Critical Isotherm Analysis Research Articles

Critical behavior of quasi-two-dimensional semiconducting ferromagnet Cr2Ge2Te6

The critical properties of the single-crystalline semiconducting ferromagnet CrGeTe$_3$ were investigated by bulk dc magnetization around the paramagnetic to ferromagnetic phase transition. Critical exponents $\beta = 0.200\pm0.003$ with critical temperature $T_c = 62.65\pm0.07$ K and $\gamma = 1.28\pm0.03$ with $T_c = 62.75\pm0.06$ K are obtained by the Kouvel-Fisher method whereas $\delta = 7.96\pm0.01$ is obtained by the critical isotherm analysis at $T_c = 62.7$ K. These critical exponents obey the Widom scaling relation $\delta = 1+\gamma/\beta$, indicating self-consistency of the obtained values. With these critical exponents the isotherm $M(H)$ curves below and above the critical temperatures collapse into two independent universal branches, obeying the single scaling equation $m = f_\pm(h)$, where $m$ and $h$ are renormalized magnetization and field, respectively. The determined exponents match well with those calculated from the results of renormalization group approach for a two-dimensional Ising system coupled with long-range interaction between spins decaying as $J(r)\approx r^{-(d+\sigma)}$ with $\sigma=1.52$.

Critical behavior of the van der Waals bonded high TC ferromagnet Fe3GeTe2

Fe3GeTe2 is a promising candidate for van der Waals bonded ferromagnet because of its high Curie temperature and the prediction that its ferromagnetism can maintain upon exfoliating down to single layer. Here, we have reported the critical behavior to understand its ferromagnetic exchange. Based on various techniques including modified Arrott plot, Kouvel-Fisher plot, and critical isotherm analysis, a set of reliable critical exponents (β = 0.327 ± 0.003, γ = 1.079 ± 0.005, and δ = 4.261 ± 0.009) has been obtained. The critical behavior suggests a three-dimensional long-range magnetic coupling with the exchange distance decaying as J(r) ≈ r−4.6 in Fe3GeTe2. The possible origin of three-dimensional magnetic characteristics in van der Waals bonded magnets is discussed.

Critical Behavior Near the Paramagnetic-Ferromagnetic Phase Transition in La1−x Sr x CoO3 (x = 0.3 and x = 0.5) Cobaltites

In this work, we present the study of structural, magnetic and critical behavior of La1−x Sr x CoO3 (x = 0.3 and 0.5) cobaltites. X-ray diffraction analysis using the Rietveld refinement shows that the sample La0.7Sr0.3CoO3 crystallizes in a rhombohedral distorted perovskite structure with $R\overline {3} c$ space group, while La0.5Sr0.5CoO3 exhibits both phases rhombohedric and cubic, with $R\overline {3} c$ and $Pm\overline {3} m$ space groups. Our samples exhibit thermo-magnetic irreversibility behavior in the magnetization data under a magnetic applied field of 0.05 T. The temperature dependence of magnetic susceptibility at higher temperatures for La0.7Sr0.3CoO3 sample reveals the presence of the Griffiths’ phase above the Curie temperature, while it follows the Curie–Weiss law perfectly for La0.5Sr0.5CoO3. Different techniques including the modified Arrott plot, the Kouvel-Fisher method and the critical isotherm analysis have been used to analyze the paramagnetic-ferromagnetic phase transition. The values of the critical exponents associated to this transition were close to those expected in the mean-field model. The validity of the calculated critical exponents has been confirmed by the scaling equation of state. The determined critical exponents using various methods are found to follow the scaling equation with the magnetization data scaled into two different curves below and above the critical temperature.

Studies on the structure, critical behavior and magnetocaloric effect in (LaBi)SrCoO cobaltite

In the present paper, we report the structure, the magnetic, the critical behavior, the magnetocaloric properties and the universal curve of La0.45Bi0.15Sr0.4CoO3 cobaltite. Polycrystalline sample was synthesized in air by the solid state reaction method at a sintering temperature of 1200 °C. Phase purity, structure, size, and crystallinity were investigated using X-ray diffractometry (XRD) and scanning electron microscopy (SEM). The Reitveld refinement of XRD pattern shows that the sample adopts a rhombohedral system with \(R\overline 3 c\) space group. Critical exponents obtained by the modified Arrott plot technique, Kouvel–Fisher method, and critical isothermal analysis are close to the theoretical prediction of 3D Heisenberg model values, revealing the characteristic of short-range ferromagnetic interactions. With these values, the M(T,µ0H) relations below and above the curie temperature collapse into two universal branches in the asymptotic critical region following the scaling equation. Moreover, the experimental magnetic entropy changes measured for various fields collapse onto a master curve, confirming the universal behavior of the magnetocaloric effect in this system. Particularly, its magnetic-field dependence obeys a power-law fitting, where the exponent n = 0.98 is quite far from the value calculated at Curie temperature from the critical exponents. This difference is related to the magnetic inhomogeneity in the sample.

Tricritical behavior of the two-dimensional intrinsically ferromagnetic semiconductor CrGeTe3

CrGeTe3 recently emerges as a new two-dimensional (2D) ferromagnetic semiconductor that is promising for spintronic device applications. Unlike CrSiTe3 whose magnetism can be understood using the 2D-Ising model, CrGeTe3 exhibits a smaller van der Waals gap and larger cleavage energy, which could lead to a transition of magnetic mechanism from 2D to 3D. To confirm this speculation, we investigate the critical behavior CrGeTe3 around the second-order paramagnetic-ferromagnetic phase transition. We obtain the critical exponents estimated by several common experimental techniques including the modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, which show that the magnetism of CrGeTe3 follows the tricritical mean-field model with the critical exponents \b{eta}, {\gamma}, and {\delta} of 0.240, 1.000, and 5.070, respectively, at the Curie temperature of 67.9 K. We therefore suggest that the magnetic phase transition from 2D to 3D for CrGeTe3 should locate near a tricritical point. Our experiment provides a direct demonstration of the applicability of the tricritical mean-field model to a 2D ferromagnetic semiconductor.

Influence of Ag Substitution on Structural, Critical Behavior and Magnetocaloric Characteristics in La0.5Ca0.5−x Ag x MnO3 (0 ≤ x ≤ 0.1) Systems

Based on the modified Arrott plots and the critical isotherm analysis, the critical exponents of both La0.5Ca0.5MnO3 and La0.5Ca0.4Ag0.1MnO3 systems have been comprehensively studied and investigated near the paramagnetic-ferromagnetic (PM-FM) phase transition. Critical exponents, β, γ, and δ, which are corresponding to the thermal variation of the spontaneous magnetization, initial susceptibility, and the isothermal magnetization, respectively, obeyed the Widom scaling law δ = 1 + γ/β. It indicates thus the validity the obtained results. We performed then another analysis of the critical behavior via the magnetocaloric effect presenting an accord with the analysis of magnetization measurements. Relative cooling power (RCP) follows the evolution of the magnetic field as RCP ∼ H 1+1/δ . The acquired critical exponents were in conformity with those gotten from the modified Arrott plots and Widom scaling law.

Correlation between critical properties and magnetocaloric effect using phenomenological model in La0.7Ca0.2Ba0.1MnO3 compound

This paper presents a theoretical work on magnetic and magnetocaloric properties of La0.7Ca0.2Ba0.1MnO3 powder sample elaborated by high energy ball milling. Using the phenomenological model, the magnetocaloric parameters such as the magnetic entropy change ΔSM and the relative cooling power RCP, have been determined from the magnetization data as a function of temperature at several magnetic applied fields. In addition, from the magnetocaloric results, such as ΔSmax ≈ a (μ0H) n and Tpeak-TC ≈ b (μ0H)1/Δ, the critical exponents values related to the magnetic transition have been determined. The estimated results are close to those expected by the tricritical mean-field model. Furthermore, using other various techniques (such as the modified Arrott plots, the Kouvel-Fisher method and the critical isotherm analysis), the values of the ferromagnetic transition temperature TC, as well as the critical exponents β, γ and δ are simulated and compared with those obtained by the theoretical model. A good agreement has been found in the vicinity of the Curie temperature. The field and temperature dependent magnetization follow the scaling theory, and all data fall on two distinct branches, one for T < TC and the other for T > TC, indicating that the critical exponents obtained in this work are accurate.

Critical properties in Dy-doped La 0.7−x Dy x Sr 0.3 MnO 3 (x=0.00, 0.03) manganites

Abstract The critical properties of La0.7−xDyxSr0.3MnO3 (x=0.00 and 0.03) manganites elaborated using sol-gel method have been investigated around the ferromagnetic-paramagnetic (FM-PM) phase transition. The X-ray diffraction characterizations show that the parent compound (x=0.00) crystallized in the rhombohedral structure with R-3C space group without any detectable impurity, while doped sample with (x=0.03) crystallized in the orthorhombic structure with Pnma space group. Various techniques such as modified Arrott plots, Kouvel-Fisher method and critical isotherm analysis were used to determine the values of the Curie temperature TC, as well as the critical exponents β (corresponding to the spontaneous magnetisation), γ (corresponding to the initial susceptibility) and δ (corresponding to the critical magnetisation isotherm). The estimated results are close to those expected by the mean-field model for both samples. The obtained values from critical isotherm M(TC, μ0H) are close to those determined using the Widom scaling relation, and all data fall on two distinct branches, one for T TC, indicating that the critical exponents obtained in this work are accurate.

Static critical behavior of the ferromagnetic transition in La0.67Ba0.33–xZnxMnO3 manganites

ABSTRACTThe critical properties of perovskite manganite La0.67Ba0.33−xZnxMnO3 around the paramagnetic–ferromagnetic phase transition are investigated through various techniques such as modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis. Through the nature of this transition was found to be in second order. The value of critical exponent, derived from the magnetic data using the Kouvel-Fisher method, yields , and 4.600.034.66 ± 0.06 with a TC of 259.98 ± 0.32 − 221.73 ± 0.05 K. The exponent values are close to those expected for three-dimensional Heisenberg ferromagnets with short-range magnetic interaction.

Phenomenological model of the magnetocaloric effect and its correlation with critical behavior near room temperature in La0.7Ca0.2Sr0.1MnO3 manganite

In this paper, we investigate the field dependence of the magnetocaloric properties of La0.7Ca0.2Sr0.1MnO3 powder sample using a phenomenological model. Our compound was elaborated by the conventional solid state reaction. The model parameters were determined from the magnetization data and were used to give better fits to magnetic transition and to calculate the magnetocaloric quantities. The magnetocaloric parameters such as the maximum of the magnetic entropy change \(\Delta S_M^{max}\) and the relative cooling power RCP, have been determined from the calculation of the magnetization as a function of temperature under several magnetic applied field. Thus, from the magnetocaloric results, such as RCP ≈ b(μ0H)1+1/δ and Tpeak − TC ≈ b (µ0H)1/Δ, the critical exponents values related to the magnetic transition have been determined. The estimated results are close to those expected by the tricritical mean-field model. Furthermore, the values of the ferromagnetic transition temperature TC, as well as the critical exponents β, γ and δ obtained by the theoretical model, are compared with those obtained by other various techniques (such as the modified Arrott plots, the Kouvel–Fisher method and the critical isotherm analysis). A good agreement has been found in the vicinity of the Curie temperature.

Estimation of the spontaneous magnetization and the universal curve in MnCo1−xNbxGe alloys with long-range interactions

We report on the critical behavior, spontaneous magnetization, and universal curve of MnCo1−xNbxGe alloys. Results demonstrate that the replacement of Co by Nb produces a decrease in the Curie temperature, TC, from 259.3 K to 240.3 K. Critical exponents obtained by the modified Arrott plot technique, Kouvel-Fisher method, and critical isothermal analysis are close to the theoretical prediction of the mean-field model values, revealing the characteristic of long-range ferromagnetic interactions. The values of spontaneous magnetization Msp estimated from |ΔSM| vs. M2 plots are in good agreement with the classical extrapolation from the Arrott curves, proving the validity and usefulness of this approach to estimate the Msp. Moreover, the experimental magnetic entropy changes measured for different fields collapse onto a universal curve, confirming the universal behavior of the magnetocaloric effect in this system.

Magnetic phase transition and magnetocaloric properties of Mn1-xSnxCoGe alloys

Magnetic phase transitions and the magnetocaloric effect of the polycrystalline Mn1-xSnxCoGe samples with x=0.02, 0.04, and 0.06 have been investigated. The transformation temperature, room temperature magnetocaloric effect and magnetic orders of these alloys could be effectively tuned by Sn substitution. A detailed study on the critical behavior was performed by various techniques, such as modified Arrott plot, Kouvel–Fisher method, and critical isotherm analysis, to determine the critical exponents (β, γ, and δ). Those exponents calculated from the critical magnetization isotherm were found to obey the Widom scaling relation remarkably well and the validity of the obtained critical exponents was also verified by the prediction of the scaling theory in the critical region. All the results are similar to the theoretical values of the mean-field model, indicating the presence of long-range ferromagnetic coupling in this system. Moreover, we made an attempt to contrast the universal curve for magnetocaloric effect in the present system, finding that rescaled entropy data for all alloys collapse into a universal curve.

Investigation of the critical properties near room temperature in La0.7Ca0.2Ba0.1MnO3 manganite

In this paper, we studied in detail the critical behavior of La0.7Ca0.2Ba0.1MnO3 manganite sample synthesized by the solid state reaction. The critical properties near the ferromagnetic–paramagnetic phase transition have been analyzed from the magnetic data, using various techniques such as the modified Arrott plots, the Kouvel fisher method, critical isotherm analysis and the field dependence of magnetocaloric results. Moreover, the validity of the critical exponents using the various methods has been confirmed by the scaling equation of state and all data fall on two distinct branches, one for T TC, indicating that the critical exponents obtained in this work are accurate. The exponents determined in this study have revealed that the β value is in between the theoretically expected values for 3D-Ising and tricritical mean field model. While, the γ value is close to the mean field theory. The third exponent δ was determined independly from the critical isotherms satisfying the Widom scaling relation. Our systematic result reflects the coexistence of a long range and short range ferromagnetic order in our sample. The temperature variation in the effective exponents βeff and γeff resemble with those for disordered ferromagnet around Curie temperature.

Critical scaling analysis for displacive-type organic ferroelectrics around ferroelectric transition

The critical scaling properties of displacive-type organic ferroelectrics, in which the ferroelectric–paraelectric transition is induced by spin-Peierls instability, are investigated by Green’s function theory through the modified Arrott plot, critical isothermal and electrocaloric effect (ECE) analysis around the transition temperature TC. It is shown that the electric entropy change −ΔS follows a power-law dependence of electric field E:−ΔS∼En with n satisfying the Franco equation n(TC)=1+(β−1)/(β+γ)=0.618, wherein the obtained critical exponents β=0.440 and γ=1.030 are not only corroborated by Kouvel–Fisher method, but also confirm the Widom critical relation δ=1+γ/β. The self-consistency and reliability of the obtained critical exponents are further verified by the scaling equations. Additionally, a universal curve of −ΔS is constructed with rescaling temperature and electric field, so that one can extrapolate the ECE in a certain temperature and electric field range, which would be helpful in designing controlled electric refrigeration devices.

Study of Critical Behavior in Amorphous Fe85Sn5Zr10 Alloy Ribbon

We have investigated the critical behavior in amorphous Fe85Sn5Zr10 alloy ribbon prepared using a single-roller melt-spinning method. This alloy shows a second-order magnetic transition from paramagnetic to ferromagnetic (FM) state at the Curie temperature TC (∼306 K). To obtain more information on the features of the magnetic transition, a detailed critical exponent study was carried out using isothermal magnetization M (H, T) data in the vicinity of the TC. Modified Arrott plot, Kouvel–Fisher plot, Widom’s scaling relation and critical isotherm analysis techniques were used to investigate the critical behavior of this alloy system around its phase transition point. The values of critical exponents determined using the above methods are self-consistent. The estimated critical exponents are fairly close to the theoretical prediction of the three-dimensional (3D) Heisenberg model, implying that short-range FM interactions dominate the critical behavior in amorphous Fe85Sn5Zr10 alloy ribbon.

Magnetic, magnetocaloric and critical behavior study of La0.78Pb0.22MnO3 manganite near room-temperature

We report the magnetic, magnetocaloric properties and critical behavior near transition temperature in La0.78Pb0.22MnO3 perovskite manganite, prepared by solid state reaction method. Temperature dependence of magnetization exhibits a ferromagnetic to paramagnetic transition (TC) at 292K. Close to this transition, a maximum magnetic entropy change (−ΔSMmax) of 4.31J/kg K and a relative cooling power (RCP) of 238J/K are determined at a field of 5T. Experimental results revealed that this sample underwent a second-order phase transition. Using the modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, the critical parameters (TC, β, γ, and δ) have been determined. The estimated critical exponents (β=0.246, γ=0.929 and δ=4.49 at TC=290.01K) are found to be close to the tricritical mean-field model. These critical exponents fulfill the Widom scaling relation δ=1+γ/β, implying the reliability of our values. On the basis of the obtained critical exponents, the magnetization-field -temperature (M-μ0H-T) data around TC are found to collapse into two curves by obeying the single scaling equation M(μ0H,ε)=(ε)βf±(μ0H/εβ+γ) where ε=T−TC/TC, f+ above and f− below TC, and confirm that the obtained values of β, γ and δ are reasonably accurate and unambiguous.

Critical properties in La0.7Ca0.2Sr0.1MnO3 manganite: A comparison between sol-gel and solid state process

The critical properties of La0.7Ca0.2Sr0.1MnO3 manganite elaborated using two different methods have been investigated around the ferromagnetic-paramagnetic phase transition. Our sample was successfully synthesized by solid-state reaction (S1) and sol-gel route (S2). The X-ray diffraction characterizations show that (S1) crystallized in orthorhombic structure with Pnma space group without any detectable impurity. Though, (S2) crystallized in rhombohedra structure with R-3C space group. Various techniques such as modified Arrott plots, Kouvel-Fisher method and critical isotherm analysis were used to determine the values of the Curie temperature TC, as well as the critical exponents β (corresponding to the spontaneous magnetization), γ (corresponding to the initial susceptibility) and δ (corresponding to the critical magnetization isotherm). The estimated results are close to those expected by the mean-field model for sample S2; while for sample S1, the exponents values are close to those expected using the tricritical model. The obtained values from critical isotherm M (TC, μ0H) are close to those determined using the Widom scaling relation, and also there are found to follow scaling equation with the magnetization data scaled into two different curves below and above TC, which implies the reliability and accuracy of the exponents. The change of the universality class can be explained by the reduction of the grain size. These results show that the critical behavior of our sample depend havely on the synthesis technique.

Ferromagnetic Order of Amorphous Fe-Ni-Zr Alloy Ribbons at Magnetic Field Below 10 kOe

In this work, we have presented a detailed study on the critical behavior at a magnetic field below 10 kOe of the ferromagnetic-paramagnetic (FM-PM) phase transition in amorphous Fe90−x Ni x Zr10 ribbons with x= 0–15, which were prepared by a rapid-quenching method. The results revealed that all the samples undergo a second-order phase transition. The Curie temperature (T C) depends strongly on Ni concentration (x), which increases from 245 K (for x= 0) up to 403 K (for x= 15). Based on isothermal magnetization data, M(H), the values of the critical exponents β, γ, and δ around the FM-PM phase transition were determined by using the modified Arrott plots, the Kouvel-Fisher methods, and the critical isotherm analyses. In the case of low Ni concentrations (x= 0 and 5), the β value is close to that expected for the 3D Heisenberg model revealing an existence of short-range FM interactions in these ribbons. In contrast, the falling of the β values in between the values of mean-field theory and the 3D Heisenberg model for high Ni concentration (x= 10 and 15) indicates a coexistence of short- and long-range FM interactions. It means that the presence of Ni favors establishing FM long-range order.

Critical behavior of the quasi-two-dimensional semiconducting ferromagnet CrSiTe3

The semiconducting ferromagnet CrSiTe3 is a promising candidate for two-dimensional magnet simply by exfoliating down to single layers. To understand the magnetic behavior in thin-film samples and the possible applications, it is necessary to establish the nature of the magnetism in the bulk. In this work, the critical behavior at the paramagnetic to ferromagnetic phase transition in single-crystalline CrSiTe3 is investigated by bulk magnetization measurements. We have obtained the critical exponents (β = 0.170 ± 0.008, γ = 1.532 ± 0.001, and δ = 9.917 ± 0.008) and the critical temperature TC = 31.0 K using various techniques such as modified Arrott plot, Kouvel-Fisher plot, and critical isotherm analysis. Our analysis suggests that the determined exponents match well with those calculated from the results of renormalization group approach for a two-dimensional Ising model coupled with long-range interaction.

Critical Behavior Near the Paramagnetic to Ferromagnetic Phase Transition in (La 0 . 6 Ca 0 . 4 MnO3)/x(CuO) Ceramic Composites

We report magnetic properties of (La0.6Ca0.4MnO3)/x(CuO) compounds in terms of isothermal magnetization analysis and scaling behavior with various critical exponents in the vicinity of the PM to FM transition region. According to the Banerjee criterion, we found that the FM–PM phase transition is the type of SOMT. Moreover, the critical exponents β, γ, and δ are determined using the modified Arrott-plot (MAP), the Kouvel–Fisher method (KF), as well as the critical isotherm (CI) analysis. The estimated results are close to those expected for three-dimensional Heisenberg class with short-range interaction. From the field dependence of RCP and ΔSM, it was possible to evaluate the critical exponents of the magnetic phase transition. These values are in good agreement with those obtained from the critical exponents using the Kouvel–Fisher plot (KF). Finally, we used the scaling hypotheses to scale the heat capacity changes to a single universal curve for (La0.6Ca0.4MnO3)/x(CuO) composites.