## 3. Results and Discussion

Figure 1 Shows XRD patterns, while the refined cell parameters for all of the samples are listed in the table (inset

Figure 1). The variations in cell parameters were previously discussed in terms of the differences in ionic radii of Mn

^{3+} and Co

^{3+} ions and internal stress induced by Co substitution [

26]. The ionic radius of Mn

^{3+} (0.645 Å) is higher than that of Co

^{3+} (0.61 Å); consequently, it was estimated that the Co doping results in a decrease in the lattice parameters and volume. However, contrary to expectations, increasing the Co content up to x = 0.2 results in an increase in cell volume, then a slight decrease was observed in the sample with x = 0.3 (table inset of

Figure 1). This sharp deviation can be ascribed to the distortion mesh-induced substitution of Mn by Co, as evidenced by the fluctuations of the angles (B-O-B) and length (B-O) of ionic bonds of cobalt or manganese with oxygen ions [

27]. For the non-doped compound, FM DE interactions are more dominant than AFM interactions of Mn

^{3+}-Mn

^{3+}and Mn

^{4+}-Mn

^{4+}, and the presence of Co favours AFM SE interactions. Since Co replaces Mn, which causes the increase of the Mn

^{3+}/Mn

^{4+}ratio, the level of hopping electrons and the number of available hopping sites decrease. The DE interaction is thus suppressed, resulting in the reduction of ferromagnetism when Co content increases and so the decrease of both

T_{C} and magnetization.

Figure 2 shows the SEM images of the prepared sample. All of the images of the samples show particles having aspherical shape, and most of them connect with each other. Corresponding EDX analysis (insets

Figure 2) revealed the presence of La, Ca, Mn, O and Co elements.

In a previous work [

26], we reported that the PM-FM transition in La

_{0.8}Ca

_{0.2}Mn

_{1−x}Co

_{x}O

_{3} compounds occurs at 256 K and 176 K for x = 0 and x = 0.2, respectively. For the sample with x = 0, the existence of the Griffiths phase was reported, as well.

Figure 3 shows the isothermal magnetization curves for La

_{0.8}Ca

_{0.2}Mn

_{1−x}Co

_{x}O

_{3} (x = 0, 0.1, 0.2 and 0.3). The

H/M vs.

M^{2} Arrott plots displayed in

Figure 4wereanalyzed using the Banerjee criterion [

28] to determine the type of magnetic phase transition in these samples. According to this criterion, La

_{0.8}Ca

_{0.2}MnO

_{3} nanopowder exhibited negative slopes, indicating that this sample is characterized by a first-order magnetic transition (FOMT) (

Figure 4 x = 0). However Zhang et al. [

15] reported second-order magnetic transition (SOMT). The same composition was prepared by Khlifi et al. [

29] using the solid state reaction at two different annealing temperatures and found to be FOMT at 1073 K and SOMT at 1473 K. Similarly, Phan et al. [

16] reported FOMT in La

_{0.7}Ca

_{0.3}MnO

_{3} manganite. Lynn et al. [

30] conclude that La

_{0.7}Ca

_{0.3}MnO

_{3} manganite does not exhibit a continuous second-order phase transition. This observation is also confirmed by Lin et al. [

31] when they clarify the breakdown of critical scaling in La

_{0.7}Ca

_{0.3}MnO

_{3} owing to a transition that is not an ordinary second-order ferromagnetic transition. Recently, Bonilla et al. [

32] observed a breakdown of the universal behaviour for La

_{2/3}Ca

_{1/3}MnO

_{3}, which is a sign of the first-order nature of the phase. With Co doping, the PM-FM phase transition is changed from first order to second order. As reported by Kim et al. [

33] the critical exponents cannot be defined for the first-order ferromagnetic phase transition, because the magnetic field induces a shift in the transition temperature, leading to a field-dependent phase boundary near

T_{C}. However, the second-order magnetic phase transition near the Curie point is characterized by a set of critical exponents,

β (associated with the spontaneous magnetization),

γ (relevant to the initial magnetic susceptibility) and

δ (associated with the critical magnetization isotherm). The nature of the magnetic phase transition in La

_{0.8}Ca

_{0.2}Mn

_{1−x}Co

_{x}O

_{3} samples with x = 0.1, 0.2 and 0.3 was investigated by evaluating their critical exponents near

T_{C}. The critical exponents were determined from the commonly-known modified Arrott plots (MAP) based on the Arrott–Noakes equation of state [

34]:

where

T_{1} and

M_{1} are some constantsthatare characteristic of the material.

Figure 5 shows the modified Arrot tplots

M^{1/β} vs. (

H/M)

^{1/γ} using four different values of trial exponents corresponding to mean field theory (

β = 0.5,

γ = 1), the 3D-Heisenberg model (

β = 0.365,

γ = 1.336), the 3D-Ising model (

β = 0.325,

γ = 1.241) and the tricritical mean-field model (

β = 0.25,

γ = 1). The critical exponents corresponding to the appropriate model that best describes magnetic interactions in a material should give quasi-straight lines that are nearly parallel in the high magnetic field region at temperatures around

T_{C}. The MAPs in

Figure 5 clearly indicate that the long-range mean-field model and the 3D-Ising model are inappropriate to describe magnetic interactions in the samples under investigation. To determine which of the other two models is the most appropriate, the relative slope RS = S(

T)/S(

T_{C}) in the high-field region was evaluated at different temperatures around

T_{C} and presented in

Figure 6. For a series of perfectly parallel straight lines in the modified Arrott plots, one should obtain a constant value of one for RS, irrespective of temperatures [

35]. According to this criterion, the 3D-Heisenberg model turned out to be the model that best describes the magnetic interactions in all doped samples as demonstrated by

Figure 6.

The values of the spontaneous magnetization

M_{S}(

T) and

χ_{0}^{−1}(

T) are obtained by extrapolating the linear parts of the MAPs in the high-magnetic field region to intersect the

M^{1/β} and the (

H/M)

^{1/γ} axes, respectively. According to the scaling hypothesis, the temperature dependences of the spontaneous magnetization (

M_{S}) and inverse initial susceptibility (

χ_{0}^{−1}) obey the asymptotic relations.

where

ε = (

T –

T_{C})/

T_{C} is the reduced temperature and

M_{0},

h_{0} and

D are the critical amplitudes. Here,

β,

γ and

δ are the critical exponents characteristic of FM short-range order or long-range order, depending on their values. More accurately, the critical exponents can be also obtained from the Kouvel–Fisher (KF) method [

36] using the following equations:

This method suggests that the curves of M_{S}(T) (dM_{S}(T)/dT)^{−1} and χ_{0}^{−1}(T)(dχ_{0}^{−1} (T)/dT)^{−1} plotted versus temperature should yield straight lines with slopes (1/β) and (1/γ), respectively, and the intercepts of the T axis give T_{C}.

Figure 7 shows the spontaneous magnetization

Ms(

T) as a function of temperature and fitted using Equations (2) and (5). This fitting gives values of

β and

T_{C}. The values obtained by fitting the curves of the samples with x = 0.1, 0.2 and 0.3 are, respectively: (

β_{MAP} = 0.204 ± 0.06 with

T_{C} = 188.271 K and

β_{KF} = 0.123 ± 0.04 with

T_{C} = 182.384 K), (

β_{MAP} = 0.401 ± 0.02 with

T_{C} = 176.32 K and

β_{KF} = 0.418 ± 0.04 with

T_{C} = 176.34 K) and (

β_{MAP} = 0.333 ± 0.03 with

T_{C} = 172.61 K and

β_{KF} = 0.404 ± 0.01 with

T_{C} = 174.35 K).

Figure 8 shows the inverse initial susceptibility

χ_{0}^{−1} (

T) as a function of temperature. Least-squares fitting of the straight lines gave the following values of

γ and

T_{C} for the samples with x = 0.1, 0.2 and 0.3, respectively: (

γ_{MAP} = 1.969 ± 0.01 with

T_{C} = 153.72 K and

γ_{KF} = 1.351 ± 0.08 with

T_{C} = 166.655 K); (

γ_{MAP} = 1.332 ± 0.05 with

T_{C} = 175.21 K and

γ_{KF} = 1.303 ±0.01 with

T_{C} = 175.14 K) and (

γ_{MAP} = 1.298 ± 0.02 with

T_{C} = 171.59 K and

γ_{KF} = 1.27 ± 0.04 with

T_{C} = 171.89 K).

These results show that the obtained critical exponents for the sample with x = 0.2 are consistent with the 3D-Heisenberg exponents. However, the exponents for the sample with x = 0.3 are between 3D-Ising and 3D-Heisenberg exponents. The evaluated exponents for x = 0.1 are inconsistent with any known universality class, and the convergence was reached with different values of transition temperature

T_{C}. It can be clearly seen that the values of

T_{C} for x = 0.1 and 0.3 deviate from that obtained by magnetic measurement at 5 K (

Table 1) [

19]. This deviation is generally due to the incomplete polarization of spins [

37].

The third critical exponent δ is directly obtained by fitting the high field region of the critical isotherm

M(

T_{C},H) plotted on a log-log scale (

Supplementary, Figure S1). The value of the third critical exponent evaluated in this manner was then compared with that calculated from the Widom scaling relation [

44]:

The values of critical exponents for our samples are listed in

Table 1 and compared with some other reported values in the literature. In fact,

δ(cal) obtained from Equation (7) is close to

δ(CI) deduced from the critical isotherm for x = 0.2 and 0.3. However, for x = 0.1, we found that

δ(CI) = 7.51 is close to that obtained for La

_{0,9}Te

_{0,1}MnO

_{3} (

δ(CI) = 7.14) [

38], and

δ(cal) = 11.983 is close to

δ(cal)= 15 obtained for La

_{0.75}Ca

_{0.25}MnO

_{3} [

39].

Assessing the reliability of the critical parameters can be carried out by means of the scaling hypothesis [

45], which predicts that in the asymptotic critical region, the magnetic equation of state can be expressed as:

where

f_{±} are regular functions with

f_{+} for

T >

T_{C} and

f_{−} for

T <

T_{C}. It means that plotting

M/|

ε|

^{β} vs.

H/|

ε|

^{β}^{+γ} makes all high-field data points fall into two universal branches corresponding to

T >

T_{C} and

T <

T_{C}. Using the values of

β and

γ obtained by the MAP (the case of x = 0.2 and 0.3) and both MAP and KF for x = 0.1 (to check the validity of the exponents), the scaled data are plotted in

Figure 9. The inset shows the same plots in a log-log scale. It is found that experimental data for x = 0.2 and 0.3 fall on two independent branches even in a low field region, indicating that the obtained values of critical exponents are reasonable for the description of magnetic interactions in these samples. Nevertheless, the critical exponents exhibited by the sample La

_{0.8}Ca

_{0.2}Mn

_{0.9}Co

_{0.1}O

_{3} are obviously anomalous and do not belong to any universality class previously reported, indicating that the ferromagnetic transition in this system is unconventional. In fact, in highly-disordered systems, the critical exponents are found to be almost abnormal and do not belong to any class of universality [

12,

46]. Similar results have been deduced for La

_{0.7}Sr

_{0.3}Mn

_{1−x}Co

_{x}O

_{3} [

40]. Phan et al. [

11] demonstrated that the substitution of Co dopants dilutes the Mn lattice and changes the disorder of the Mn-related magnetic lattice. Furthermore, they concluded that with the same composition, critical exponent values are dependent on the sample type (single-crystal or polycrystalline form), and on the nature of dopants and the concentration. Additionally, the doped site (A or B-site), as well as the type of atoms also affect the values of critical temperature and critical exponents and a precise determination of the type of magnetic transition becomes difficult when the magnetic transition is a mixture of FOMT and SOMT [

41,

42,

46]. Moreover, several studies show that the presence of FM cluster in manganites can also modify the critical exponents [

47,

48].

The field dependence of the entropy change related to the local exponent

n has been analyzed using the results of magnetic entropy change. Indeed, according to Oesterreicher et al. [

24], the field dependence of the magnetic entropy change of materials with a second-order phase transition is expressed as:

where the exponent

n depends on the magnetic state of the compound (

Supplementary, Figure S2). Moreover, it can be locally calculated as follows:

Some authors claim that second-order phase transitions materials should be represented by the mean field model with the corresponding

n = 2/3 [

24], and any deviation from that value should be explained by the distribution of the Curie temperatures of the material [

49,

50]. Using the power law (Equation (9)) represented in

Supplementary Figure S2, the obtained values of

n are 0.833, 0.731 and 0.845 for x = 0.1, 0.2 and 0.3, respectively. These values deviate significantly from the mean field predictions [

36]. From

Figure 10, it can be clearly seen that

n evolves with the field in the entire studied temperature range and exhibits minimum values around their peak temperatures [

51,

52].

The value of

n obtained for x = 0.2 is in the vicinity of the value

n = 0.75, being close to that obtained for the soft magnetic amorphous alloys [

53,

54].

In addition, we calculated

n from the critical exponent

β and

γ using the relation [

25]:

Using Equation (7), Relation (11) can be rewritten as:

The exponents

n are calculated using Equation (11) or (12). The obtained values are 0.405, 0.654 and 0.591 for x = 0.1, 0.2 and 0.3, respectively. These values are lower than those deduced from the power law. We believe that a large deviation of the

n values obtained from two routes is because the exponent values

β and

γ are much different from those expected for mean field theory, especially for x = 0.1. Thus, the variation of

n(

T,

H) indicates the inhomogeneous character of our compounds, although the presence of a minority magnetic phase would alter the value of the magnetic entropy change exponent. In this context, Shen et al. [

55] show that in single-phase materials,

n is field independent especially at the temperature corresponding to the minimum of

n. In a multi-phase system or a system with magnetic inhomogeneity,

n(

T) is field dependent at any temperature. Caballero et al. [

56] studied the critical behavior of Pr

_{0.5}Sr

_{0.5}MnO

_{3} and concluded that for small applied magnetic fields, the magnetic field response of magnetization is weak in materials exhibiting FM clusters, thus resulting in a small change in magnetic entropy, while for high applied magnetic fields, all of the magnetic moments are aligned with the magnetic field resulting in an overall change in magnetic entropy that is close to the theoretically predicted value. In short, the local exponent

n gives further insight into the critical behavior in the compound.