Widom Scaling Law Research Articles

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 in the ferromagnet MgMn6Sn6 single crystal with HfFe6Ge6-type structure

A thorough investigation was carried out to analyze the magnetic critical properties of single crystals belonging to the HfFe6Ge6-type structure. MgMn6Sn6 exhibit a ferromagnetic transition below TC ≈ 300 K. The magnetization measurements were analyzed around the phase transition temperature of this ferromagnetic material for two different orientations: H//ab and H//c. Using Kouvel‐Fisher method, we can determine critical exponents β, γ, and δ in this study. To evaluate accuracy of these values, the scaling equation m=f±(h) and Widom scaling law δ=1+γ/β were applied. Critical exponents γ = 1.191 ± 0.007 at TC = 300.92 K, β = 0.326 ± 0.003 at TC = 300.96 K, and δ = 4.653 with TC ≈ 300 K indicate that the three-dimensional Ising model (β = 0.325, γ = 1.24, and δ = 4.82) is the better model for H//ab. The mean‐field and 3D‐Heisenberg models were the best models below TC and above TC for H//c, respectively. The MgMn6Sn6 single crystal exhibits a notable anisotropic transition in its magnetic phase according to all experimental findings. The presence of an anisotropic state with short-range order in the ferromagnetic system are revealed by analyzing electron spin resonance (ESR) spectroscopy. Our research provides experimental proof of a holistic comprehension regarding the magnetic phase-transition characteristics demonstrated by MgMn6Sn6.

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.

Long‐Range Magnetic Exchange Coupling in Quasi‐2D CrTe Ferromagnetic Thin Films

The 2D chromium telluride family (CrxTey) is an outstanding candidate for creating high‐density and dissipationless nanodevices because of its high Curie temperature, chiral spin patterns, and large saturation magnetic moment. However, the precise magnetic exchange mechanism and crucial phase transitional property of CrxTey must be fully analyzed. Herein, a large‐area CrTe (x:y = 1:1) single‐crystalline films deposited on an Al2O3 substrate. Based on the analysis of critical isothermal magnetization around the Curie temperature TC = 201 K and the modified Arrott plot, the precise critical exponents β = 0.386(3) and γ = 1.391(2) are obtained. Both their reliability and accuracy are also verified by Kouvel–Fisher theory, Widom scaling law, and scaling equation. Moreover, using the renormalization group theory, it is confirmed that CrTe belongs to a quasi‐2D Heisenberg‐like behavior with long‐range interactions. Finally, the density functional theory calculations indicate that, near the Fermi level, the CrTe band structure is mainly composed of Cr atom. The spin‐polarized density of states shows that the total magnetic moments are determined mainly by the polarized spin‐up t2g electron of Cr atom. This work is an important asset for a series of CrxTey materials in future spintronic applications.

Critical Behavior of the Competing Magnetic Nature in the Antiperovskite Compound

Antiperovskite materials are deemed as a promising candidate for data storage and efficient energy conversion, exemplified by giant magnetoresistance (GMR) and magnetocaloric effect (MCE). In particular, the abundant magnetic configurations of antiperovskites endow them with a high degree of diversity in probing novel phenomena. Herein, we report the critical behavior of the antiperovskite Sn0.6NFe3.4 with an obvious second-order magnetic transition around Curie temperature (TC = 230 K). The obtained critical exponents (β, γ, and δ) are well consistent for different theoretical methods, which are also verified by the Widom scaling law and scaling equation. It is found that the critical exponent of β is much in accordance with the value of the mean-field model, indicating a long-range FM coupling in the system, whereas the other γ and δ are accessible to the three-dimensional (3D) Heisenberg model. Our work promotes the understanding of antiperovskite magnetism, providing insight for further fundamental research and resulting applications.

Magnetic and Critical Properties of Cr1/3NbS1.86 with TC = 56 K

The magnetic and critical properties of a TeBr‐transported CrNbS single crystal using magnetic susceptibility χ(T) and isothermal magnetization M(H) measurements are studied. A paramagnetic‐to‐ferromagnetic phase transition temperature (T C) is determined to be 56 K from χ(T) and an Arrott–Noakes’ plot. At high temperatures (T ≥ 85 K), χ(T) is well captured by the Curie–Weiss law and the effective magnetic moment is derived as P eff = 3.77 μ B per Cr. The signature of helical and chiral soliton spin structures is absent, as inferred from the (H, 2 K) data. The spontaneous magnetic moment P = 1.66 μ B per Cr compared with P eff, leading to the Rhodes–Wohlfarth ratio to be 1.75, is rather small, suggesting a slightly itinerant magnetism. However, the spin‐wave (SW) model describes the temperature‐dependent magnetization better than the single‐particle (SP) model. Further, analysis of the critical exponent yields β = 0.3852(1) and γ = 1.3575(4), which are close to the theoretical prediction of the 3D Heisenberg model and therefore suggests dominant short‐range ferromagnetic interactions. Finally, the deviation of the critical exponents from the theoretical model and the violation of Widom scaling law is discussed.

Critical behavior of ferromagnetic transition in La0.85Co1.68As2 crystal

The critical behavior of ferromagnetic transition is investigated by bulk magnetization measurements for La0.85Co1.68As2 crystal with a ThCr2Si2-type structure, grown by using NaAs as a flux, in which the vacancies on the both La and Co sites exist. The critical exponents [β = 0.388(3), γ = 1.44(2) and δ = 4.855(9)] with Curie temperature Tc = 99.4 K are obtained, whose self-consistency and reliability are verified by the Widom scaling law and the scaling equation. The critical exponents of ferromagnetic transition occurring in La0.85Co1.68As2 belong to the three-dimensional (3D) Heisenberg model, and suggest that the magnetic interaction decays as J(r)∼r−4.975 with a short range coupling. These results provide helpful information for understanding the mechanism of ferromagnetism emerging in La0.85Co1.68As2 compound.

Magnetocaloric effect and critical behavior of the Mn-rich itinerant material Mn3GaC with enhanced ferromagnetic interaction**Project supported by National Key Research and Development Program of China (Grant Nos. 2018YFA0305700 and 2018YFA0305800), One Hundred-Talent Program in Institute of Physics (Grant No. Y7K5031X61), Youth Promotion Association of CAS (Grant No. 2018010).

We revisit the reversible magnetocaloric effect of itinerant ferromagnet Mn3GaC near the ferromagnetic to paramagnetic phase transition by adopting the experimental and theoretical methods and critical behavior of Mn-rich Mn3GaC with an enhanced FM interaction. Landau theory model cannot account for temperature dependent magnetic entropy change which is estimated from thermal magnetic methods only considering magnetoelastic coupling and the electron–electron interaction, apart from molecular mean-field model. Critical behavior is studied by adopting the modified Arrott plot, Kouvel–Fisher plot, and critical isotherm analysis. With these critical exponents, experimental data below and above Tc collapse into two universal branches, fulfilling the single scaling equation m = f±(h), where m and h are renormalized magnetization and field. Critical exponents are confirmed by Widom scaling law and just between mean-field model and three-dimensional Heisenberg model, as the evidence for the existence of long-range ferromagnetic interaction. With increasing the Mn content, Tc increases monotonously and critical exponents increases accordingly. The exchange distance changes from J(r) ∼ r–4.68 for x = 0 to J(r) ∼ r–4.71 for x = 0.08, respectively, which suggests the competition of the Mn–Mn direct interaction and the itinerant Mn–C–Mn hybridization. The possible mechanism is proposed.

Ferromagnetic frustration in ternary nitride ZnFe3N.

As a new antiperovskite nitride, ZnFe3N was synthesized and characterized by almost completely substituting iron atoms at corner positions of γ'-Fe4N. The magnetic interactions of the system with the space group Pm3[combining macron]m are fully investigated. The critical behavior was investigated based on the measured magnetic data around the ferromagnetic phase transition temperature. In this work, the values of critical exponents (β, γ and δ) were obtained systematically using the Kouvel-Fisher method in the critical region. The Widom scaling law (δ = 1 + γβ-1) and the scaling equation (m = f±(h)) were used to reveal the reliability of these values. The values of the critical exponents (β = 0.325, γ = 1.228, and δ = 4.778) are different from those predicted by the three-dimensional (3D) Heisenberg model and mean-field model, and are very close to those of the 3D-Ising model. Combined with ESR analysis, the spin clusters induced by changes in chemical bonds are considered to be the cause for the existence of an anisotropic short-range ordered state in this ferromagnetic system.

The critical behavior in antiperovskite compound Fe3.4In0.6N

Abstract The critical behavior of the antiperovskite crystal Fe3.4In0.6N has been systematically studied in this paper. We have used a modified Arrott plot, the Kouvel-Fisher method, and critical isotherm analysis to obtain the critical exponents (β, γ and δ) which suggest that a long-range magnetic coupling described by mean-field (MF) theoretical model is dominant in Fe3.4In0.6N. The experimental M−T−H relations below and above Curie temperature (Tc) collapse into two branches by the single scaling equation m = f±(h) (m and h are renormalized magnetization and field), respectively. The critical exponents are confirmed by the Widom scaling law δ = 1 + γβ−1. In addition, it is suggested that the ferromagnetic (FM), anti-ferromagnetic (AFM) interactions and the magnetic disorders result to the exchange distance which is obtained just as J(r)∝ r - 4.4 .

Novel critical behavior of magnetization in URhSi: Similarities to the uranium ferromagnetic superconductors UGe2 and URhGe

We study the critical behavior of dc magnetization in the uranium ferromagnet URhSi around the paramagnetic to ferromagnetic phase transition at ${T}_{\mathrm{C}}\ensuremath{\sim}10$ K with a modified Arrott plot, a Kouvel-Fisher plot, the critical isotherm analysis, and the scaling analysis. URhSi is isostructural to uranium ferromagnetic superconductors URhGe and UCoGe. The critical exponent $\ensuremath{\beta}$ for the temperature dependence of the spontaneous magnetization below ${T}_{\mathrm{C}}$, $\ensuremath{\gamma}$ for the magnetic susceptibility, and $\ensuremath{\delta}$ for the magnetic isotherm at ${T}_{\mathrm{C}}$ in URhSi have been determined as $\ensuremath{\beta}=0.300\ifmmode\pm\else\textpm\fi{}0.002$, $\ensuremath{\gamma}=1.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\delta}=4.38\ifmmode\pm\else\textpm\fi{}0.04$ by the scaling analysis and the critical isotherm analysis. These critical exponents fulfill the Widom scaling law $\ensuremath{\delta}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1+\phantom{\rule{0.16em}{0ex}}\ensuremath{\gamma}/\ensuremath{\beta}$. Magnetization has strong uniaxial magnetic anisotropy in the ferromagnetic state of URhSi. However, the universality class of the ferromagnetic transition does not belong to the three-dimensional Ising system with short-range exchange interactions between magnetic moments ($\ensuremath{\beta}=0.325$, $\ensuremath{\gamma}=1.241$, and $\ensuremath{\delta}=4.82$). The obtained exponents in URhSi are similar to those in the uranium ferromagnetic superconductors ${\mathrm{UGe}}_{2}$ and URhGe, and uranium ferromagnets UIr and $\mathrm{U}({\mathrm{Co}}_{0.98}{\mathrm{Os}}_{0.02}$)Al. We have previously reported the unconventional critical behavior of magnetization in the uranium ferromagnetic superconductors [N. Tateiwa et al., Phys. Rev. B 89, 064420 (2014)]. The universality class of the ferromagnetic transition in URhSi may belong to the same one in the uranium ferromagnetic superconductors and the uranium ferromagnets. The unconventional critical behavior of the magnetization in the uranium compounds cannot be understood with previous theoretical interpretations of critical phenomena. The absence of the superconductivity in URhSi is discussed from several viewpoints. The improvement of the sample quality in URhSi could provide a good opportunity to gain a deeper understanding of the ferromagnetic superconductivity in the uranium ferromagnets.

Magnetocaloric effect and critical behavior in ternary equiatomic magnesium compounds REPtMg (RE = Tb, Dy and Ho)

The magnetocaloric effect (MCE) and critical behavior in the REPtMg (RE = Tb, Dy and Ho) intermetallic compounds around Curie temperatures TC were investigated by magnetization measurements. All the REPtMg compounds exhibit pronounced MCE performance with the maximum magnetic entropy change (−ΔSMmax) values up to 5.1, 7.2, 10.2 J/kg K under the magnetic field change (ΔH) of 0–5 T around TC of 58, 29, and 20 K for RE = Tb, Dy and Ho, respectively. The reasonable and accurate critical exponents for the present REPtMg series compounds for RE = Tb, Dy and Ho obtained by the ΔH dependence of ΔSM method are evaluated to be β= 0.4691, 0.4681, 0.4141, γ= 1.0412, 1.0482, 1.1067 and δ= 3.2196, 3.2393, 3.6728 around TC, which are verified by the Widom scaling law and scaling equations, respectively.

Effect of Gd substitution on the critical scaling of the ferromagnetic transition of La0.6-x Gdx Sr0.4 MnO3(x=0, 0.05, 0.1) manganite

This paper presents a detailed study on the critical properties of perovskite manganite La0.6-x Gdx Sr0.4 MnO3(x = 0, 0.05, 0.1) synthesized by Sol-gel method. The critical exponents near the PM-FM phase transition are determined through various methods such as modified Arrot plot, Kouvel-Fisher methods, and critical isothermal analysis based on the data of magnetic measurements recorded around the Curie temperature. These specimens are in agreement with their corresponding values proved by Widom scaling law, as well as the scaling theory. According to the values of critical exponents, the presence of the phase transition is found to be not relevant to the common transition classes but they can be attributed to some abnormal variations. It is suggested that the induced lattice disordering and magnetic disordering, as a result of Gadolinium incorporation, to be potential reasons for the presence of a large magnetocaloric effect, in addition to an anomalous ferromagnetic phase transition in the studied material.

Critical properties of the quasi-two-dimensional metallic ferromagnet Fe2.85GeTe2

We have investigated the critical behavior of Fe2.85GeTe2 single crystals near the paramagnetic-ferromagnetic transition by bulk dc magnetization measurements. The critical exponents β, γ and δ, obtained from modified Arrott plot, Kouvel–Fisher method, and critical isothermal magnetization analysis, could fulfill the Widom scaling law. The self-consistency and reliability of these exponents are further verified by the magnetic state equations below and above the Curie temperature at high magnetic field. In addition, the exchange distance deduced from the susceptibility exponent is shown to decay as . Based on the observations, we suggest that the competition between direct magnetic exchange and Coulombic and/or RKKY interactions should be responsible for the intrinsic magnetism in this system.

Critical behavior and magnetocaloric effect in monoclinic Cr5Te8

The critical behavior and magnetocaloric effects of the monoclinic Cr5Te8 single crystal have been studied by using magnetization measurements. Experimental results show that the paramagnetic (PM) to ferromagnetic (FM) phase transition is a second-order phase transition at Tc= 222 K. The detailed critical behavior analysis indicates that the values of critical exponents β, γ, and δ determined by the modified Arrott plots, the Kouvel-Fisher method as well as the critical isotherm analysis agree well with each other and fulfill the Widom scaling law. The critical exponents are close to those expected for the three-dimensional Ising model and the ferromagnetic interaction is at the boundary between long-range type and short-range one. In addition, the magnetocaloric property of monoclinic Cr5Te8 is consistent with the analysis of critical behavior.

Critical behavior of magnetization in URhAl: Quasi-two-dimensional Ising system with long-range interactions

The critical behavior of dc magnetization in the uranium ferromagnet URhAl with the hexagonal ZrNiAl-type crystal structure has been studied around the ferromagnetic transition temperature T_C. The critical exponent beta for the temperature dependence of the spontaneous magnetization below T_C, gamma for the magnetic susceptibility, and delta for the magnetic isotherm at T_C have been obtained with a modified Arrott plot, a Kouvel-Fisher plot, the critical isotherm analysis and the scaling analysis. We have determined the critical exponents as beta = 0.287 +- 0.005, gamma = 1.47 +- 0.02, and delta = 6.08 +- 0.04 by the scaling analysis and the critical isotherm analysis. These critical exponents satisfy the Widom scaling law delta=1+gamma/beta. URhAl has strong uniaxial magnetic anisotropy, similar to its isostructural UCoAl that has been regarded as a three-dimensional (3D) Ising system in previous studies. However, the universality class of the critical phenomenon in URhAl does not belong to the 3D Ising model (beta = 0.325, gamma = 1.241, and delta = 4.82) with short-range exchange interactions between magnetic moments. The determined exponents can be explained with the results of the renormalization group approach for a two-dimensional (2D) Ising system coupled with long-range interactions decaying as J(r)~r^-(d+sigma) with sigma = 1.44. We suggest that the strong hybridization between the uranium 5f and rhodium 4d electrons in the U-Rh_I layer in the hexagonal crystal structure is a source of the low dimensional magnetic property. The present result is contrary to current understandings of the physical properties in a series of isostructural UTX uranium ferromagnets (T: transition metals, X: p-block elements) based on the 3D Ising model.

3D-Heisenberg magnetic coupling in the skyrmion system Fe1.5−xCoxRh0.5Mo3N ([formula omitted

Critical behaviors of Fe1.5−xCoxRh0.5Mo3N (x=0.5,1.0,1.2) are investigated, which are well known as a new family hosting skyrmion phase. The critical exponents β, γ, and δ, which correspond to the magnetization, are obtained by the modified Arrott plot and critical isothermal analysis. The reliability of these critical exponents are testified by the Widom scaling law and scaling equation. The critical exponents of Fe1.5−xCoxRh0.5Mo3N (x=0.5,1.0,1.2) are close to the universality class of 3D-Heisenberg model, which suggests that the magnetic coupling in this system is of an isotropic short-range type.

Scaling study of magnetic phase transition and critical behavior in Nd0.55Sr0.45Mn0.98Ga0.02O3 manganite

Microscopic magnetic interactions and magnetism can be probed by studying the critical paramagnetic–ferromagnetic phase transition of magnetic materials. In this work, the critical behavior of the lightly Ga-doped manganites Nd0.55Sr0.45Mn0.98Ga0.02O3 (NSMGO) has been investigated. Scaling analysis reveals the critical exponents β=0.308, γ=1.197, and δ=4.916, where their self-consistency and reliability are verified by the Widom scaling law and the scaling equation. These exponents indicate that the magnetic phase transition in NSMGO belongs to the anisotropic 3D-Ising universality class. Nevertheless, the asymptotic critical exponents βeff and γeff are found to approach to the values of 3D-Heisenberg model, implying the existence of isotropic short-range magnetic interactions. We suggest that, although the Nd-based manganites possess a strong magnetocrystalline anisotropy, this effect can be partly averaged out in view of the present polycrystalline NSMGO. Therefore, in the asymptotic critical region, the phase transition reveals the crossover from the scaling of 3D-Ising toward 3D-Heisenberg universality class.

Critical Phenomena and Estimation of Spontaneous Magnetization by Magnetic Entropy Analysis of La0.7Sr0.3Mn0.94Cu0.06O3

This paper presents a detailed study on the structural, magnetic, and critical behavior and magnetocaloric effect of La0.7Sr0.3Mn0.94Cu0.06O3 prepared by solid-state reaction method. X-ray diffraction patterns indicate that Cu-doped manganite crystallizes to a single phase with a rhombohedral structure. Critical components β, γ, and δ were determined by using modified Arrott plot, Kouvel–Fisher method, and critical isotherm analyses. These components are consistent with their corresponding values and were validated by Widom scaling law and scaling theory. The deduced critical exponents (β = 0.456, γ = 0.9912, and δ = 3.206) are also close to their corresponding theoretical values predicted by mean-field model, indicating that long-range interaction dominates the critical behavior of the LSMCO system. Spontaneous magnetization was determined by two methods. One method is based on magnetic entropy change under different magnetic fields, and the other method is based on classical extrapolation from the Arrott plot. The two methods are in excellent agreement, confirming the validity of the deduction of spontaneous magnetization using the magnetic entropy change.

Griffiths-like phase, critical behavior near the paramagnetic-ferromagnetic phase transition and magnetic entropy change of nanocrystalline La0.75Ca0.25MnO3

In this work, we report the structural and magnetic properties of La0.75Ca0.25MnO3 nanoparticles synthesized by the sol-gel route. Rietvield refinement of X-ray powder diffraction confirms that our sample is single phase and crystallizes in orthorhombic system with Pnma space group. The facts that effective magnetic moment is large and the inverse susceptibility deviates from the Curie Weiss lawn indicate the presence of Griffiths-like cluster phase. The critical exponents have been estimated using different techniques such as modified Arrott plot, Kouvel-Fisher plot and critical isotherm technique. The critical exponents values of La0.75Ca0.25MnO3 are very close to those found out by the mean-field model, and this can be explained by the existence of a long-range interactions between spins in this system. These results were in good agreement with those obtained using the critical exponents of magnetic entropy change. The self-consistency and reliability of the critical exponent was verified by the Widom scaling law and the universal scaling hypothesis. Using the Harris criterion, we deduced that the disorder is relevant in our case. The maximum magnetic entropy change (ΔSM) calculated from the M-H measurements is 3.47 J/kg K under an external field change of 5 T. The ΔSM-T curves collapsed onto a single master curve regardless of the composition and the applied field, confirming the magnetic ordering is of second order nature. The obtained result was compared to ones calculated based on the Arrott plot and a good concordance is observed. Moreover, the spontaneous magnetization obtained from the entropy change is in excellent agreement with that deduced by classically extrapolation the Arrott curves. This result confirms the validity of the estimation of the spontaneous magnetization using the magnetic entropy change.