Critical Isotherm Analysis Research Articles

Critical Behavior of Cobalt-doped van der Waals Ferromagnet (Fe0.74Co0.26)3GeTe2

Two-dimensional (2D) van der Waals (vdW) ferromagnet Fe3GeTe2 (FGT) demonstrates considerable potential for applications in spintronics due to its relatively high Curie temperature (TC) and strong perpendicular magnetic anisotropy. Moreover, these demanding magnetic properties are sensitive to the Fe contents and can be effectively tuned by substituting Fe with other elements, especially the cobalt (Co) doping. Here, we delve into the magnetic critical behavior of the Co-doped vdW ferromagnet (Fe0.74Co0.26)3GeTe2 (FCGT). Utilizing a range of techniques including modified Arrott plots, Widom scaling method, the Kouvel-Fisher approach, and critical isotherm analysis, we derived reliable and self-consistent critical exponents: β = 0.360, γ = 1.043, δ = 3.87, with a TC of 134.3K. These critical exponents of FCGT do not conform to a single theoretical model but instead lie between the Mean-field model and the three-dimensional Heisenberg one. Despite the distinct critical exponents, Co doping does not alter the critical behavior intrinsic to the FGT. Further investigation implies the FCGT sustains long-range magnetic interactions, as evidenced by the magnetic exchange distance decay of J(r) ≈ r-4.566. Our work could offer insights for the understanding of the macroscopic magnetic behavior of vdW ferromagnets doped with other transition metals.

3D-Ising-type magnetic interactions stabilized by the extremely large uniaxial magnetocrystalline anisotropy in layered ferromagnetic Cr2Te3

We investigate the magnetocrystalline anisotropy, critical behavior, and magnetocaloric effect in ferromagnetic-layered Cr2Te3. We have studied the critical behavior around the Curie temperature (TC) using various techniques, including the modified Arrott plot (MAP), the Kouvel-Fisher method (KF), and critical isothermal analysis (CI). The derived critical exponents β = 0.353(4) and γ = 1.213(5) fall in between the three-dimensional (3D) Ising and 3D Heisenberg type models, suggesting complex magnetic interactions by not falling into any single universality class. On the other hand, the renormalization group theory, employing the experimentally obtained critical exponents, suggests 3D-Ising-type magnetic interactions decaying with distance as J(r) = r−4.89. We also observe an extremely large uniaxial magnetocrystalline anisotropy energy (MAE) of Ku = 2065 kJ/m3, the highest ever found in any CrxTey based systems, originating from the noncollinear ferromagnetic ground state as predicted from the first-principles calculations. The self-consistent renormalization theory (SCR) suggests Cr2Te3 to be an out-of-plane itinerant ferromagnet. Further, a maximum entropy change of -ΔSMmax≈ 2.08 J/kg − K is estimated around TC for the fields applied parallel to the c-axis.

Study on the magnetic properties and critical behavior of CoFe2−x Al x O4 (x = 1.0 and 1.2) spinel ferrite

Abstract We have examined the behavior of CoFe2−x Al x O4 (x = 1.0 and 1.2) ferrite close to the transition from ferromagnetic to paramagnetic phases (T C). The findings indicate that at a temperature of T C = 314 K (x = 1.0) and T C = 224 K (x = 1.2), there is a second order magnetic phase transition. We used various methods, including modified Arrott plot, Kouvel–Fisher method, and critical isotherm analysis, to determine the critical exponents which were found to be similar to those expected for the Tricritical Mean-field model (β = 0.288, γ = 1.057, and δ = 4.665) for the x = 1.0 sample. The critical exponents for the x = 1.2 sample (β = 0.771, γ = 1.081, and δ = 2.403) belonged to a different universality class. These results suggest that the replacement of Fe ions with non-magnetic Al ions decreases the Co–Fe, Fe–Fe, Co–Co interaction sites in the CoFe2O4 spinel ferrite, increasing magnetic disorder.

Experimental and computational Insights Into the magnetic anisotropy and magnetic behaviour of layered room-temperature ferromagnet Cr1.38Te2

We investigate the structural, magnetocrystalline anisotropy, critical behaviour, and magnetocaloric effect in the layered room-temperature monoclinic ferromagnet Cr1.38Te2. The critical behavior is studied by employing various techniques such as the modified Arrott plot (MAP), the Kouvel-Fisher method (KF), and the critical isothermal analysis (CI) around the Curie temperature (T C ) of 316 K. The derived critical exponents are self-consistent and obey the rescaling analysis. The Monte-Carlo simulations reproduce the experimentally obtained critical exponents. However, the derived critical exponents do not suggest any single universality class of the magnetic interactions. On the other hand, the renormalization group (RG) theory suggests 3D-Ising type long-range exchange interactions [J(r)], decaying with distance (r) as J(r) = r −(d+σ) = r −4.73. Further, magnetocrystalline anisotropy energy density (K u ) is found to be temperature dependent. The ground state magnetic easy-axis (b-axis) is identified by analyzing the magnetocrystalline anisotropy energy (MAE) using the density functional theory calculations. Maximum entropy change - ΔSmmax ≈2.51 J/kg-K is found near the T C .

Critical behavior of quasi-two-dimensional ferromagnet Cr1.04Te2

Abstract The self-intercalation of Cr into pristine two-dimensional (2D) van der Waals ferromagnetic CrTe2, which forms chromium tellurides (Cr x Te2), has garnered interest due to their remarkable magnetic characteristics and the wide variety of chemical compositions available. Here, comprehensive basic characterization and magnetic studies are conducted on quasi-2D ferromagnetic Cr1.04Te2 crystals. Measurements of the isothermal magnetization curves are conducted around the critical temperature to systematically investigate the critical behavior. Specifically, the critical exponents β = 0.2399, γ = 0.859, and δ = 4.3498, as well as the Curie temperature T C = 249.56 K, are determined using various methods, including the modified Arrott plots, the Kouvel–Fisher method, the Widom scaling method, and the critical isotherm analysis. These results indicate that the tricritical mean-field model accurately represents the critical behavior of Cr1.04Te2. A magnetic phase diagram with tricritical phenomenon is thus constructed. Further investigations confirm that the critical exponents obtained conform to the scalar equation near T C, indicating their self-consistency and reliability. Our work sheds light on the magnetic properties of quasi-2D Cr1.04Te2, broadening the scope of the van der Waals crystals for developments of future spintronic devices operable at room temperature.

Critical behaviors of van der Waals itinerant ferromagnet Fe3.8GaTe2

The critical behavior of the van der Waals ferromagnet Fe3.8GaTe2 was systematically studied through measurements of isothermal magnetization, with the magnetic field applied along the c-axis. Fe3.8GaTe2 undergoes a non-continuous paramagnetic to ferromagnetic phase transition at the Curie temperature T c ∼ 355 K. A comprehensive analysis of isotherms around T c utilizing the modified Arrott diagram, the Kouvel–Fisher method, the Widom scaling law, and the critical isotherm analysis yielded the critical exponent of β= 0.411, γ = 1.246, and δ = 3.99. These critical exponents are found to be self-consistent and align well with the scaling equation at high magnetic fields, underscoring the reliability and intrinsic nature of these parameters. However, the low-field data deviates from the scaling relation, exhibiting a vertical trend when T < T c. This discrepancy suggests the occurrence of a first-order phase transition in the crystal under a low magnetic field when T< T c. Mössbauer spectra were employed to provide insights into the critical behaviors of magnetic moments at different sites, including (Fe1)A, (Fe1)B, and Fe2. The results are consistent with the isothermal magnetization analysis. Additionally, the renormalization group theory analysis reveals that the spin coupling within the Fe3.8GaTe2 crystal follows the three-dimensional Heisenberg ({d:n} = {3:3}) type with long-range magnetic and exchange interaction decays with a distance as J(r) ≈ r −4.80.

Critical behavior and magnetocaloric effect in quasi-two-dimensional Mn3Si2Se6

The ferrimagnetic nodal-line semiconductor Mn3Si2Te6, featuring a quasi-two-dimensional crystal structure, has attracted significant attention owing to its rich physical properties. Here, we systematically investigate the magnetic properties of its isostructural compound, Mn3Si2Se6. The magnetism of this material is attributed to the Mn atoms, which exhibit antiferromagnetic interactions among the Mn atoms. The competition between antiferromagnetic exchange interactions results in an overall ferrimagnetic state, with a second-order transition from paramagnetic to ferrimagnetic occurring at 68 K. Critical exponents are derived using various common experimental techniques, including the modified Arrott plot, Kouvel-Fisher method, and critical isotherm analysis. The results indicate that the magnetism of Mn3Si2Se6 closely aligns with the mean-field model, with critical exponents β, γ, and δ determined as 0.475(3), 1.059(3), and 3.20(1), respectively, at the critical temperature of 68.3(2) K. Moreover, the curves of the magnetic entropy change − ΔSM(T, H) exhibit a peak around Tc, with −ΔSMmax amounts to 4.61 (3.63) J kg−1 K−1 for H∕∕ab (∕∕c) with a field change of 7 T. The estimated magnetic entropy value near Tc through specific heat is remarkably saller than the expected spin entropy of Mn2+ ions, indicating the presence of short-range interactions in the paramagnetic region.

Investigation of Magnetocaloric Effect and Critical Field Analysis of Nd0.7−xLaxSr0.3MnO3 (x = 0.0−0.3) Manganites

This research explored the temperature dependence magnetization, magnetocaloric effect, and critical field analysis of Nd0.7−xLaxSr0.3MnO3 (x = 0.0, 0.1, 0.2 and 0.3) manganites samples synthesized through solid state reaction route. The substitution of La3+ at Nd3+ site enhances the A-site average ionic radius ( rA ) from 1.321 Å (for x = 0.0) to 1.348 Å (for x = 0.3). An increase in rA reduced the internal chemical pressure inside the lattice such that the Mn–O–Mn bond angle approached 180°. In addition, an increase in Mn–O–Mn bond angle supresses the spin-lattice interaction, which consequently reduces the change in maximum magnetic entropy (ΔSMmax) at the magnetic transition temperature. This implies that variation of A-site average ionic radius influences the change in maximum magnetic entropy. The critical behavior of the manganite samples was addressed based on the data of magnetization measurements around the transition temperature TC. Various techniques such as modified Arrott plot, a Kouvel–Fisher method, and critical isotherm analysis were used to measure the values of the ferromagnetic transition temperature TC, as well as the critical exponents of β, γ, and δ. The magnetic order parameter n is very well obeyed with scaling relation with change in magnetic entropy.

Critical behavior in the ferromagnet

High-Curie-temperature ferromagnets are promising candidates for designing new spintronic devices. Here we have successfully synthesized the single crystal of the itinerant ferromagnet Mn5Ge3 using flux method. Its critical properties were investigated by means of bulk dc magnetization at the boundary of the paramagnetic (PM) and ferromagnetic (FM) phase to determine intrinsic magnetic interactions. Critical exponents with a critical temperature K and with K are acquired by the modified Arrott plot, whereas is deduced by a critical isotherm analysis at K. The self-consistency and reliability of these critical exponents are verified by the Widom scaling law and the scaling equations. Further analysis reveals that the spin interaction in Mn5Ge3 exhibits three-dimensional Ising-like behavior. The magnetic exchange interaction is found to decay as , meaning that the spin interactions exceed the nearest neighbors, which may be related to the different Mn-Mn interactions with inequal exchange strengths.

Coexisting magnetic interaction, critical behavior and magnetocaloric effect at high temperature in Fe50Ni50 soft ferromagnetic alloy

Herein, we report the structural, magnetic and magnetocaloric properties of Fe50Ni50 alloy prepared through the planetary ball-mill. Rietveld analysis of the X-ray profiles confirmed the cubic crystallographic phase with the Fm-3 m space group. The magnetic characterization demonstrates that Fe50Ni50 alloy exhibits the coexistence of magnetic interaction within the ferromagnetic phase with Curie temperature 762 K. The Arrott plot results confirmed the second-order nature of the alloy. Critical exponents were also determined using the modified Arrott plot, Kouvel-Fisher plot and critical isotherm analysis. Obtained results indicated coexisting magnetic behavior, i.e., long-range ferromagnetism and inhomogeneous magnetic state. This alloy also noted a significant magnetocaloric parameters, i.e., magnetic entropy change (−ΔSM ∼ 2.3 J/kg-K) and relative cooling power (RCP ∼ 230 J/kg) at 30 kOe applied field. Both parameters followed power law dependence of magnetic field, and the exponent value supports the ferromagnetic nature along with the coexisting magnetic state. By normalizing the ΔSM curve with rescaled temperature, a universal master curve is also constructed for this alloy, reiterating the second-order nature of the magnetic transition.

Magnetic critical behavior of van der Waals Fe3GaTe2 with above-room-temperature ferromagnetism

Fe3GaTe2 is a promising van der Waals material for future spintronic applications because of its intrinsic above-room-temperature ferromagnetism. Herein, high quality Fe3GaTe2 single crystals were successfully grown by the chemical vapor transport method. Its magnetic critical behavior in the vicinity of paramagnetic to ferromagnetic phase transition region was systematically investigated. The critical exponents β = 0.350(1) and γ = 1.385(5) were obtained by using the Kouvel–Fisher method, and δ = 4.92(1) was obtained by a critical isotherm analysis at critical temperature TC = 358 K, which is consistent with the Widom scaling relation δ = 1 + γ/β. The critical exponents determined using different methods are self-consistent and obey the scaling equations predicted by the scaling theory. The renormalization group theory analysis based on the extracted critical exponents further reveals that the magnetic interaction in Fe3GaTe2 is of the 3D Heisenberg type with long-range magnetic coupling.

Critical behavior and its correlation with magnetocaloric effect in amorphous Fe68+xCr12-xSi8B12 (x=0, 4 and 10) ribbons

Using the melt-spinning technique, ribbons made of amorphous Fe68+xCr12-xSi8B12 with varying compositions of x (0, 4, and 10) were fabricated. The transition temperature can be tuned from 360 (for x = 0) to 610 K (for x = 10) with decreasing Cr content. Around the Curie temperature, there is a magnetic transition known as second-order paramagnetic to ferromagnetic (PM-FM) magnetic transition (SOMT) that the ribbons experience. Our exploration of magnetic interaction involves analyzing critical behavior through various methods including the modified Arrott plot (MAP), Kouvel-Fisher method (K–F), and critical isotherm analysis (CIA). Upon determining critical exponents values (β, γ, and δ), we found them to be in proximity to those anticipated for the short-range 3D-Heisenberg model. Furthermore, an independent analysis of critical behavior was performed using the magnetocaloric effect (MCE) to confirm these findings.

Uncovering the universality of critical phenomenon in a ferromagnet Gd5Si4 prepared under high pressure

Manipulation of phase transition is of great interest not only because it can enhance the application ability of accompanying physical effects, but also because it allows us to uncover the physical connotations near the critical temperature point. In this work, we systematically examine the influence of high-pressure annealing (HPA) on critical phenomena in Gd5Si4 ferromagnet. A second-order magnetic phase transition is determined in the annealed sample under 3 GPa with a reduced Curie temperature. Various techniques including modified Arrott plot, Kouvel-Fisher method, and critical isotherm analysis are adopted to calculate critical exponents β, γ, and δ of HPA compound. Using the obtained parameters, we employ the renormalization group theory to calculate the spin exchange range, which satisfies J(r) ∼ r−4.454, exhibiting a longer-range-ordered ferromagnetic interaction. Our work indicates that HPA can manipulate the degree of order inside matter with a novel universality class, thus acquiring desired functionality through reasonable HPA treatment.

Unusual critical behavior of Griffiths ferromagnet Ho2NiMnO6 at magnetic phase transition

The critical behavior at the ferromagnetic to paramagnetic phase transition of a Griffiths ferromagnet, Ho2NiMnO6, is studied using modified Arrott plot, Kouvel–Fisher, and critical isotherm analysis. Here, we report a second-order phase transition and conclude from the estimated critical exponents that it is unusual and do not belong to conventional universality classes. However, they obey scaling relationships, which indicates the renormalization of interactions around the phase transition temperature. The presence of Griffiths phase in the system accounts for the unusual critical exponents observed.

Crystal structure, critical phenomena and magnetocaloric properties of Ni-substituted ferromagnetic Pr2Fe17 intermetallic compound around room temperature

The structural, critical behavior and magnetocaloric properties of Pr2Fe16.9Ni0.1 alloy have been studied. Rietveld's refinement of the X-ray powder diffraction patterns shows that the sample crystallizes in the rhombohedral structure with an R3¯m space group. Based on thermomagnetic measurements, critical behavior was investigated using the modified Arrott plot, Kouvel-Fisher methods, and critical isotherm analysis. The critical exponents (β, γ, and δ) were found to be similar to the 3D-Ising type indicating that the magnetic interactions extend beyond the nearest neighbors. Furthermore, Pr2Fe16.9Ni0.1 exhibits a direct magnetocaloric effect around room temperature with moderate magnetic entropy change (−ΔSMmax = 6.2J.kg−1.K−1 at μ0ΔH=5T) and a wide working temperature range. Additionally, the Relative Cooling Power (RCP) is about 89% of the well-known gadolinium (Gd) for μ0ΔH=2T. The good magnetocaloric response of this compound is comparable or even better than that of many other intermetallic alloys.

Structural and critical properties of Pr0.5Sr0.5MnO3 nanoparticle

In this work, a sample of Pr0.5Sr0.5MnO3 nanoparticles with an average crystalline size of <D> = 58 ± 2 nm was prepared by a combination of reactive milling method for 6 h at room temperature and heat treatment at the 1100 °C for 0.5 h. The x-ray diffraction analysis revealed the existence of a Pr0.5Sr0.5MnO3 single phase with the tetragonal structure (I4/mcm space group). Temperature and magnetic field dependences of magnetisation measurements indicated a coexistence of two magnetic phase transitions. One is the antiferromagnetic-ferromagnetic transition at T N = 150 K. The other is the second-order ferromagnetic-paramagnetic phase transition at T C = 273.5 K. Using the modified Arrott plots and the Kouvel-Fisher methods, the critical isotherm analysis, and the scaling relation, the magnetic order in Pr0.5Sr0.5MnO3 nanoparticle sample has been pointed out. Accordingly, the critical exponents were found to be β = 0.486, γ = 1.181, and δ = 3.249. These values are quite close to the allowable exponents of the mean field model, suggesting an existence of the long-range ferromagnetic order. A slight deviation from the mean field model has been explained by the formation of the core/shell structure in Pr0.5Sr0.5MnO3 nanoparticle.

Magnetic, magnetocaloric and critical exponent properties of Eu3(PO4)2 prepared by sol gel method

The Eu3(PO4)2 compound was prepared by the sol gel chemical route and its magnetic, magnetocaloric, and critical exponent properties have been investigated. Combining different models, a detailed analysis of the magnetic phase transitions in the Eu3(PO4)2 compound has been actually performed. At 5.9 K, the phase transition from the AFM (Antiferromagnetic) state to the PM (Paramagnetic) state is 2nd order. A phenomenological model is now used to comprehend the MCE (magnetocaloric effect) behavior of this sample. The CEs (Critical Exponents) are clearly obtained employing MAPs (Modified-Arrott-Plots), CIA (Critical Isotherm Analysis) and also WSR (Windom Scaling Relation) techniques based upon the data of magnetic measurements registered around the TN (Néel temperature). Besides, an analysis of critical performance on the basis of the MCE is also provided. These exponents are β (beta), γ (gamma), and δ (delta) were properly revealed to be nearby those of the MFM (Mean-Field-Model) values. This simply reflects the existence of an AFM long-range order in our sample. A single equation of state also can be used to scale the magnetization below and above TN. Nevertheless, it should be noted that the scaling relations are respected, which indicates a renormalization of the interactions nearby the TN. The forecasting of the J(r) (exchange interaction) validated the long-range AFM order in this sample.

In-field critical behaviour of [formula omitted]-Mn type Co[formula omitted]Zn[formula omitted]Mn[formula omitted] skyrmion-host

An elaborate in-field critical behaviour study based on magnetic isotherms measured in the vicinity of ordering temperature of a polycrystalline cubic β-Mn type Co7Zn7Mn6 alloy is presented. The Rhodes-Wohlfarth analysis suggests the itinerant character of this chiral magnet. The critical exponents β, γ and δ for spontaneous magnetization, initial magnetic susceptibility and critical M-H isotherm, respectively are evaluated through the modified Arrott plots, Kouvel–Fisher method and critical isotherm analysis. The estimated critical exponents β = 0.382(1), γ = 1.386(2) and δ = 4.628(1) corroborate that the critical behaviour of the Co7Zn7Mn6 system is akin to that of a three-dimensional (3D) Heisenberg ferromagnet. Based on the renormalization group theory predictions, the magnetic moments in the Co7Zn7Mn6 chiral magnet are found to interact through the attractive extended-type short-range interactions between spins.

Magnetic critical behavior and room temperature magnetocaloric effect in Ba1.9Pr0.1FeMoO6 double perovskite compound

In this paper, we report the critical behavior related to the magnetocaloric effect of Ba1.9Pr0.1FeMoO6 double perovskite. A maximum magnetic entropy change, |ΔSMmax| ∼ 0.68 Jkg−1K−1 and relative cooling power, RCP ∼ 62 Jkg−1 were observed under applied field of H = 2.5 T. The critical exponents were evaluated by using modified Arrott plots, Kouvel-Fisher, critical isotherm analysis and Widom scaling relation methods. A second order phase transition (paramagnetic to ferromagnetic) was observed at Curie temperature, TC = 319.73 K with critical exponents β = 0.45, γ = 0.84, and δ = 2.91. The value of these critical exponents shows that this sample obeys mean field model and Widom scaling relation. Also, we observed that the different isothermal magnetization data are able to collapse into the two branches below and above TC based on the single scaling equationM(H,ε)=εβf±(H/εβ+γ) that affirms the authenticity of the critical exponents.

Microstructure, Critical Behavior and Magnetocaloric Properties of Melt-Spun Ni51.82Mn32.37In15.81

Heusler alloy with an atomic composition of Ni51.82Mn32.37In15.81 was prepared by melt spinning from arc-melted ingots. X-ray diffraction, scanning electron microscopy and magnetic measurements were used to study the structural, microstructural and magnetic properties. The crystal structure consists of a mixture of B2 austenite (~50%) and 14M martensite (~50%). The alloy undergoes a second order magnetic transition at a Curie temperature of TcA=194.2 K. The hysteresis loop reveals the occurrence of exchange bias phenomenon at room temperature. The critical exponents β, γ and δ were estimated using modified Arrott plots, Kouvel–Fisher curves and critical isothermal analysis. The respective values are β=0.500±0.015, γ=1.282±0.055 and δ=3.003±0.002. The critical behaviour in ribbons is governed by the mean field model with a dominated long-range order of ferromagnetic interactions. The maximum entropy change, ∆SMmax, for an applied magnetic field of 5 T reaches an absolute value of 0.92 J/kg·K. The experimental results of entropy changes are in good agreement with those calculated using Landau theory.