Study of the Critical Behavior in La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ Manganite Oxide

Abstract

In order to study the critical behavior of La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 near room temperature, magnetization measurements were performed. It can be seen from the findings that the specimens show a second order phase transition. Given the relative slope, the 3D-Ising model was deduced as the most suitable model. It was found that the evaluated critical exponents were β = 0.320, γ = 1.296 and δ = 4.965 at TC = 319 K. These verify the Broadom’s scaling equation δ = 1 + γ/β, which demonstrates that our values have been proved valid. Near TC magnetization-strength-temperature, (M-μ0H-T) results fell in two curves according to critical exponents, obeying the single scaling equation Mμ0Hε=εβʄ±μ0H/εβ+γ with ε=T−Tc/Tc as the reduced temperature.

1. Introduction

Extensive research on Ln_1 − xA_x MnO₃ (Ln $=$ La; A $=$ Ca, Sr, Ba, Pb, etc.) hole-doped manganite has attracted significant attention, for its specific transport and magnetic properties and its potential for being used over the years for future technological purposes, which are based on the colossal magnetoresistance effect (CMR) [1, 2] and magnetocaloric behavior [3, 4]. For a more detailed understanding of the relationship that exists between the insulator-to-metal transition and the CMR effect, it is necessary to answer two fundamental issues concerning the ferromagnetic (FM)-paramagnetic (PM) phase transition temperature: firstly, the order of the phase transition and, secondly, the common universality class. The detailed study of the critical exponents around the FM-PM transition is essential to address this point [5, 6]. In this way, it can be considered that this transition can be explained through the double exchange phenomenon (DE), and by the percolation process, phase separation [7], electron–phonon pairing [8], the effect of quenched disorder [9] and the observation of the Griffiths phase (GP) [9]. Initially, on the DE model, the description of the critical behavior relied on a long-range mean field theory [10, 11]. Later, the work of Motome-Furulawa [12, 13] which suggested a short-range Heisenberg model for critical behavior comprising only nearest-neighbor exchange, was carried out by taking into account the existence of a short-range interaction at the level of localized spins. In addition, various pertinent experimental studies on critical phenomena confirm that view also the resulting critical exponent data. In accordance with the one obtained by conventional ferromagnetic Heisenberg model. Taking the data from the DC magnetic study, researchers Ghosh et al. [14] have pointed out that critical exponent β is found to be 0.37 at ${\mathit{La}}_{0.7}{\mathit{Sr}}_{0.3}\mathit{Mn}{O}_3$ ferromagnetic manganite. (β = 0.365 for the Heisenberg model, in which case the critical exponent β was in close contact to the temperature dependency over the spontaneous magnetization under the Curie temperature T_C) as well as the critical case value of β = 0.374 has been also referred in the ferromagnet DE ${\mathit{Nd}}_{0.6}{\mathit{Pb}}_{0.4}\mathit{Mn}{O}_3$ [15]. Nevertheless, a rather elevated value that was β = 0.5 achieved in La_0.8Sr_0.2 MnO₃ polycrystals was consistent to the one found in the mean field pattern [16]. In contrast, the small value of the critical exponent in terms of β = 0.14 that was found in La_0.7Ca_0.3MnO₃ monocrystal indicates a first-order rather than second-order PM-FM transition occurring in this system [17]. Between, there was finding of moderate critical value of β = 0.25 in the polycrystalline ${\mathit{La}}_{0.6}{\mathit{Ca}}_{0.4}\mathit{Mn}{O}_3$ was in a fair agreement with the values of the tricritical point [18, 19]. Thus, various critical exponents have been seen β ranging between 0.1 and 0.5, now, there are four different types of theoretical patterns, mean-field model (β = 0.5), three-dimensional (3D) Heisenberg (β = 0.365), 3D-Ising (β = 0.325), as well as tricritical mean-field (β = 0.25), which have been employed in order to have some explanation of the critical characteristics provided by manganites. Given the discrepancy between the reported critical values, it is necessary to examine the critical behavior of similar manganite perovskites.

In this chapter, we concentrate on a more detailed evaluation of the critical exponents α, β and γ as well as the Curie temperature T_C for La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ compound near the FM-PM phase transition temperature by carrying out analyses using different techniques.

2. Experimental details

The polycrystalline sample La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ was prepared using the sol–gel method. The sol–gel process is a very well known and proven method to generate homogeneous particles having a great quality and thin material. We have produced the polycrystalline La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ through sol–gel process. During this method, the basic reagents, C₆H₉LaO₆, H₂O, C₄H₆CaO₄,₂O, C₄H₆MnO₄, 4H₂O and NiCl₂6H₂O have been weighed using stoichiometric ratios which were dissolved in distilled water under continued stirring. By adding an excess of citric acid, a sol was obtained by slow evaporation from the addition of ethylene glycol, which rendered the solution complex (with a molar ratio of 1:2:2 with regard to the cations:citric acid:ethylene glycol). Subsequently, every reagents have been diluted, and the mixture has been heated up on a hot plate, which gives a gel form. This gel was then dried and calcined for 5 h at 600°C. After, resultant powders have been annealed at 1000°C during 24 h and compressed to form circular pellets. The resulting pellets have been then sintered in air through 1000°C for approximately 10 h. Using X-ray diffraction, the crystalline structure of the studied powder was verified, confirming further that the specimen actually crystallizes into the rhombohedral structure having R$\overline{3}$c space group, this was achieved through the use of X-ray diffraction (Siemens D5000 X-ray diffractometer, using monochromatic Cu Kα radiation λ = 1.5406 A°). We also measured the magnetic isotherms of the sample over a 0–5 T range, approximately the PM-FM, phase transition, through a vibrating sample magnetometer designed in the laboratory of Louis Neel in Grenoble, for precise extraction of ceramic specimen’s critical exponent. Isotherm value was adjusted by means of a demagnetization coefficient D that was calculated according to a standardized procedure based on measurement of low field DC magnetization under low temperature (H = H_app − DM).

3. Scaling analysis

The following power law relation was cited by the scaling hypothesis near the critical region defined by:

$$M_s\left(T\right)=M_0{\left(-\epsilon \right)}^{\beta },\epsilon <0$$E1

$${\mathrm{\chi }}_0^{-1}\left(T\right)=\left(\frac{h_0}{M_0}\right){\epsilon }^{\gamma },\epsilon >0$$E2

$$M=D{H}^{\frac{1}{\delta }},\mathrm{\epsilon }=0$$E3

In which$M_0$, $h_0$ and D represent the critical amplitudes while ε stands for the reduced temperature$\epsilon =\left(T-T_{\mathrm{c}}\right)/T_{\mathrm{c}}$. Besides, within the asymptotic critical zone, line with the prediction of the scale formula, the equation of magnetic state is written in the following form:

$$M\left({\mu }_{0}H,\epsilon \right)={\epsilon }^{\beta }\mathrm{ʄ}\pm \left(\frac{{\mu }_{0}H}{{\epsilon }^{\beta +\gamma }}\right)$$E4

In this case, ${ʄ}_{-}$ and ${ʄ}_{+}$ represent regular analytical functions and above $T_{\mathrm{c}}$ [20, 21]. This last Eq. (4) shows that for the correct choice of the values of β, γ and δ as well as true scaling relations, the scaled $M/{\left|\epsilon \right|}^{\beta }$ plotted as a function of the scaled ${\mu }_{0}H/{\left|\epsilon \right|}^{\beta +\gamma }$ will fall in two universal curves for $T>T_{\mathrm{c}}\left(\epsilon >0\right)$ and the other for $T<T_{\mathrm{c}}\left(\epsilon <0\right)$ [22]. This makes possible to have a fairly important critical regime criterion.

More generally, it is possible to point to four models that are based on the critical exponent values; Mean-field model (β = 0.5 and γ = 1), 3D-Heisenberg model (β = 0.365 and γ = 1.336), 3D-Ising model (β = 0.325 and γ = 1.24), in addition to Tricritical mean-field model (β = 0.25 and γ = 1). The different models can be seen below in Table 1.

Composition	Techniques	$T_{\mathrm{c}}\left(K\right)$	β	γ	δ	Ref.
${\mathit{La}}_{0.67}{\mathit{Ca}}_{0.33}{\mathit{Mn}}_{0.98}{\mathit{Ni}}_{0.02}{O}_3$	MAP	318.300	0.320	1.269	4.965	This work
	KF	317.900	0.324	1.238	4.820	[23]
Mean field model	Theory		0.5	1	3	[23]
3D-Ising model	Theory		0.325	1.24	4.82	[23]
Tricritical mean field model	Theory		0.25	1	5	[23]
3D-Heisenberg model	Theory		0.365	1.336	4.8	[24]
${\mathit{La}}_{0.7}{\mathit{Ca}}_{0.15}{\mathit{Sr}}_{0.15}\mathit{Mn}{O}_3$		335	0.32	1.21	4.36	[25]
${\mathit{Nd}}_{0.7}{\mathit{Ca}}_{0.05}{\mathit{Sr}}_{0.25}{\mathit{Mn}}_{0.98}{\mathit{Ga}}_{0.02}{O}_3$		247	0.308	1.197	4.91	[26]

Table 1.

Critical exponents of the La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ specimen compared with different theoretical models and previous manganite results.

4. Critical behavior

More generally, it is necessary to understand the way in which the ferromagnetic-paramagnetic (“FM-PM”) phase transition occurs (either first or second order), according to scaling postulate. The magnetic system whose phase transition behavior of second order in the vicinity of the Curie temperature spot is driven by an ensemble of interdependent critical exponents [19] β (linked to the spontaneous magnetization $M_S$), γ (associated with the starting magnetic susceptibility ${\chi }_0$), δ (linked to the critical magnetization isotherm to $T_{\mathrm{c}}$). As already known, it is impossible to find the critical exponents in case of a first-order ferromagnetic phase transition; in fact, the external magnetic field allows this transition to be shifted, leading to a phase depending on the field strength $T_c\left(H\right)$ [27]. Magnetization measurement exponents are mathematically defined and given us by the following relationships:

$$M_s\left(T\right)=M_0{\left|\epsilon \right|}^{\beta },\epsilon <0,T<T_{\mathrm{c}}$$E5

$${\chi }_0^{-1}\left(T\right)=\left(\frac{h_0}{M_0}\right){\epsilon }^{\gamma },\epsilon >0,T>T_{\mathrm{c}}$$E6

$$M=D{H}^{1/\delta },\epsilon =0,T=T_{\mathrm{c}}$$E7

Where 𝜀 represents the reduced temperature$\left(T-T_{\mathrm{c}}\right)/T_{\mathrm{c}}$, and$M_0$, $h_0/M_0$, and D, correspond to the critical amplitudes.

In general, both the critical exponents and the critical temperature may be found easily from the Arott curve. From the Arott-Noakes state equation, ${\left(H/M\right)}^{1/\gamma }=\left(T-T_{\mathrm{c}}\right)/T_{\mathrm{c}}+{\left(M/M_1\right)}^{1/\beta }$ [28], where $M_1$ represents a material constant, the Normal Arrott has demonstrated that the relationship between $M^2$ as a function of H/M was essentially a function of the average field model in terms of the critical exponent β = 0.5 and γ = 1.0. As a result, the $M^2$ versus H/M curves must exhibit linear behavior around $T_{\mathrm{c}}$ as well as the line at T = $T_{\mathrm{c}}$ must exactly get through the origin. In addition, one can determine the magnetic transition order using the slope of those lines, following Banerjee’s criterion [29]. From these straight lines, we can identify the magnetic transition order. Given that as a result, there is a positive slope associated with the second-order transition, whereas the negative slope is related to the first-order transition.

Figure 1 displays the Arrott plot $M^2$ versus H/M for the La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ sample at the temperature range close to$T_{\mathrm{c}}$. Clearly, currently, the positive slope of curves $M^2$ versus H/M curves allows to the conclusion of a second order ferromagnetic phase transition. However, the nonlinearity and the appearance of increasing curvature of all Arrott traces, even at high fields, which means that the average field theory is not, satisfied the current phase transition.

Figure 1.

Typical pattern of M² versus ${\mu }_0$H/M traces, for La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ compound.

Most of the time, charge effect, differential orbital degrees of freedom and lattice, which exists in high field regions, are removed in a ferromagnetic so that the ordering parameter is usually confused with macroscopic magnetization. Thus, in order to achieve the correct values β and γ, a modified Arrott trace requires producing quasi-linear lines of the $M^2$ versus H/M plots. As illustrated by the Figure 2(a)–(c), it was found that three types of test exponents of tricritical mean field (β = 0.25, γ = 1.0), 3D Ising (β = 0.325, γ = 1.24) as well as 3D Heisenberg model (β = 0.365, γ = 1.336) were employed in order to create a modified Arrott plot.

Figure 2.

Modified Arrott graphs: (M^1/β versus (${\mu }_0$H/M)^1/γ) over compound La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃.

To match these results, the related slopes (RS) were calculated, given by $\mathit{RS}=S\left(T\right)/S\left(T_{\mathrm{c}}=319K\right)$. Thus, the relative slopes must be kept at 1 apart from the temperatures, whether the modified Arrott graph displays a set which are absolute parallels.

As already seen in Figure 3, the mean field RS, the 3D-Heisenberg and the Tricritical mean filed certainly deviate from the RS = 1 straight line; however the RS of the 3D-Ising model approximated it. As a result, Arrott’s third plot shows the best result, amongst these three patterns, reporting that the critical properties of the La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ specimen can be depicted using the 3D-Ising model.

Figure 3.

Variation of RS with temperature of specimen La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃.

Subsequently, in the Figure 2(b), the linear extrapolation from the high field region two interceptions with axes ${\left(H/M\right)}^{1/\gamma }$ and ${\left(M\right)}^{1/\beta }$ gives credible values of inverse susceptibility ${\chi }_0^{-1}\left(T,0\right)$ and spontaneous magnetization$M_s\left(T,0\right)$, respectively. These temperature dependent values, ${\chi }_0^{-1}\left(T,0\right)$ as a function of T and $M_s\left(T,0\right)$ versus T, are displayed in Figure 4. As already mentioned by Eqs (5) and (6), based on the experimental results (open line) may be adjusted to two solid graphs (continuous line). It allows to give two novel cases in β = 0.320 with $T_{\mathrm{c}}=318.300K$ and γ = 1.296 with$T_{\mathrm{c}}=317.819K$. Therefore, these findings are very similar to the 3D-Ising model.

Figure 4.

M_S (left) with ${\chi }_0^{-1}$ (right) versus temperature for La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ specimen (solid lines are model fits).

Susceptible, these critical exponents and $T_{\mathrm{c}}$ may be more accurately determined by the Kouvel-Fisher (KF) approach [20]:

$$\frac{M_s\left(T\right)}{\mathrm{d}{M}_s\left(T\right)/\mathrm{d}T}=\frac{T-T_{\mathrm{c}}}{\beta }$$E8

$$\frac{{\chi }_0^{-1}\left(T\right)}{\mathrm{d}{\chi }_0^{-1}\left(T\right)/\mathrm{d}T}=\frac{T-T_{\mathrm{c}}}{\gamma }$$E9

By conforming to this method, $M_s{\left(\mathrm{d}{M}_s/\mathrm{d}T\right)}^{-1}$ as a function of T with ${\chi }_0^{-1}{\left(\mathrm{d}{\chi }_0^{-1}/\mathrm{d}T\right)}^{-1}$ as a function of T is expected to give straight lines of slopes 1/β and 1/γ, respectively. T-axis intercepts fit correctly to $T_{\mathrm{c}}$ if those lines are extrapolated to the zero ordinate. As shown by Figure 5, adjustment results using KF method yield the exponents as well as$T_{\mathrm{c}}$ to the deviation: β = 0.324 with$T_{\mathrm{c}}=319.900K$ and γ = 1.238 with $T_{\mathrm{c}}=317.120K$.

Figure 5.

K–F graphs of La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ specimen (solid lines represent model fits).

Certainly, using the KF method, the resulting critical exponent values and $T_{\mathrm{c}}$ and the values obtained by using the modified Arrott of tricritical mean-field model are in agreement. For a better verification on the dependability as regards the above critical exponents, we may consider how the three critical exponents β, γ, and δ relate to each other.

In this place, first we should find out δ value. By the terms of Eq. (7), we can directly get δ value by tracing the critical isotherm at $T_{\mathrm{c}}$. According to Figure 6, we distinguished M versus H plot at 319 K as being critical isothermal from the above discussion. Moreover, it is illustrated through the slide of Figure 6, which displays the graph on M versus H in the form of a log–log scale. Using Eq. (7), there is an appropriate result of the full straight edge and a slope of 1/δ is obtained. Based on a straight-line fit, a third critical exponent δ = 4.863 was derived. Following statistical theory, all three of these critical exponents should complete Widom’s scale relationship:

Figure 6.

M versus ${\mu }_0$H graph for La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ specimen that occurs on T_C = 319 K. the inset the enclosed shows the similar graph scaled to ln–ln.

$$\delta =1+\frac{\gamma }{\beta }$$E10

Using the above obtained data β and γ As a result, Eq. (10) provides values of δ = 4.965 β and γ as obtained from Figure 4, as well as δ = 4.820 β and γ as evaluated using Figure 5. Respectfully, the above values are very similar to the ones calculated using the critical isotherm. As a result, the above relationship was proved with a plot of $M\left(T=T_0\right)$ as a function of ${\mu }_0{H}^{\beta \left(\beta +\gamma \right)}={\mu }_0{H}^{1/\delta }$ and proving that the curve is linear as illustrated in Figure 6.

Given that two critical exponents δ are similar to δ estimated based on critical isotherms near $T_{\mathrm{c}}$. As a result, the critical exponents obtained in this study absolutely obey the Widom scaling relationship, which entails the implication that both β and γ resultant data agree. In the critical region, the magnetic equation is expressed as follows:

$$M\left({\mu }_{0}H,\epsilon \right)={\epsilon }^{\beta }ʄ\pm \left({\mu }_{0}H/{\epsilon }^{\beta +\gamma }\right)$$E11

Where ${ʄ}_{+}$ for $T>T_{\mathrm{c}}$ and ${ʄ}_{-}$ for $T<T_{\mathrm{c}}$ are regular functions [30]. As indicated by the Eq. (11), $M{\epsilon }^{-\beta }$ as a function of $H{\epsilon }^{-\left(\beta +\gamma \right)}$ leads to two universal plots: The first plot for temperature $T>T_{\mathrm{c}}$ and the second plot for temperature$T<T_{\mathrm{c}}$. Therefore, there is a comparison between the obtained results and the scaling theory prediction by Eq. (11). As already seen in Figure 7, both experimental data fall on two curves, one over $T_{\mathrm{c}}$ and one under$T_{\mathrm{c}}$, by correspondence with the scaling theory. This same graph is shown inside the inset in Figure 7 in the form of a log–log scale. Same, all points fall in two different curves. We can conclude from these results that the obtained values of the critical exponents and those of $T_{\mathrm{c}}$ are reliable. On the other hand, there is accurate characterization of critical properties in our current system with the 3D-Ising model.

Figure 7.

Renormalized magnetization $M/{\left|\xi \right|}^{\beta }$ versus ${\mu }_{0}H/{\epsilon }^{\beta +\gamma }$ for the La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ sample.

5. Conclusion

In this paper, it was extensively studied by DC magnetization the critical process towards the FM-PM state transition in La_0.67Ca_0.18Sr_0.15Mn_0.98Ni_0.02O₃ polycrystalline manganite. We have extracted the values of critical exponents of our sample by the Modified Arott plot method and the Kouvel-Fisher method. It is interesting to note that the found exponents are quite similar to those expected from the universality class of 3D-Ising model.

Acknowledgments

K. Laajimi would like to express her gratitude and thanks for its funding by the Tunisian Ministry of Higher Education and Scientific Research, which allowed her to carry out her research and provided her with all the necessary assistance.