Critical behavior near room temperature in La_0.75Ca_0.05Na_0.20MnO₃ sample

Abstract

The critical behavior of La_0.75Ca_0.05Na_0.20MnO₃ was studied at around room temperature via magnetization measurements. From the relative slope, we deduced that the Tricritical Mean-Field model was the most suitable model. The estimated critical exponents were found to be β = 0.24 ± 0.004, γ = 0.98 ± 0.065 and δ = 5.08 at T_C = 300 K. These critical exponents satisfied the Widom scaling relation δ = 1 + γ/β, implying the reliability of our values. Based on the critical exponents, the magnetization–field–temperature (M–μ₀H–T) data around T_C collapsed into two curves, obeying the single scaling equation with ε = (T − T_C)/T_C being the reduced temperature. These results suggest short-range interaction in our sample.

---

1. Introduction

Over the past few years, the colossal magnetoresistance (CMR) effect in doped manganites has attracted the attention of many researchers owing to their promising physical properties and potential applications around the Curie temperature T_C, such as memory recording, field sensors and magnetic reading heads.^1–3 In general, the perovskite structure shows a distortion from the ideal structure (cubic) to a rhombohedral or orthorhombic structure, which is mainly caused by the Jahn–Teller (JT) effect following the deformation of the MnO₆ octahedron. Particularly, the low-doped La_1−xSr_xMnO₃ and La_1−xCa_xMnO₃ samples have extraordinary electronic and magnetic properties and are good candidates for technological applications.^4,5 These materials are characterized by an insulator (I)–metal (M) transition and paramagnetic (PM)–ferromagnetic (FM) transition, which leads to the CMR effect. These properties have been explained by the double-exchange (DE) interaction,⁶ phase separation⁷ and JT effects, which represent the strong electron–phonon interaction.⁸ Recently, many studies have investigated the critical behaviors of manganites around the Curie temperature (T_C) to understand their transition nature (first or second order) and the universality class. Historically, the transition from PM-I to the FM-M state has been elucidated by the Zener DE theory.⁶ The analysis of the critical behaviors around the T_C is a powerful tool to understand the interaction mechanism responsible for the transition. Therefore, it is important to study in detail the critical exponents in the PM–FM region. Initially, critical phenomena in the DE model were associated with the long-range Mean-Field theory.⁹ Subsequently, Motome and Furulawa proposed that the critical behavior could be associated to the short-range Heisenberg universality class.¹⁰ Several experimental studies on critical phenomena are still controversial concerning their Mean-Field values,¹¹ short-range Heisenberg interaction,¹² 3-D Ising values,¹³ and others models which are not classified into any universality class ever known.¹⁴ In general, the critical exponent β varied from 0.3 to 0.5, while the reported values for exponent γ varied from 1 to 1.4. Currently, there are four models that can be defined and used to explain the critical properties in magnetic compounds. These models are the 3D-Heisenberg (β = 0.365), Mean-Field (β = 0.5), Tricritical Mean Field (β = 0.25) and 3D-Ising (β = 0.325) models. It is noteworthy that the critical phenomenon in the PM–FM region raised important fundamental discussions. Previously,¹⁵ we studied the magnetic and magnetocaloric properties of La_0.75Ca_0.05Na_0.20MnO₃ samples. We noticed that our sample displays a FM–PM transition around room temperature. According to the magnetocaloric study, a large magnetic entropy change was observed, with a maximum of 6.01 J kg⁻¹ K⁻¹. In addition, it has an important relative cooling ability, which was found to be 225 J kg⁻¹.¹⁵ These results indicate that the material can be a good potential candidate in the refrigeration domain.

In this study, we present the first detailed dc magnetization study of the critical phenomena in a DE FM, La_0.75Ca_0.05Na_0.20MnO₃. We determined the T_C and the critical exponents γ, β and δ and tested them for scaling.

2. Experimental details

The La_0.75Ca_0.05Na_0.20MnO₃ polycrystalline sample was prepared by the flux method, using NaCl as a flux.¹⁵ The crystal structure of the studied powder was checked by X-ray diffraction, which confirmed that the sample crystallized in the rhombohedral structure with the R3̄c space group and had a detectable secondary phase of Mn₃O₄.¹⁵ The relative amount of Mn₃O₄ as determined from the XRD patterns of the sample was found to be 10%. To extract the critical exponent of the samples accurately, the magnetic isotherm for the sample was measured in the range of 0–5 T, in the vicinity of the PM to FM phase transition. The isothermals were corrected by a demagnetization factor D that was determined by a standard procedure from the low-field dc magnetization measurement at low temperatures (H = H_app − DM).

3. Results and discussion

In our earlier work,¹⁵ we showed that for La_0.75Ca_0.05Na_0.20MnO₃, the field-cooled (FC) magnetization versus temperature under an applied magnetic field of 0.05 T demonstrated a magnetic transition from the PM to FM state at its T_C, with the decrease in temperature; the T_C value was found to be 300 K. To introduce the FM state in manganite oxide type perovskite, the existence of the Mn³⁺–Mn⁴⁺ mixed valence is necessary. Ferromagnetism is caused by the exchange energy; this favors electrons with parallel spins. A deeper insight into the magnetic phase transition may be obtained by analyzing the critical phenomena. Based on Banerjee's criterion, a positive slope reflects a second order magnetic transition, while a negative one reflects a first order magnetic transition. From Fig. 1, we obtained a positive slope for our sample. This indicates that La_0.75Ca_0.05Na_0.20MnO₃ undergoes a second order PM–FM transition. It is well known that a magnetic system showing a second order PM–FM transition, around T_C, can be described by a set of interrelated critical exponents.

Fig. 1 A set of typical M² vs. μ₀H/M plots, for the La_0.75Ca_0.05Na_0.20MnO₃ sample.

According to the scaling hypothesis, for a second-order phase transition around T_C, the critical exponents β (associated with the spontaneous magnetization just below T_C), γ (relevant to the initial magnetic susceptibility χ₀, just above T_C), and δ (associated with the critical isotherm M(T_C, μ₀H) at T_C) are given as:¹⁶

M_S(T) = M₀(−ε)^β, ε < 0, T < T_C (1)

where β is associated with the spontaneous magnetization M_S, γ is associated with the initial magnetic susceptibility χ₀, δ is associated with the critical magnetization isotherm at T_C, ε = (T − T_C)/T_C is the reduced temperature, μ₀H is the demagnetization adjusted magnetic field and M₀, h₀/M₀, and D the critical amplitudes. According to the critical region theory, the reduced magnetic equation of state can be expressed aswhere f _± are the regular analytical functions with f₊ for T > T_C, and f₋ for T < T_C.^17,18

Eqn (4) implies that M/|ε|^−β as a function μ₀H/|ε|^−(β+γ), falls on two universal curves, one for T > T_C, and the other for T < T_C. In general, four models can be defined according to the values of the critical exponents; 3D-Heisenberg model (β = 0.365 and γ = 1.336), Mean-Field model (β = 0.5 and γ = 1), Tricritical Mean-Field model (β = 0.25 and γ = 1) and the 3D-Ising model (β = 0.325 and γ = 1.24). The different models are collected in Table 1.

Table 1 Comparison of critical exponents of the La_0.75Ca_0.05Na_0.20MnO₃ sample with various theoretical models and earlier results on manganites

Samples	Technique	TC (K)	β	γ	δ	Ref.
La0.75Ca0.05Na0.20MnO3	MAP	299.48 ± 0.12K	0.24 ± 0.004	0.98 ± 0.065	5.08	This work
	KF	299.04 ± 0.03	0.22 ± 0.006	1.034 ± 0.063	5.70	This work
	CI (exp)	300	—	—	4.86 ± 0.004	This work
Mean Field model	Theory	—	0.5	1	3	18
3D-Heisenberg model	Theory	—	0.365	1.336	4.8	18
3D Ising model	Theory	—	0.325	1.24	4.82	18
Tricritical Mean Field model	Theory	—	0.25	1	5	19
La0.1Nd0.6Sr0.3MnO3		249.36 ± 0.04	0.248 ± 0.006	1.066 ± 0.002	5.298	25
Nd0.7Sr0.3MnO3		238	0.271 ± 0.006	0.922 ± 0.016	4.5	26
La0.7Ba0.3MnO3		311.2	0.54 ± 0.02	1.04 ± 0.04	3.08 ± 0.04	27
La0.7Ca0.3Mn0.91Ni0.09O3		199.5 ± 0.1	0.171 ± 0.006	0.976 ± 0.012	6.7	28
La0.67Pb0.33Mn0.97Co0.03O3		345	0.233 ± 0.002	1.06 ± 0.06	5.52 ± 0.04	29
La0.6Ca0.4MnO3		265.5	0.25	1.03	5	30
La0.7Sr0.3MnO3		354	0.37 ± 0.04	1.22 ± 0.03	4.25 ± 0.2	2

---

The Modified Arrott Plot (MAP) is a conventional method to estimate the critical exponents and critical temperature.²⁰ According to this method, isotherms plotted in the form of M² as function of μ₀H/M constitute a set of parallel straight lines around T_C. The line at T = T_C should pass through the origin. It is worth mentioning that the Arrott plot assumes the critical exponents following the Mean-Field theory. In the present case, the curves in the Arrott plot are non-linear and concave as shown in Fig. 1. For this reason, we tried to analyze our data according to the MAP method, based on the Arrott–Noakes equation. This method is based on the equation of state:²¹

where a and b are constants.

Fig. 2 displays MAP at different temperatures for our sample using three models of critical exponents: Fig. 2(a) 3D-Heisenberg model, Fig. 2(b) Tricritical Mean-Field model, and Fig. 2(c) 3D-Ising model. One can see that all models yield quasi straight lines. Here, it is difficult to define which model is the most appropriate for determining the critical exponents. To distinguish which model better describes this system, we calculated the relative slope (RS) defined at the critical point: RS = S(T)/S(T_C). The S(T) and S(T_C) are the slope for a given T close to T_C and the slope at T = T_C, respectively. The most appropriate model should be close to 1 (unity).²² The RS as function of the temperature curve for all models is shown in Fig. 3. From this figure, it can be seen that the Tricritical Mean-Field model is the best to describe our system.

Fig. 2 Modified Arrott plots: (M^1/β vs. μ₀H/M) for the La_0.75Ca_0.05Na_0.20MnO₃ sample.

Fig. 3 RS versus temperature for the La_0.75Ca_0.05Na_0.20MnO₃ sample.

The high field straight line portions of the isotherms in Fig. 2(b) can be linearly extrapolated to obtain the spontaneous magnetization M_S(T, 0) and the inverse susceptibility (χ₀⁻¹(T)). The temperature dependence of M_S(T, 0) and χ₀⁻¹(T) for our sample are shown in Fig. 4. The continuous curves in Fig. 4 denote the power law fitting of M_S(T, 0) and χ₀⁻¹(T) as a function of temperature according to eqn (1) and (2), respectively. This gives the values of β = 0.24 ± 0.004 with T_C = 299.48 ± 0.12 K and γ = 0.98 ± 0.065, with T_C = 299 ± 0.66 K. Moreover, β, γ and T_C can be also determined from the Kouvel–Fisher (KF) method²³ based on the following expressions:

Fig. 4 M_S (left) and χ₀⁻¹ (right) as function of temperature for the La_0.75Ca_0.05Na_0.20MnO₃ sample (solid lines are fits to the model).

According to eqn (6) and (7), we plotted M_S(T)[dM_S(T)/dT]⁻¹ and χ₀⁻¹(T)[dχ₀⁻¹(T)/dT]⁻¹ from the obtained M_S(T) and χ₀⁻¹(T) curves, which should yield straight lines with slopes 1/β and 1/γ respectively. The T_C values are obtained from the intercepts on the T axis. The KF curve for our sample is shown in Fig. 5. Besides, the parameters obtained from the linear fitting are found to be β = 0.220 ± 0.006 with T_C = 299.04 ± 0.03 K and γ = 1.034 ± 0.063 with T_C = 298.53 ± 0.06 K for our sample. Obviously, the values of the critical exponents and T_C obtained using the KF method are in agreement with those obtained using the MAP of the Tricritical model. The value of δ can be determined from the linear fitting of the critical isotherm M(T_C, μ₀H). Fig. 6 displays M(μ₀H) curve at 300 K, chosen as the critical isotherm based on the previous discussion. The inset (a) of Fig. 6 shows the same curve M vs. μ₀H on the ln–ln scale. The solid straight line with a slope 1/δ is the fitting result using eqn (3). From the linear fit, we obtained the third critical exponent δ, which is found to be 4.86 ± 0.004. According to the statistical theory, these three critical exponents must fulfill the Widom scaling relation:²⁴

δ = 1 +  γ/β (8)

Fig. 5 K–F plots for the La_0.75Ca_0.05Na_0.20MnO₃ sample (solid lines are fits to the model).

Fig. 6 M vs. μ₀H plot for the La_0.75Ca_0.05Na_0.20MnO₃ sample at T_C = 300 K. The inset (a) displays the same plot in ln–ln scale and the inset (b) shows M(T_C) as function of μ₀H^1/δ.

Using this scaling relation and the estimated values of β and γ, we found that δ value is close to the value directly determined from the critical isotherm at T_C, implying that the obtained β and γ values are consistent. All the obtained critical exponents determined from the various methods are listed in Table 1 and compared with other works.^2,25–30

Furthermore, the relationship between critical exponents can also be verified by plotting M(T_C) as a function of μ₀H^1/δ (inset (b) of Fig. 6). The plot shows a linear curve, which proves that the different critical exponents and T_C obey the Widom scaling relation, and that the determined β and γ values are reliable. In order to make sure of the reliability of the determined critical exponent values and check our data in the critical region, we plotted M|ε|^β as a function of μ₀H|ε|^(β+γ) in the low and high field region. The plots are shown in Fig. 7, using the critical exponents obtained via the KF method. From this figure, we observe that all the points fall into two curves, one below T_C and another one above T_C. This suggests that the values of the exponents and that of the T_C are reasonably accurate. The inset of Fig. 7 displays the same plot on a log–log scale. One can see the same behavior is observed in Fig. 7. This confirms that our obtained critical exponents and T_C are in agreement with the scaling hypothesis.¹⁶ Besides, the reliability of the critical exponents can be tested by other rigorous methods. The first method is based on the m² vs. μ₀h/m curve.¹⁸ Here, m = M(μ₀H, ε)ε^−β and h = μ₀Hε^−(β+γ) are the renormalized magnetization and field, respectively. The m² vs. μ₀h/m curve for our sample is shown in Fig. 8. We can see from this figure that all the data still fall on two separate curves. These results confirm that the critical exponents can generate the scaling equation of state for La_0.75Ca_0.05Na_0.20MnO₃.

Fig. 7 Scaling plot below and above T_C for the La_0.75Ca_0.05Na_0.20MnO₃ sample. The inset shows the same plots on a ln–ln scale.

Fig. 8 The renormalized magnetization m² versus μ₀h/m for the La_0.75Ca_0.05Na_0.20MnO₃ sample.

The second method is shown in Fig. 9. According to the scaling equation of state:³¹

where h is the scaling function. Based on this, we can confirm the reliability of the used T_C and the critical exponents. The hypothesis from the scaling eqn (9) is that all the experimental data points should collapse onto a universal curve.³² Fig. 9 displays M/(μ₀H)^1/δ vs. ε/(μ₀H)^1/β+γ, using the critical exponents and the T_C determined from the KF method and the critical isotherm. It is clear that our curves collapse onto a universal curve, which confirms that the critical exponents are appropriate.

Fig. 9 M(μ₀H)^−1/δ as function of ε(μ₀H)^−1/(β+γ) for the La_0.75Ca_0.05Na_0.20MnO₃ sample in the critical region, at different temperatures.

Moreover, the reliability of the critical exponents can be checked by an equation proposed by Franco et al.³³ In fact, for magnetic systems, the scaling equation of state takes the form:

Eqn (10) is based on the relationship between the critical exponents and the magnetic entropy change, where Δ = β + γ  and α + 2β + γ = 2 are the usual critical exponents. According to eqn (10) and using the critical exponents from MAP, Fig. 10 shows vs. plot. We can see that all (−ΔS_M) curves collapse on a single curve. This indicates excellent agreement between the critical exponents and the scaling hypothesis near the magnetic transition.

Fig. 10 The variation of vs.

4. Conclusion

In the present study, we have comprehensively studied the critical phenomenon at the FM–PM phase transition in a polycrystalline manganite of La_0.75Ca_0.05Na_0.20MnO₃ via dc magnetization. The values of the critical exponents for our sample were extracted using the MAP method, KF method and critical isotherm analysis. Interestingly, the obtained exponent values are very close to the values expected for the universality class of the Tricritical Mean Field model. The validity of the obtained critical exponents using the various methods was confirmed by the Widom scaling relation and the universal scaling hypothesis.

Conflicts of interest

The authors declare no conflict of interest.

References

1. K. Chahara, T. Ohno, M. Kasai and Y. Kosono, Appl. Phys. Lett., 1993, 63, 1990–1992.
2. K. Ghosh, C. J. Lobb, R. L. Greene, S. G. Karabashev, D. A. Shulyatev, A. A. Arsenov and Y. Mukovskii, Phys. Rev. Lett., 1998, 81, 4740–4743.
3. M. S. Kim, J. B. Yang, Q. Cai, X. D. Zhou, W. J. James, W. B. Yelon, P. E. Parris, D. Buddhikot and S. K. Malik, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 71, 014433.
4. A. Urushibara, Y. Morimoto, T. Arima, A. Masamitsu, G. Kido and Y. Tokura, Phys. Rev. B: Condens. Matter Mater. Phys., 1995, 51, 14103–14109.
5. M. McCormack, S. Jin, T. H. Teifel, R. M. Fleming, J. M. Phillips and R. Ramesh, Appl. Phys. Lett., 1994, 64, 3045–3047.
6. C. Zener, Phys. Rev., 1951, 82, 403–405.
7. M. Uehara, S. Mori, C. H. Chen and S. W. Cheong, Nature, 1999, 399, 560–563.
8. X. G. Li, R. K. Zheng, G. Li, H. D. Zhou, R. X. Huang, J. Q. Xie and Z. D. Wang, Europhys. Lett., 2002, 60, 670–676.
9. K. Kubo and N. Ohata, J. Phys. Soc. Jpn., 1972, 33, 21–32.
10. Y. Motome and N. Furulawa, J. Phys. Soc. Jpn., 2001, 70, 1487–1490.
11. S. Taran, B. K. Chaudhuri, S. Chatterjee, H. D. Yang, S. Veeleshwar and Y. Y. Chen, J. Appl. Phys., 2005, 98, 103903.
12. A. Tozri, E. Dhahri and E. K. Hlil, Phys. Lett. A, 2011, 375, 1528–1533.
13. L. Chen, J. H. He, Y. Mei, Y. Z. Cao, W. WXia, H. F. Xu, Z. W. Zhu and Z. A. Xu, Phys. B, 2009, 404, 1879–1882.
14. D. Kim, B. L. Zink, F. Hellman and J. D. Coey, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 65, 214424–214431.
15. S. Bouzidi, M. A. Gdaiem, J. Dhahri and E. K. Hlil, RSC Adv., 2019, 9, 65–76.
16. H. E. Stanley, Introduction to Phase Transitions and critical Phenomena, Oxford University, London, 1971.
17. H. E. Stanley, Rev. Mod. Phys., 1999, 71, 358–366.
18. S. N. Kaul, J. Magn. Magn. Mater., 1985, 53, 5–53.
19. K. Huang, Statistical Mechanics, Wiley, New York, 2nd edn, 1987.
20. A. Arrott, Phys. Rev., 1957, 108, 1394–1396.
21. A. Arrott and J. E. Noakes, Phys. Rev. Lett., 1967, 19, 786–789.
22. A. Schwartz, M. Scheffler and S. M. Anlage, Phys. Rev. B: Condens. Matter Mater. Phys., 2000, 61, R870–R873.
23. J. S. Kouvel and M. E. Fisher, Phys. Rev., 1964, 136, 1626–1632.
24. B. Widom, J. Chem. Phys., 1965, 43, 3898–3905.
25. J. Fan, L. Ling, B. Hong, L. Zhang, L. Pi and Y. Zhang, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 144426.
26. T. L. Phan, T. A. Ho, P. D. Thang, Q. T. Tran, T. D. Thanh, N. X. Phuc, M. H. Phan, B. T. Huy and S. C. Yu, J. Alloys Compd., 2014, 615, 937–945.
27. M. Ziese, J. Phys.: Condens. Matter, 2001, 13, 2919–2934.
28. T. L. Phan, Q. T. Tran, P. Q. Thanh, P. D. H. Yen, T. D. Thanh and S. C. Yu, Solid State Commun., 2014, 184, 40–46.
29. N. Dhahri, J. Dhahri, E. K. Hlil and E. Dhahri, J. Magn. Magn. Mater., 2012, 324, 806–811.
30. D. Kim, B. Revaz, B. L. Zink, F. Hellman, J. J. Rhyne and J. F. Mitchell, Phys. Rev. Lett., 2002, 89, 227202.
31. V. Franco, R. Caballero-Flores, A. Conde, K. E. Knipling and M. A. Willard, Appl. Phys. Lett., 2011, 98, 102505.
32. R. M'nassri, N. Chniba-Boudjada and A. Cheikhrouhou, J. Alloys Compd., 2015, 640, 183–192.
33. V. Franco and A. Conde, Int. J. Refrig., 2010, 33, 465–473.

---