Relationship between magnetic and electric

best of our knowledge, much of the research undertaken on manganites have emphasized the effect of CMR. Nevertheless, little research has sufficiently dealt with the anomalies of the different thermophysical properties, as the particular heat near the magnetic phase transition, shedding the light on the values of the universal critical parameters.
The experiments carried out to predict the critical exponents of FM manganites have shown that a second-order metal insulator phase transition complies with one of the common universality classes. Indeed, the universality class hinges on both the microscopic system details and the global information, specifically the dimension of the order parameter and space [8]. Actually, the ferromagnetism and CMR of manganites have been explicated through the magnetic double exchange model. It noteworthy to mention that the critical behavior in the double-exchange (DE) model was first described with long-range mean-field theory [9-11]. However, the latest research works have underlined the ferromagnetic phase transition itself [8, 12-14], and therefore for a supplementary understanding of this matter, the critical behavior at the phase transition temperature should be examined in detail. It is trustworthy to note that critical phenomena in manganites have been described earlier within the framework of mean-field theory [8]. However, there is still an argument not only about the experimental estimations pertaining to the critical exponents, but also about the experimental ones concerning the order of the magnetic transitions. The latter include three dimensional 3D-Heisenberg interaction, 3D-Ising values, mean-field values, and those which cannot be categorized into any identified universality class [15-17].
Previously [18], we have comprehensively explored the structural, magnetic and MCE of our sample. In this research work, the objective is to prove the correlation, if there exists any, between magnetic and electric properties based on the magnetocaloric effect and critical behavior of nanocrystalline sample. Nevertheless, the obtained results revealed that the determination of MCE and critical behavior accorded well with critical exponents and MCE based on magnetic and electric measurements.

2. Experimental details:

Nanocrystalline sample of nominal composition was prepared according to the citrate gel method as reported in Ref. [18]. Magnetic measurements were realized with the BS1 magnetometer developed at Néel Institute (CNRS-Grenoble). The isothermals M vs. H data used for the present survey are corrected by a demagnetization factor that has been determined by a standard procedure from low-field dc magnetization measurement. Indeed, the internal field H used for the scaling analysis has been corrected for demagnetization, , where D is the demagnetization factor obtained from M vs. H measurements in the low-field linear-response regime at low temperature. Electronic transport and magneto-transport properties have been performed by the standard four-probe method in a commercial Physical Property Measurement System (PPMS Quantum Design) under different magnetic applied fields.

3. Magnetocaloric effect:
The structure, magnetic and magnetocaloric effect of nanocrystalline sample have been recently studied, and its detailed basic physical properties are reported in [18]. The crystal structure of the final product was checked by X-ray diffraction, confirming that the sample with a single phase is crystallized in the rhombohedral structure with space group [18]. From the magnetic measurement, a second-order magnetic phase transition from ferromagnetic to paramagnetic state was noticed at the Curie temperature TC = 325 K. On the other hand, the magnetocaloric effect (MCE) was estimated using the Maxwell relation:

(1)

It was found that the maximum of magnetic entropy change ΔSMax is equal to 1.6485 J Kg-1
K-1 at a magnetic field of 2 T [18].

3. 1. Magnetic Entropy from resistivity:
Many researchers have found a strong correspondence between the electrical and magnetic properties [19, 20]. In manganites, the CMR and MCE effects are commonly noticed near the magnetic phase transition temperature. It is evident that a relationship exists between the change in magnetic entropy and resistivity. In this context, Xiong et al [7] have proposed a relationship between ΔSM and  given by:
, α = 21, 74 emu/g (2)
where the parameter α determines the magnetic properties of the sample. Different functional dependences between M and  have been used to estimate α directly where [7], O’Donnell et al. [19] have indicated that the exact equation should be , while Chen et al. [20] have suggested the equation for small and intermediate magnetization near and above the Curie temperature.

The temperature (T) dependence of resistivity  (T) of sample measured at different applied magnetic field is presented in Fig. 1.
It is worth noting that the electrical resistivity reveals a complex variation, decreasing from low temperature, going through a minimum at around 50K (Tmin) and a maximum near room temperature, showing the existence of metal-insulator transition at around TMI = 270K.
According to the (H, T) curves plotted in Fig. 1, the magnetic entropy change (ΔSM) were estimated using Eq. (2), under an applied magnetic field of 2T. For this aim, the variation of versus M (a), M2 (b) and M2/T (c) are presented in Fig. 2. These studies have shown that resistivity intensely relies on M/T. For , we found that α = 17.07 (emu) /g. This is determined from the fitting of  versus M/T curve around the transition temperature TC with the equation: (Fig. 2).
According to Eqs. (1) and (2), the entropy changes determined by the resistivity measurement and the Maxwell relation, described in our former research work [18] are presented in Fig. 3.
The estimated values were found to accord well with the experimental ones in the behavior of magnetic entropy under 2T magnetic field with an offset in the maximum of magnetic entropy change (ΔSMmax). Such difference between these two curves may be explicated by the presence of an extrinsic nature of the resistivity caused by the effect of nanometric grain size on the behavior of the electrical response of nanocrystalline sample [21]. Therefore, a sharp drop of the resistivity was noted at low temperature explained essentially by taking into consideration the intergranular spin polarized tunneling happening through the grain boundaries [21, 22] which noticeably elucidate the difference between TMI and Tc values, and particularly the offset found between the experimental magnetic entropy change (ΔSM) and the estimated one using Xiong relationship.

3. 2. Magnetic Entropy from Landau theory:
In the manganites case, the free energy (G) can be developed as a function of magnetization that is the order parameter in our case. In the vicinity of the transition temperature TC, G of a ferromagnetic system can be written as follows [23, 24]:
(3)
with A(T), B(T) and C(T): the Landau parameters. According to the equilibrium condition, (∂G/∂M = 0), we obtain the following equation in the vicinity of TC:
(4)
The Landau parameters are identified according to relation (4) by fitting the curves giving the evolution of μ0H as a function of M. Fig. 4 shows the variation of A, B and C parameters as a function of temperature. The variations of A(T) curves reveal a minimum decrease in the vicinity of TC, and then rises linearly. The B(T) curves are negative below TC and positive above it, which implies that the transition is of a second order, which asserts our earlier published results from the Arrott plots [18]. The magnetic entropy change can be expressed by the following relation:
(5)
where A\’ (T), B\’ (T) and C\’ (T) are the derivatives of Landau coefficients pertaining to temperature.
Using the a(T), b(T), and c(T) parameters, the temperature dependence of the magnetic entropy change (ΔSM) is calculated through Eq. (5) under 2T magnetic field as shown in Fig. 5. For the sake of comparing the obtained findings, Fig. 5 lists the variations of the experimental and theoretical values of magnetic entropy as a function of temperature under 2T magnetic field.
It can be noted that while the curves reveal an accordance only above Tc, they show a small deviation below TC, which can be interpreted by the existence of some antiferromagnetic domains in most FM phase [25].

4. Critical parameters:
4.1. Critical parameters determined from magnetization data:
To determine the magnetic transition nature in sample, the critical behavior near the Curie temperature TC is explored. According to the scaling hypothesis, a continuous phase transition near the critical temperature TC indicates a power law dependence of spontaneous magnetization MS(T), and inverse initial susceptibility on the reduced temperature with a set of interdependent critical exponents ,  and  etc. as given below:

, (6)

where and is a critical amplitude. γ is the isothermal magnetic susceptibility exponent defined as:

, (7)

where is the inverse zero-field susceptibility, and is a critical amplitude. δ is the critical isotherm exponent:

, (8)

where is the demagnetization adjusted applied magnetic field, and D is a critical amplitude. It should be noted that Eqs. (6) and (7) are rigorously valid in the limit , i.e., in the asymptotic critical region. Eq. (8) is valid precisely at T = TC. Evidently, the exact determination of TC is of a paramount importance.
For a better understanding of the nature of the second-order magnetic phase transition near the Curie point of our samples, we used Arrott plots [26] to identify the Curie temperature and the critical exponents in the vicinity of the phase transition temperature. The exponents  and can be obtained from spontaneous magnetization Ms(T) and initial susceptibility below and above TC. The reliable values of the critical exponents and the Curie temperature TC were determined from the Arrott-Noakes plots (also called modified Arrott plots (MAP)). In this technique, the M = f(H) data are transformed into series of isothermals (M1/ = f((H/M))1/) depending on the following relationship [27]:

(9)

According to the MAP method, the vs. plots at different temperatures are parallel to each other at elevated magnetic fields. Critical isotherm at T = TC is the line which passes through the origin. The initial values of β and γ are selected as such a way that they provide straight lines in the MAP.
Based on these curves, all models render quasi-straight lines and approximately parallel to the high field region. Hence, it is relatively difficult to distinguish which one of them is the best for the identification of critical exponents. In order to compare these findings and select the best model describing the system, their relative slopes (RS) that are defined as RS = S(T)/S(TC) are calculated. Fig. 6 presents the RS vs. T curve for the four models: Mean field model, 3D-Heisenberg, 3D-Ising and Tricritical mean-field model. The most adequate model should be the first to have an adjacent RS value very close to the unity. Thus, we can deduce that the Mean-field model is the eventual model for the determination of critical exponent for sample. This model describes long-range exchange magnetic interactions with negligible critical fluctuations near the Curie temperature TC. Interestingly, this result is in agreement with our previous study predicted with the field dependence of entropy change [18].
The values of spontaneous magnetization and are obtained from a linear extrapolation of MAP at fields above 0.2 T to the intercept with the and axes, respectively. Only the high-field linear region is used for the analysis since MAP diverges from linearity at low field due to the reciprocally misaligned magnetic domains [28]. The critical exponents of the Mean-field model (β = 0.5, γ = 1and δ = 3) [29] are used as trial critical exponents for this data (Fig. 7).
The values of and are plotted as a function of temperature. By fitting these plots with Eqs. (6) and (7), we get the new values of β and γ. By using these new values of β and γ, new MAP is created (Fig. 7). In an iterative process, we get the constant values of β, γ and TC. In Fig. 8, the and versus temperature curves are plotted. The continuous curve presents the power-law fits of Eqs. (6) and (7) to and , respectively. Eq. (6) provides the value of β = 0.72 ± 0.006 with Tc = 332.30 ± 0.07 and Eq. (7) gives that of γ = 1.10 ± 0.002. These values agree well with those obtained by Mean field model.
These critical exponents can also be identified more clearly according to the Kouvel-Fisher (KF) [30], according to which vs. T and vs. T yield straight lines with slopes 1/β and 1/γ, respectively. The KF plot are presented in Fig. 9. Table 1 lists the critical exponents obtained from both Arrott-Noakes plots and the KF method together with TC. It is worthy to note that the values of critical exponents along with the calculated TC using both methods match soundly well, which proposes that the estimated values are self-consistent and irrefutable.
With an exact TC, the critical exponent δ can be directly calculated from the magnetization data on the critical isotherm with a log-log plot of M vs. H (see Fig. 10). The high field region of the plot is a straight line with slope 1/δ. The value of the obtained δ is given in Table 1. The exponent δ has also been calculated from Widom scaling relation according to which the critical exponents β, γ, and δ are related in the following way [31, 32]:

(10)
Using this scaling relation and the estimated values of β and γ, we obtain a δ value that is very close to the δ estimates from the critical isotherms at TC. Consequently, the estimates of the critical exponents are reliable.
In the critical region, magnetization and internal field should obey the universal scaling behavior. Fig. 11 reveals the plots of vs. for the considered sample. The two curves illustrate temperatures below and above TC. The inset indicates the same data in log-log scale. It is obviously seen from Fig. 11 that the scaling is well complied, i.e., all the points fall on two curves, one of which is for T TC. Hence, the obtained values of the critical exponents and TC are trustworthy and in accordance with the scaling hypothesis.

4.2. Calculation of critical exponents from resistivity
It is reported that one of the fundamental characteristics of manganites materials relates to the solid correlation between the magnetic and electrical properties. For example, a good accordance has been demonstrated in the calculation of the magnetocaloric properties using magnetic and electrical measurements [33, 34]. In addition, according to the aforementioned results, the critical exponents were determined based on the data of static magnetic measurements. The estimated critical exponents agree well with the prediction of the Mean field theory sample. To further understand the correlation among magnetic and electrical properties in manganites, the critical exponents determined from the electrical measurements were examined.
The systems that show second-order metal-insulator phase transition obey one of the common universality classes such as Mean field, Three-dimensional Heisenberg, three-dimensional Ising and Tricritical mean field [35]. These models have unique values for a set of critical exponents as given in Table 3.
Therefore, the determination of the values of critical exponents close to second-order Metal-Insulator transition, and the assignment of one of these models to second-order systems have been tremendously useful for the better understanding of the nature of phase transition.

C. 1. Specific heat critical exponent ()
According to the Fisher-Langer theory [36], specific heat at constant pressure (Cp) and at the phase-transition temperature is proportional to the temperature derivative of the resistivity at T = TMI. The thermal derivative of the resistivity is given by Fisher-Langer [37] as:

(11)

where is the specific heat critical exponent, Cp is the specific heat and is the reduced temperature. The two power law forms of Eq. (11) below and above TMI given by Geldart et al [38] are:

(12)

(13)

where A and B are constants, and \’ are specific heat critical exponents below and above TMI. The temperature derivative of resistivity normalized according to resistivity at TMI [(1/(TIM)) (d/dT)] against is shown in Fig. 12, and Eqs. (12) and (13) are fitted below and above TMI in the same figures. The solid lines passing through the data are best fit in the two regions. As listed in Table 2, the values for constants A, B, specific heat critical exponents below and above TMI (and \’) are obtained from the fitting analysis.
It is obvious from this table that the specific heat critical exponents for x = 0 sample (and \’) are -0.001 and 0.003, respectively, and these values agree well with those obtained using the Mean field model [35].

C.2. Calculation of and 
Furthermore, other critical exponents are calculated from the Suezaki-Mori model [39], which relates the temperature derivative of the electrical resistivity to the reduced temperature () magnetic ordering as follows:

(14)

With B+ a constant and  the reduced temperature [(T-TMI)/TMI], as already designated.
Taking natural logarithm on both sides, the Eq. (14) can be rewritten as,

(15)

The slope of lnd/ dtversus ln plot determines the value of ). As is obtained from the fit with Fisher-Langer method, thus the value of  can be obtained.
Fig. 13 shows lnd/ dtversus ln above TMI and gives the value of ) for nanocrystalline sample. The slope was found to be 0.0800.02. Substituting the value of (derived from Fig. 12) in this relation, the critical exponent is obtained as 1.0770.02 for nanoparticle sample. Finally, to obtain the critical exponent , we use the Rushbrooke scaling relation, 22As a results, the values of is found to be 0.460.02.
The estimated critical exponents from these experimental data and critical exponents from the theoretical models as the Mean field, 3D Heisenberg, Ising and Tricritical mean-field are listed in Table 3. It is to be noted that the estimated values for sample is completely consistent with the Mean field model. These results accord well with the analysis of critical exponents from the magnetization measurements. This study shows a descriptive report of the correlation between magnetic and electrical transport properties. In light of this qualitative agreement, a strong correlation between electrical and magnetic properties in nanocrystalline sample near the phase transition temperature has been proven.

5. Conclusion:
This research work has investigated the correlation between the magnetic and electrical properties of nanocrystalline sample based on magnetocaloric effect and critical behavior. It is noted that there is a strong correlation between ΔSM extracted from magnetic and electrical measurements in manganite in the vicinity of the phase-transition temperature. A little difference of the maximum of entropy change has been explained by the effect of nanometric grain size on the behavior of the electrical response of nanocrystalline sample. Moreover, we have examined the critical behavior from resistivity for the sample using magnetic and electric measurements magnetic field. It was noted that the good quantitative agreement between calculated and experimental data from such analysis would be quite unexpected. However, the fact of observing qualitative agreement between them reveals a strong correlation of electrical and magnetic properties in the manganites of type near the phase transition temperature.