key tools which allow us to interpret the performing properties of different magnetocaloric materials. The Landau theory was used to evaluate the importance of magnetoelastic coupling and electron interaction in the magnetocaloric effect. 29,30 The Mean-eld theory was created to establish direct relations between magnetic entropy change and magnetization. [31][32][33] The theory of critical phenomena was exploited to justify the existence of a universal magnetocaloric behavior in second-order magnetic phase transition materials. 34,35 In the present work, a detailed investigation was conducted on magnetocaloric properties of La 0.6 Ca 0.3 Sr 0.1 MnO 3 compound and its potential application in the cooling elds. Landau mean-eld analysis was performed to estimate the magnetic entropy change (DS M ) near the Curie temperature. Results are then compared to those obtained using the classical Maxwell relation. A phenomenological universal curve was used as a simple method for extrapolating the magnetic entropy change to conrm the order of the magnetic transition. From the eld dependence of isothermal entropy change data, critical exponents were calculated and then veried by the scaling law. From the magnetic entropy change (DS M vs. M 2 ), spontaneous magnetization (M spont ) was estimated and then compared to that estimated from the classical extrapolation of the Arrott curves (m 0 H/M vs. M 2 ). The La 0.6 Ca 0.4 MnO 3 and La 0.6 Sr 0.4 MnO 3 samples were prepared by citric-gel method 36,37 using nitrate reagents:

Experiment
Sr(NO 3 ) 2 . The precursors were dissolved in distilled water. Citric acid and ethylene glycol were added to prepare a transparent stable solution. The solution was heated at 80 C in order to eliminate water excess and to obtain a viscous glassy gel. The solution on further heating at 120 C led to the emergence of dark grayish akes which were calcined at 700 C for 12 h. Then, the powder was pressed into pellets and nally sintered at 900 C for 18 h.

Characterization
The structure and phase purity of La 0.6 Ca 0.3 Sr 0.1 MnO 3 were examined by powder X-ray diffraction technique with CuKa radiation (l ¼ 1.5406 A), at room temperature, by a step scanning of 0.015 in the range of 20 # 2q # 80 . The morphology of the surface was observed by the scanning electron microscopy (SEM). This technique was employed also to prepare a histogram of particle size. The elemental composition of the prepared specimen was checked by the energy dispersive X-ray analysis (EDAX). The magnetization curve versus temperature was obtained under an applied magnetic eld of 0.05 T with a temperature ranging from 5 to 400 K. Isothermal magnetization data as a function of magnetic eld was performed with dc magnetic elds from 0 to 5 T.

Results and discussion
3.1. Structural study Fig. 1a illustrates the X-ray diffraction (XRD) pattern of La 0.6 -Ca 0.3 Sr 0.1 MnO 3 sample. Rietveld renement is performed by using Fullprof program. 38 The tting between the observed and the calculated diffraction proles shows an excellent agreement, taking into consideration the low value of the t indicator (c 2 ¼ 1.897). We notice that the sample is of a single phase without any trace of foreign impurity conrming the high purity of the product material. All the diffraction peaks are indexed in the orthorhombic structure with Pbnm space group. The crystal structure of La 0.6 Ca 0.3 Sr 0.1 MnO 3 is schematically depicted in Fig. 1b. The renement results are gathered in Table 1.
Goldschmidt's tolerance factor t G as a criterion for the formation of a perovskite structure is calculated using the following expression: 39 where r A , r B and r O are the radii of A, B and O site ions in ABO 3 structure, respectively. Oxide-based manganite compounds have a perovskite structure if their tolerance factor is between 0.78 and 1.05. 40 In the present study, the obtained tolerance factor of La 0.6 Ca 0.3 -Sr 0.1 MnO 3 is 0.925 which is within the stable range of the perovskite structure.
The average crystallite size is obtained from the XRD peaks using the Debye-Scherrer formula: 41 where l ¼ 1.5406 is the wavelength of CuKa radiation, K ¼ 0.9 is the shape factor, b is the full-width at half-maximum of an XRD peak in radians and q is the Bragg angle. The mean value of the crystallite size of La 0.6 Ca 0.3 Sr 0.1 MnO 3 corresponds to 30 nm which conrms the nanometric size of our compound.
The Halder-Wagner (H-W) method is another method to determine the crystallite size. It is expressed as follows: is a coefficient related to strain effect on the crystallites.
The plot of (b * /d * ) 2 (axis-y) as a function of (b * /d *2 ) (axis-x) corresponding to the 5 strongest peaks of La 0.6 Ca 0.3 Sr 0.1 MnO 3 is shown in Fig. 1c. The crystallite size D HW is achieved from the slope inverse of the linearly tted data and the root of the yintercept gives the microstrain 3. The values of D HW and 3 are found to be respectively 31.9 nm and 0.0062. It is worth noting that the crystallite size calculated by H-W method is slightly higher than that calculated by Debye-Scherrer method because the broadening effect due to the microstrain is completely excluded in Debye-Scherrer technique. 43 Fig . 2a shows the SEM micrograph of our synthesized sample. The particles are largely agglomerated with a broad size distribution. The size distribution of particles presented in the inset of Fig. 2a is analyzed quantitatively by tting the histogram using a Lorentzian function. The mean diameter of La 0.6 Ca 0.3 Sr 0.1 MnO 3 is mostly 59 nm. The particle size obtained by SEM image is larger than that calculated by XRD data which indicates that each particle observed by SEM is formed by several crystallized grains.  (18) Bond angles and bond lengths

Magnetic measurements
The temperature dependence of magnetization curve is carried out under an applied magnetic eld of 0.05 T (Fig. 3). With decreasing temperature, La 0.6 Ca 0.3 Sr 0.1 MnO 3 exhibits a single magnetic transition from PM to FM phase at Curie temperature (T c ¼ 304 K) dened as the temperature at which dM/dT shows a minimum (inset Fig. 3). Curie temperature near room temperature has a great importance in terms of the cooling technology. 44 In order to better understand the magnetic behavior of our sample in the PM region above T c , we studied the inverse magnetic susceptibility as a function of temperature c À1 (T). Fig. 3 shows that c À1 (T) follows the Curie-Weiss law dened as: 45 where q cw is Curie-Weiss temperature and C is Curie constant.
It is known that the tting of c À1 (T) curve using Curie-Weiss law provides a valuable information about the magnetic character of material. [46][47][48][49] In our case, by tting the high temperature region of c À1 (T), the Curie-Weiss temperature q cw proves to be equal to 310 K. The obtained value of q cw is positive, validating the FM character of our sample. Generally, q cw is slightly higher than T c which refers basically to the presence of a magnetic inhomogeneity. 50 The experimental effective paramagnetic moment m exp eff can be estimated from the Curie constant by the relation: 51 where N A is the Avogadro number, m B is the Bohr magneton and k B is the Boltzmann constant. In this paper, the magnetization is expressed in m B /Mn. The Curie constant is thus reduced to: The calculated effective paramagnetic moment m cal eff is calculated as follows: 52 where m eff (Mn 3+ ) ¼ 4.9 m B and m eff (Mn 4+ ) ¼ 3.87m B . 53 The obtained values of m exp eff and m cal eff are found to be equal to 5.57m B and 4.51m B , respectively. The difference between the experimental effective paramagnetic moment and the calculated one can be explained by the existence of FM clusters within the PM phase. 54 The isothermal magnetizations versus applied magnetic eld M(m 0 H,T) measured at various temperatures with a maximum magnetic eld of 5 T are depicted in Fig. 4a. Below T c , M(m 0 H,T) data increases sharply at low elds and then shows a tendency to saturation as eld value increases which is typical for FM  materials. Above T c , a dramatic decrease of M(m 0 H,T) is observed with an almost linear behavior as a feature of PM materials. Fig. 4b presents the Arrott plots of (M 2 vs. m 0 H/M) which are derived from the isothermal magnetizations. According to the criterion proposed by Banerjee, 55 the order of the magnetic phase transition can be checked through the sign of the slope of Arrott curves (M 2 vs. m 0 H/M). The positive slope observed for all studied temperatures indicates that the magnetic transition between the FM and PM phase is of the second order.

Magnetocaloric properties
In order to enquire about the efficiency of our compound in the magnetic refrigeration systems, the magnetic entropy change (DS M ) can be determined indirectly from the isothermal magnetization curves using the approximated Maxwell equation: 56 Depending on the magnitude of (ÀDS M ) and its full-width at half maximum (dT FWHM ), the magnetocaloric efficiency can be determined through the relative cooling power (RCP). 59 The latter, dened as the heat transfer between the hot and the cold sinks in one ideal refrigeration cycle, can be described by the following formula: The calculated RCP is 98.17 J kg À1 for m 0 H ¼ 2 T and 262.53 J kg À1 for m 0 H ¼ 5 T, which stands for about 60 and 64% of that observed in pure Gd, respectively. (ÀDS max M ) and RCP constitute a good initial approximation to the potential performance of a material used as a magnetic refrigerator. To evaluate the applicability of our compound as magnetic refrigerant, the obtained values of (ÀDS max M ) and RCP in our study, compared to other magnetic materials, 5,58,60-66 are summarized in Table 2.
For the theoretical modeling of the MCE, Amaral et al. 29 attempted to explore in depth the MCE in terms of Landau theory of phase transition which takes into account the electron interaction and magnetoelastic coupling effects.
According to Landau theory, Gibb's free energy is expressed as: 67 where a(T), b(T) and c(T) are Landau coefficients. These coefficients are temperature-dependent parameters containing the electron condensation energy, the elastic and the magnetoelastic coupling terms of the free energy. 29,68 Using the equilibrium condition at T c (vG/vM ¼ 0), the obtained relation between the magnetization of the material and the applied eld is expressed as follows: Landau's parameters a(T), b(T) and c(T) determined from a polynomial t of the experimental isothermal magnetizations are shown in the inset of Fig. 5.
The magnetic entropy change is theoretically obtained from the differentiation of the free energy with respect to temperature as follows: 69 where a 0 (T), b 0 (T) and c 0 (T) are the temperature derivatives of the landau coefficients. Fig. 5 shows the magnetic entropy behavior of our sample, obtained by comparing the results coming from the Maxwell relation integration of the experimental data and the one calculated by using the Landau theory. An excellent concordance is found between the experimental magnetic entropy change and the theoretical one in the vicinity of the magnetic phase transition. The result indicates that both magnetoelastic coupling and electron interaction can account for the MCE properties of this sample. 70 From physical point of view, the efficiency of magnetic refrigerant materials can be assessed by the nature of the phase transition that they undergo. 71 The phase transition can be of the rst order in which the rst derivative of the Gibb's free energy is discontinuous. Therefore, magnetization shows an abrupt change at the transition temperature. Although this change causes a correspondingly giant magnetic entropy change, this appears at the cost of thermal and magnetic hysteresis, which should be avoided in refrigerators appliances. However, if the magnetic phase transition is of the second order, no thermal and magnetic hysteresis are observed which is much more suitable for refrigerators applications.
To further investigate the nature of the phase transition in samples, Bonilla et al. 72 have suggested a phenomenological universal curve. The construction of the phenomenological universal curve is based on the collapse of all DS M (T,m 0 H) data measured at different m 0 H into one single new curve. This procedure was performed by normalizing the magnetic entropy change curves DS M with respect to their peak DS max M (DS M / DS max M ) and rescaling the temperature axis using two additional reference temperatures in a different way below and above T c . The positions of these additional reference temperatures in the curve correspond to q ¼ AE1: where q is the rescaled temperature, T r1 and T r2 are the temperature values of the two reference points of each curve. For the present work, T r1 and T r2 have been selected as temperatures corresponding to DS M (T r1,2 ) ¼ (1/2)DS max M .
Departing from Fig. 6, it is obvious that all normalized entropy change curves collapse into a single curve conrming that the PM/FM transition observed in our sample is of the second order, which is in good agreement with the analysis of the Banerjee criterion.

Critical behavior determination through magnetic entropy change
Generally, the common methods to identify the critical behavior of materials undergoing second order phase transition are the modied Arrott plots 73 and the Kouvel-Fisher method. 74 The choice of model to rst construct some tentative Arrott plots and determine initial values of the critical exponents affects systematically their nal values. Since several researchers make different choices, a considerable uncertainty is unavoidable. To eliminate the drawbacks arising from the conventional method, 75 another method based on the eld dependence of magnetic entropy change can be used to show the intrinsic relation between MCE and the universality class. According to the approach suggested by Oesterreicher and Paker, 76 the eld dependence of the magnetic entropy change of second order phase transition magnetic materials can be approximated by a universal law of the eld: The exponent n which is dependent on m 0 H and T, can be calculated as follows: At T ¼ T c , the exponent n becomes an independent eld: 77 where b and g are the critical exponents. Using bd ¼ b + g 78 the relation (16) can be rewritten as: By tting DS M vs. m 0 H data on the ln-ln scale (Fig. 7a), the value of n obtained from the slope around T c is 0.58 AE 0.04. On the basis of the mean-eld approach, the eld dependence of the magnetic entropy change at the Curie temperature corresponds to n ¼ 2/3. 79,80 The deviation of n value from the mean-eld behavior refers basically to the presence of magnetic inhomogeneities in the vicinity of transition temperature. 81 The eld dependence of RCP for our sample can be expressed as a power law: 64 where d is the critical exponent of the magnetic transition.
The value of d obtained from the tting of RCP vs. m 0 H plot is 5.07 AE 0.06 (Fig. 7b). By combining the value of n and d according to eqn (16) and (17), the obtained values of the critical exponents b and g are 0.319 AE 0.026 and 1.302 AE 0.010, respectively. It is noticed that the values of the critical exponents calculated using the magnetic entropy change match reasonably well within the 3D-Ising model To check the reliability of the obtained critical exponents, Franco et al. 71 used the scaled equation of state which is expressed as: where a ¼ 2 À 2b À g and D ¼ b + g are the usual critical exponents 82 and T r ¼ T c À T T c is the reduced temperature.
According to eqn (19) Fig. 7c. All the experimental data clearly collapses on a single master curve for all measured elds and temperatures indicating that the obtained values of the critical exponents for this specimen are in excellent accordance with the scaling hypothesis, which further reinforces their reliability. This result conrms that the critical behavior is well correlated with the MCE properties.

Spontaneous magnetization determination through magnetic entropy change
In the following section, the mean-eld theory is invested so as to investigate the spontaneous magnetization (M spont ) in our sample. A general result issued from a mean-eld theory reveals that the magnetic entropy as a function of magnetization can be described as: 16,83,84 SðsÞ ¼ where N is the number of spins, k B is the Boltzmann constant, J is the spin value, B J is the Brillouin function for a given J value and s ¼ M/NJgm B is the reduced magnetization. For small M values, a proportionality of magnetic entropy to s 2 can be dened as: In the FM state, the system presents a spontaneous magnetization, therefore s s 0. Consequently, considering only the rst term of eqn (21), the magnetic entropy may be written as: