Abstract
The van der Waals ferromagnet Fe_{5}GeTe_{2} has a Curie temperature T_{C} of about 270 K, which is tunable through controlling the Fe deficiency content and can even reach above room temperature. To achieve insights into its ferromagnetic exchange that gives the high T_{C}, the critical behavior has been investigated by measuring the magnetization in Fe_{5}GeTe_{2} crystal around the ferromagnetic ordering temperature. The analysis of the measured magnetization by using various techniques harmonically reached to a set of reliable critical exponents with T_{C} = 273.7 K, β = 0.3457 ± 0.001, γ = 1.40617 ± 0.003, and δ = 5.021 ± 0.001. By comparing these critical exponents with those predicted by various models, it seems that the magnetic properties of Fe_{5}GeTe_{2} could be interpreted by a threedimensional magnetic exchange with the exchange distance decaying as J(r) ≈ r^{−4.916}, close to that of a threedimensional Heisenberg model with longrange magnetic coupling.
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Introduction
A prominent virtue of the quasitwodimensional (2D) van der Waals (vdW) bonded materials is that they could be exfoliated into multi or single layer, thus making them useful in various novel heterostructures and devices. Moreover, the vdW materials in the 2D limit exhibit extraordinary physical properties, such as those observed in the intensively studied graphene and transition metal dichalcogenides^{1,2,3,4,5,6}, etc. Known as the MerinWagner theorem^{7}, intrinsic longrange magnetic order can not appear in the isotropic magnetic 2D limit because the strong thermal fluctuations in such case prohibit the spontaneous symmetry breaking and hence the longrange magnetic ordering. Nevertheless, a small anisotropy is sufficient to open up a sizable gap in the magnon spectra and consequently stabilizes the magnetic order against finite temperature. This picture has been realized by the observation of longrange ferromagnetic (FM) order in mono or fewlayer CrI_{3}^{8}, Cr_{2}Ge_{2}Te_{6}^{9}, Cr_{2}Si_{2}Te_{6}^{10}, VSe_{2}^{11}, and MnSe_{2}^{12}, etc. The vdW magnets in the 2D limit host rich magnetoelectrical, magnetooptical, or spin–lattice coupling effects that are capable of producing intriguing properties which are scarcely observed in bulk. Very recently, currentinduced magnetic switch was observed in the fewlayer Fe_{3}GeTe_{2}^{13}, demonstrating the vdW magnets a versatile platform for nanoelctronics. Moreover, heterostructures constructed by using vdW magnets have profound valleytronics and spintronics device applications^{14,15}. For example, the tunneling magnetoresistance (MR) in spinfilter magnetic vdW CrI_{3} heterostructures even approaches 1.9 × 10^{4}%, remarkably superior to that constructed by using conventional magnetic thin films^{16}. The easy exfoliation, weak interlayer coupling, and tunability of magnetic properties make the vdW magnets a model family of materials for exploring exotic phenomena and finding novel applications.
In the handful FM vdW magnets, the physical properties in the 2D limit differ from each other due to rather complex magnetic interactions. The semiconducting monolayer CrI_{3} is an Ising ferromagnet with very low Curie temperature (T_{C}) of about 45 K due to the weak superexchange interaction along the CrICr pathway^{8,17}. The similar weak FM superexchange in the Heisenberg magnet bilayer Cr_{2}Ge_{2}Te_{3} also results in a low T_{C} of ~ 30 K, and FM order is even not present in the monolayer^{9}. As a contrast, the FM exchange with an itinerant character mediated by carriers in metallic Fe_{3}GeTe_{2} monolayer is much stronger than the superexchange in CrI_{3} and Cr_{2}Ge_{2}Te_{6}, thus yielding a remarkably higher T_{C} of about ~ 130 K, which can be raised even above room temperature by using the ionic gating technique^{18,19}.
The tremendous efforts in perusing high T_{C} magnets more recently led to the discovery of a T_{C} of ~ 130–230 K in the bulk quasi2D vdW Fe_{3x}GeTe_{2}, which can even be enhanced up to room temperature^{18}. Interestingly, similar as Fe_{3x}GeTe_{2}, bulk Fe_{5x}GeTe_{2} shows a tunable T_{C} ranging from ~ 270 to ~ 363 K by controlling the Fe deficiency content x or by substituting Co for Fe, suggesting the detrimental role of Fe in the magnetic exchange^{20,21,56}. A reversible magnetoelastic coupled firstorder transition near 100 K was detected by neutron powder diffraction^{20}. Considering the exotic physical properties in exfoliated Fe_{3}GeTe_{2} nanoflakes and its heterostructures, such as the extremely large anomalous Hall effect^{22}, planar topological Hall effect^{23}, Kondo lattice physics^{24}, anisotropic magnetostriction effect^{25}, spin filtered tunneling effect^{16}, magnetic skyrmions^{26}, etc., Fe_{5}GeTe_{2} would also be expected to provide extraordinary opportunities to explore intriguing physical properties. To well understand the physical properties of Fe_{5}GeTe_{2}, the magnetic exchange model should be established first. However, the direct measurements on the magnetic structure are absent yet. Alternatively, study on the magnetic critical behavior and analysis of the critical exponents in vicinity of the paramagnetic (PM) to FM transition region could yield valuable insights into the magnetic exchange and properties. For example, the method has established the magnetic exchange models for CrI_{3}^{27}, VI_{3}^{28}, Fe_{3}GeTe_{2}^{29,30}, Co_{2}TiSe^{31}, and Fe_{0.26}TaS_{2}^{32}, etc. In this work, we have reported the investigation on the critical behavior of Fe_{5}GeTe_{2}, which finds that the obtained set of critical exponents are close to those calculated from the renormalization group approach for a longrange 3D Heisenberg model with the magnetic exchange distance decaying as J(r) ≈ r^{−4.916}.
Result and discussion
Chracterizations on the crystal structure, quality and compositions are presented in the supplementary materials (SI). Figure 1a depicts the temperature dependence of magnetization M(T) for Fe_{5}GeTe_{2} measured with zerofieldcooling (ZFC) and fieldcooling (FC) mode under the applied magnetic field H = 1 kOe along the abplane of the crystal. The magnetization displays an abrupt PM to FM transition at ~ 270 K and no clear separation between the ZFC and FC curves. The inset of Fig. 1a is the inverse temperature dependent magnetic susceptibility χ^{1}(T) with the dotted straight line representing the Curie–Weiss law fitting. It shows a deviation of \({\chi }^{1}(T)\) from the straight line near 295 K which is much higher than T_{C}. The obtained Weiss temperature is 283 K, which is also higher than T_{C}, indicating a strong FM interaction. The effective moment as μ_{eff} = 6.659 μ_{B}/Fe is also obtained. Considering the varied effective magnetic moment of Fe^{2+} with the values raging from 4.90 to 6.70 μ_{B} in various materials including sphalerite and monoclinic pyroxenes obtained from magnetic susceptibility analysis^{33} and the Fe deficiency in our crystals, the value we obtained from the Curie–Weiss law fitting is reasonable. The FM ground state can also be demonstrated by the isothermal magnetization M(H) shown in Fig. 1b measured at 2 K. The low coercive field indicates a soft ferromagnetism in Fe_{5}GeTe_{2}, which is similar as that of Fe_{3}GeTe_{2}^{29,30}. The saturation magnetic moment along the caxis is about 2.4 μ_{B}/Fe, likely unveiling the magnetic anisotropy at low temperature. The initial isothermal magnetizations in the temperature range of 261–285 K measured with H//caxis were shown in Fig. 1c and the Arrott plot^{34}, that is, M^{2} vs. H//M, is shown in Fig. 1d. The positive slope of all M^{2} vs. H/M curves, according to the Banerjee’s criterion^{35}, indicates that the PM to FM transition has a secondorder in nature. The Arrott plot was initially tried to for the analysis of the measured magnetizations, so the mean Landau meanfield theory with the critical exponents β = 0.5 and γ = 1.0 is involved. If it works, the M^{2} vs. H//M curves should be straight and parallel to each other in the high magnetic field region, and additionally, the isothermal magnetization at T_{C} should pass through the origin. However, seen in Fig. 1d M^{2} vs. H//M curves are clearly nonlinear with a downward curvature, suggesting that the fit does not work for Fe_{5}GeTe_{2}. The failure of the Arrott plot within the framework of Landau meanfield theory lies in that the itinerant ferromagnetism in Fe_{5}GeTe_{2} should have significant electronic correlations and spin fluctuations, which however are neglected in the Landau meanfield theory.
The secondorder PM to FM phase transition in Fe_{5}GeTe_{2} can be described by the magnetic equation of state and is characterized by critical exponents β, γ and δ that are mutually related. According to the scaling hypothesis, for a secondorder phase transition, the spontaneous magnetization M_{S}(T) below T_{C}, the inverse initial susceptibility χ_{0}^{–1}(T) above T_{C} and the magnetization M at T_{C} can be used to obtain β, γ and δ by using the equations^{36}:
where ε = (T—T_{C})/T_{C} is the reduced temperature, and M_{0}, h_{0}/m_{0}, and D are the critical amplitudes. Though the Landau meanfield theory can not be used, the critical isothermal magnetizations, alternatively, can be analyzed with the ArrottNoakes equation of state^{37}:
where a and b are the fitting constants. Five different models including the 2D Ising model (β = 0.125, γ = 1.75)^{38}, the 3D Heisenberg model (β = 0.365, γ = 1.386)^{38}, the 3D Ising model (β = 0.325, γ = 1.24)^{38}, the 3D XY model (β = 0.345, γ = 1.316)^{39} and the tricritical meanfield model (β = 0.25, γ = 1.0)^{40} were used for the modified Arrott plots, which are shown in Fig. 2a–e. One can see that the lines in Fig. 2d,e are not parallel to each other, thus excluding the tricritical meanfield and 2D Ising models. In Fig. 2a–c, all lines in each figure are almost parallel to each other in the high magnetic field region, thus making the choice of an appropriate model for Fe_{5}GeTe_{2} impossible in this step. As we mentioned above, the modified Arrott plot should be a set of parallel lines in the high magnetic field region with the same slope of S(T) = dM^{1/β}/d(H/M)^{1/γ}. The normalized slope NS is defined by NS = S(T)/S(T_{C}), which enables us an easy comparison of the NS of different models and to select out the most appropriate one with the ideal value of unity. The NS values versus the temperature for different models are plotted in Fig. 2f, which clearly show that the NS of the 2D Ising model has the largest deviation from unity. One can see that when T > T_{C}, NS of the 3D Ising model is close to unity, while when T < T_{C} the 3D XY model seems as the best. This indicates that the critical behavior of Fe_{5}GeTe_{2} may not belong to a single universality class. The fact also likely indicates that the magnetic character of Fe_{5}GeTe_{2} is nearly isotropic above T_{C} and the enhancement of the anisotropic exchange below T_{C}.
To achieve indepth insights into the nature of the PM to FM transition in Fe_{5}GeTe_{2}, the precise critical exponents and critical temperature should be obtained. In the modified Arrott plot, the linear extrapolation of the nearly straight curves from the high magnetic field region intercepting the M^{1/β} and (H/M)^{1/γ} axes yields reliable values of M_{S}(T) and χ_{0}^{–1}(T), respectively. The extracted M_{S}(T) and χ_{0}^{–1}(T) can be used to fit the β and γ by using Eqs. (1) and (2). The thus obtained β and γ are thereafter used to reconstruct a modified Arrott plot. Consequently, new M_{S}(T) and χ_{0}^{–1}(T) are generated from the linear extrapolation in the high field region, and a new set of β and γ will be acquired. This procedure should be repeated until β and γ are convergent. The obtained critical exponents from this method are independent on the initial parameters, thus guaranteeing the reliability of the analysis and that the obtained critical exponents are intrinsic. The final modified Arrott plot with β = 0.351(1) and γ = 1.413(5) is presented in Fig. 3, which shows that the isotherms in the high magnetic field region are actually a set of parallel straight lines. In addition, the final M_{S}(T) and χ_{0}^{–1}(T) with solid fitting curves are depicted in Fig. 4a, which yield the critical exponents β = 0.344(5) with T_{C} = 273.76(3) K and γ = 1.406(1) with T_{C} = 273.88(4) K.
It is necessary to check the accuracy of above analysis. The KouvelFisher (KF) method can also be employed to fit the critical exponents and critical temperature, which is expressed as^{40}:
where M_{S}(T)/(dM_{S}(T)/dT) and χ_{0}^{–1}(T)/(dχ_{0}^{–1}(T)/dT) are linearly dependent on temperature with the slopes of 1/β and 1/γ, respectively. As is shown in Fig. 4b, the linear fits give β = 0.346(4) with T_{C} = 273.75(7) K and γ = 1.364(9) with T_{C} = 273.97(9) K, respectively, which are consistent with those obtained from the iterative modified Arrott plot, thus confirming the reliability of the above analysis.
The iterative modified Arrott plot gives the critical exponents β and γ, while the critical exponent δ can be obtained by using Eq. (3). Figure 4c shows the isothermal magnetization M(H) at a critical temperature T_{C} = 274 K and the inset shows the plot at a log–log scale. According to Eq. (3), the M(H) at T_{C} should be a straight line in the log–log scale with the slope of 1/δ, thus giving δ = 5.02(1). To check the reliability of such analysis, δ was also calculated by using the Widom scaling relation^{41}:
which gives δ = 5.02(6) and δ = 4.94(0) by using the β and γ obtained with modified Arrott plot and KouvelFisher plot, respectively, which are consistent with those fitted by using Eq. (3).
From above analysis, a set of critical exponents are obtained, which are actually self consistent. It is of essential importance to check whether the obtained critical exponents and T_{C} can generate a scaling equation of state for Fe_{5}GeTe_{2}, i.e., to examine the reliability of these critical exponents again by using the scaling analysis. According to the scaling hypothesis, for a magnetic system in the critical asymptotic region, the scaling equation of state can be expressed as^{42}:
where M(H, ε), H, and T are variables; f_{+} for T > T_{C} and f_{˗} for T < T_{C} are the regular functions. Equation (8) can also be written as:
where \(m\equiv {\varepsilon }^{\beta }M(H,\varepsilon )\) and \(h\equiv {\varepsilon }^{(\beta +\gamma )}\). If the critical exponents β, γ and δ could be properly chosen, the scaled m(h) plot will fall onto two universal curves for T > T_{C} and T < T_{C}, respectively. In such case, the interactions are believed to be properly renormalized in the critical regime following the scaling equation of state. The scaled m and h curves are plotted in Fig. 5a, which actually show two branches below and above T_{C}, thus guarantying the reliability of the obtained critical exponents. The two branches are much clear when the same data are plotted in a log–log form, seen by the inset of Fig. 5a. To support the analysis, we used a more rigorous method by plotting m^{2} against h/m, seen in Fig. 5b in which all data apparently separate into two curves below and above T_{C}. The reliability of the obtained critical exponents and T_{C} can also be examined by checking the scaling of the magnetization curves. The scaling state equation of magnetic systems is^{42}:
where h(x) is a scaling function. From Eq. (10), the εH^{−(βδ)} vs. MH^{−1/δ}should fall on one universal curve^{43}, as seen by the inset of Fig. 5b. The T_{C} lies on the zero point of εH^{(βδ)} axis. As a result, the well rescaled curves further confirm that the obtained critical exponents and T_{C} are reliable and consistent with the scaling hypothesis.
It is valuable to compare the critical exponents of Fe_{5}GeTe_{2} with those of other layered vdW magnets and those predicted by various models. The critical exponents of Fe_{5}GeTe_{2} obtained by using different analysis techniques and different theoretical models are summarized in Table 1, together with those of other several FM vdW magnets including Fe_{3x}GeTe_{2} (x = 0, 0.15, and 0.36), Cr_{2}Si_{2}Te_{6}, and Cr_{2}Ge_{2}Te_{6}. The previous comprehensive study reached a conclusion that the critical exponent β for a 2D magnets lies in the range of ~ 0.1 ≤ β ≤ 0.25^{44}. It is apparent that the β values of Cr_{2}Si_{2}Te_{6} and Cr_{2}Ge_{2}Te_{6}, which were verified as 2D Ising magnets^{45,46}, are actually within the window, while those of Fe_{3x}GeTe_{2} and Fe_{5}GeTe_{2} are apparently larger than 0.25, thus excluding the 2D Ising model for them^{29,30}. Moreover, the γ values of Fe_{3x}GeTe_{2} and Fe_{5}GeTe_{2} are much larger than those for the tricritical meanfield and 3D Ising models^{38,39}, suggesting the two models are not appropriate. Combining the β and γ values, the magnetic critical behavior in Fe_{5}GeTe_{2} should have a 3D nature, indicating that the interlayer magnetic exchange can not be neglected. It was suggested that Fe_{3x}GeTe_{2} has a smaller vdW gap and hence a stronger interlayer magnetic exchange than that in Cr_{2}(Si,Ge)_{2}Te_{6}^{17}. It is therefore a natural hypothesis that the vdW gap in Fe_{5}GeTe_{2} is also very small. To achieve more insights, the critical exponents of Fe_{5}GeTe_{2} should be compared with the several 3D models more carefully. The β of Fe_{5}GeTe_{2} is much closer to that of the 3D XY model^{39} while the γ is closer to that of the 3D Heisenberg model^{38}, likely implying that the obtained critical exponents of Fe_{5}GeTe_{2} can not be simply categorized into any conventional universality classes.
For a homogenous magnet, it is essential to use the magnetic exchange distance J(r) to further determine the universality class of the magnetic phase transition. Within the framework of to the renormalization group theory, the magnetic exchange decays with the distance r in a form J(r) ~ e^{–r/b} for the shortrange magnetic exchange and J(r) ~ r^{–(d+σ)} for the longrange exchange, where r is the exchange distance, b is the spatial scaling factor, d is the dimensionality of the system, and the positive constant σ denotes the range of exchange interaction^{47,48}. Moreover, within this theory model the magnetic susceptibility exponent γ is defined as^{47}:
where n is the spin dimensionality, Δσ = (σ – d/2) and \(G\left(\frac{d}{2}\right)=3\frac{1}{4}{(\frac{d}{2})}^{2}\). For 3D materials (d = 3) with 3/2 ≤ σ ≤ 2, the magnetic exchange decays relatively slowly as J(r) ~ r ^{–(d+σ)} due to a longrange magnetic exchange. For σ > 2, the 3D Heisenberg model is valid for 3D isotropic magnets, where J(r) decreases faster than r ^{5} due to the shortrange magnetic exchange, while when σ ≤ 3/2, the meanfield model works and J(r) decreases slower than r^{4.5}^{47,48}. To obtain the values of d, n, and σ for Fe_{5}GeTe_{2}, a method similar to that in Ref.^{47}. was adopted. In this method, σ is initially adjusted according to Eq. (11) with several sets of {d : n} to get a proper γ that is close to the experimental value (~ 1.364). The obtained σ is then used to calculate other critical exponents by the following equations: ν = γ/σ, α = 2 − νd, β = (2 − α − γ), and δ = 1 + γ/β. Several sets of {d : n} will be tried, with the typical results being summarized Table 2, which finally achieved the critical exponents of β = 0.3851, γ = 1.3613 and δ = 4.5351, which match well with the experimental values, when {d: n} = {3: 3} and σ = 1.916. Such a result indicates that the 3D Heisenberg type magnetic exchange with longrange interaction decaying as J(r) ≈ r^{–4.916} can account for the magnetic properties of Fe_{5}GeTe_{2}, which is consistent with our analysis presented above.
The magnetic exchange in quasi2D vdW magnets has been subjected to immense investigations. For Cr_{2}Si_{2}Te_{6}, the magnetic critical behavior analysis and neutron scattering studies consistently suggest the universality class of 2D Ising model accounting for its magnetic properties^{26,49}. Because of the smaller vdW gap and hence an enhanced interlayer exchange in Cr_{2}Ge_{2}Te_{6}, its critical behavior shows a transition from the 2D Isingtype to a 3D tricritical meanfield type^{50}. It is useful to compare the magnetic critical behavior of Fe_{5}GeTe_{2} with that of Fe_{3}GeTe_{2}. The mean distance between the two adjacent Te layers that across the vdW gap in Fe_{3}GeTe_{2} is 0.423 nm^{51}, which is rather close to that of CrGeTe_{3}, 0.377 nm^{49}, which presumably can account for the 3D Heisenberg characteristics of the critical behavior. Previous studies on Fe_{5}GeTe_{2} indicate small magnetic anisotropy at high temperature^{20}, so the 3D magnetism for the critical behavior in Fe_{5}GeTe_{2} is reasonable. Moreover, it is found that the magnetic anisotropy in Fe_{3x}GeTe_{2} strongly depends on the Fe deficiency^{52}, which can be largely suppressed with increasing the deficiency content x. If we pay a close attention to the critical exponents of Fe_{5}GeTe_{2}, it is easily found that they are much closer to those of Fe deficient Fe_{3x}GeTe_{2}^{29}, likely further demonstrating the weak magnetic anisotropy in Fe_{5}GeTe_{2}. However, the possible transition between different universality classes of models of the critical behavior should be carefully checked, if we recall into our mind that a critical phase transition between 3 and 2D at the temperature of ~ 0.9T_{C} in NiPS_{3} and an anisotropic 2D to 3D magnetism below T_{C} in MnPS_{3} were experimentally confirmed^{53,54}. Though such possibility has not been examined yet in Fe_{3x}GeTe_{2}, considering that Fe_{3x}GeTe_{2} indeed shares similarities as MPS_{3} (M = Mn, Fe, and Ni) in that they all have 2D antiferromagnetic ground state with the ferromagnetic layers in them order antiferromagnetically along the caxis at low temperature, as well as the 3D critical behavior near T_{C}, the critical phase transition definitely need to be checked in Fe_{3x}GeTe_{2}. For Fe_{5}GeTe_{2}, it is somewhat different from MPS_{3} and Fe_{3x}GeTe_{2}, which behaves as an easyaxis vdW ferromagnet with the magnetic moments preferring to align along the caxis but with weak anisotropy at high temperature due to the easy polarization of moments and the interaction between the FM layers is still FM. However, the magnetism of Fe_{5}GeTe_{2} is somewhat complex due to the multiple Fe sublattices and composition tunable T_{C}. It is revealed that the magnetic moments on Fe(1) sublattice order below ~ 100–120 K while the majority of the moments order at T_{C}^{21}. Shortrange order associated with occupations of split sites of Fe(1) is also present. Additionally, the magnetic anisotropy is enhanced at low temperature. Regarding these, more studies to establish the precise spin structure at low temperature are extremely desired.
Conclusion
In summary, we have investigated the magnetic critical behavior in vicinity of the PM to FM phase transition in the quasi2D van der Waals ferromagnet Fe_{5}GeTe_{2} which has a near room temperature T_{C} of approximately 270 K. The estimated critical exponents β, γ and δ values from the various techniques and theoretical models show nice consistence with each other and follow the scaling behavior well. The critical exponents suggest a second order phase transition and they do not belong to any single universality class of model, just lying between the 3D Heisenberg model and the 3D XY model. The magnetic exchange distance is found to decay as J(r) ≈ r^{–4.916}, which is close to that of 3D Heisenberg model with longrange exchange. The critical phenomena indicate weak magnetic anisotropy of Fe_{5}GeTe_{2} at high temperature, possibly due to its small vdW gap. The very recent calculations indicate that monolayer formation energy of Fe_{5}GeTe_{2} lies inside the energy range of other 2D materials^{55}, and the synthesis of the monolayer is therefore highly expected. Moreover, considering the tunable T_{C} which can even to be ~ 350 K^{20,21,56}, the investigation on the precise magnetic structure of Fe_{5}GeTe_{2} would find extraordinary opportunities for applications in nextgeneration spintronic devices.
Methods
Single crystals were grown from chemical vapor transport (CVT) technique by using iodine as the transport agent, similar as the method described previously^{20,21}. The crystal used in this experiment is flat with a typical dimension of 2 mm * 2 mm * 0.1 mm. The crystallographic phase and crystal quality were examined on a Bruker D8 single crystal Xray diffractometer (SXRD) with Mo K_{α} (λ = 0.71073 Å) at 300 K. The chemical compositions and uniformity of stoichiometry were checked by the energy dispersive spectroscopy (EDS) at several spots on the crystals. The direct current (dc) magnetization was measured on the Quantum Design magnetic properties measurement system (MPMS3) with the magnetic field applied parallel to caxis of the crystal. Isothermal magnetizations were collected at a temperature interval of 1 K in the temperature range of 261–285 K, which is just around T_{C} (~ 270 K). It should be noted that each curve was initially magnetized. The applied magnetic field was corrected by considering the demagnetization factor, which was used for the analysis of critical behavior. The demagnetization factor is roughly estimated to be ~ 0.88 with considering the crystal size^{57}.
References
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
Zhang, Y., Tan, Y.W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in grapheme. Nature 438, 201–204 (2005).
Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thin MoS_{2}: A new directgap semiconductor. Phys. Rev. Lett. 105, 136805 (2010).
Lu, J. M. et al. Evidence for twodimensional Ising superconductivity in gated MoS_{2}. Science 350, 1353–1357 (2015).
Xi, X. et al. Strongly enhanced chargedensitywave order in monolayer NbSe_{2}. Nature Nanotechnol. 10, 765–769 (2015).
Li, L. et al. Controlling manybody states by the electricfield effect in a twodimensional material. Nature 529, 185–189 (2016).
Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one or twodimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133 (1966).
Huang, B. et al. Layerdependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270–273 (2017).
Gong, C. et al. Discovery of intrinsic ferromagnetism in twodimensional van der Waals crystals. Nature 546, 265–269 (2017).
Lin, M.W. et al. Ultrathin nanosheets of CrSiTe_{3}: A semiconducting twodimensional ferromagnetic material. J. Mater. Chem. C 4, 315–322 (2016).
Bonilla, M. et al. Strong roomtemperature ferromagnetism in VSe_{2} monolayers on van der Waals substrates. Nat. Nanotechnol. 13, 289–293 (2018).
O’Hara, D. J. et al. Room temperature intrinsic ferromagnetism in epitaxial manganese selenide films in the monolayer limit. Nano Lett. 18, 3125–3133 (2018).
Wang, X. et al. Currentdriven magnetization switching in a van der Waals ferromagnet Fe_{3}GeTe_{2}. Sci. Adv. 5, eaaw8904 (2019).
Zhong, D. et al. Van der Waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics. Sci. Adv. 3, e1603113 (2017).
Samarth, N. Magnetism in flatland. Nature 546, 216–217 (2017).
Song, T. C. et al. Giant tunneling magnetoresistance in spinfilter van der Waals heterostructures. Science 360, 1214–1218 (2018).
McGuire, M. A., Dixit, H., Cooper, V. R. & Sales, B. C. Coupling of crystal structure and magnetism in the layered, ferromagnetic insulator CrI_{3}. Chem. Mater. 27, 612–620 (2015).
Deng, Y. et al. Gatetunable roomtemperature ferromagnetism in twodimensional Fe_{3}GeTe_{2}. Nature 563, 94–99 (2018).
Li, Q. et al. Patterninginduced ferromagnetism of Fe_{3}GeTe_{2} van der Waals materials beyond room temperature. Nano Lett. 18, 5974–5980 (2018).
May, A. F. et al. Ferromagnetism near room temperature in the cleavable van der Waals crystal Fe_{5}GeTe_{2}. ACS Nano 13, 4436–4442 (2019).
May, A. F., Bridges, C. A. & McGuire, M. A. Physical properties and thermal stability of Fe_{5−}_{x}GeTe_{2} single crystals. Phys. Rev. Mater. 3, 104401 (2019).
Kim, K. et al. Large anomalous Hall current induced by topological nodal lines in a ferromagnetic van der Waals semimetal. Nat. Mater. 17, 794–799 (2018).
You, Y. et al. Planar topological Hall effect in a uniaxial van der Waals ferromagnet Fe_{3}GeTe_{2}. Phys. Rev. B 100, 134441 (2019).
Zhang, Y. et al. Emergence of Kondo lattice behavior in a van der Waals itinerant ferromagnet, Fe_{3}GeTe_{2}. Sci. Adv. 4, eaao6791 (2018).
Zhuang, H. L., Kent, P. R. C. & Hennig, R. G. Strong anisotropy and magnetostriction in the twodimensional Stoner ferromagnet Fe_{3}GeTe_{2}. Phys. Rev. B 93, 134407 (2016).
Ding, B. et al. Observation of magnetic Skyrmion Bubbles in a van der Waals ferromagnet Fe_{3}GeTe_{2}. Nano Lett. 20, 868–873 (2020).
Liu, Y. & Petrovic, C. Threedimensional magnetic critical behavior in CrI_{3}. Phys. Rev. B 97, 014420 (2018).
Liu, Y., Abeykoon, M. & Petrovic, C. Critical behavior and magnetocaloric effect in VI_{3}. Phys. Rev. Res. 2, 013013 (2020).
Liu, Y., Ivanovski, V. N. & Petrovic, C. Critical behavior of the van der Waals bonded ferromagnet Fe_{3−}_{x}GeTe_{2}. Phys. Rev. B 96, 144429 (2017).
Liu, B. J. et al. Critical behavior of the van der Waals bonded high T_{C} ferromagnet Fe_{3}GeTe_{2}. Sci. Rep. 7, 6184 (2017).
Rahman, A. et al. Critical behavior in the halfmetallic Heusler alloy Co_{2}TiSn. Phys. Rev. B 100, 214419 (2019).
Zhang, C. H. et al. Critical behavior of intercalated quasivan der Waals ferromagnet Fe_{0.26}TaS_{2}. Phys. Rev. Mater. 3, 114403 (2019).
Parks, G. A. & Akhtar, S. Magnetic moment of Fe^{2+} in paramagnetic minerals. Am. Mineralo. J. Earth Planet. Mater. 53, 406–415 (1968).
Arrott, A. Criterion for ferromagnetism from observations of magnetic isotherms. Phys. Rev. 108, 1394–1396 (1957).
Banerjee, B. On a generalised approach to first and second order magnetic transitions. Phys. Lett. 12, 16–17 (1964).
Fisher, M. E. The theory of equilibrium critical phenomenon. Rep. Prog. Phys. 30, 615 (1967).
Arrott, A. & Noakes, J. E. Approximate equation of state for nickel near its critical temperature. Phys. Rev. Lett. 19, 786 (1967).
Kaul, S. N. Static critical phenomenon in ferromagnets with quenched disorder. J. Mag. Magn. Mater. 53, 5–53 (1985).
Le Guillou, J. C. & ZinnJustin, J. Critical exponents from field theory. Phys. Rev. B 21, 3976 (1980).
Kouvel, J. S. & Fisher, M. E. Detailed magnetic behavior of nickel near its curie point. Phys. Rev 136, A1626–A1632 (1964).
Widom, B. Surface tension and molecular correlations near the critical point. J. Chem. Phys. 43, 3892 (1965).
Stanley, H. E. Phase Transitions and Critical Phenomena (Clarendon Press, Oxford., 1971).
Phan, M. et al. Tricritical point and critical exponents of La_{0.7}Ca_{0.3−x}Sr_{x}MnO_{3} (x = 0, 0.05, 0.1, 0.2, 0.25) single crystals. J. Alloy. Comp. 508, 238–244 (2010).
Taroni, A., Bramwell, S. T. & Holdsworth, P. C. Universal window for twodimensional critical exponents. J. Phys. Condens. Matter. 20, 275233 (2008).
Liu, Y. & Petrovic, C. Critical behavior of quasitwodimensional semiconducting ferromagnet Cr_{2}Ge_{2}Te_{6}. Phys. Rev. B 96, 054406 (2017).
Liu, B. et al. Critical behavior of the quasitwodimensional semiconducting ferromagnet CrSiTe_{3}. Sci. Rep. 6, 33873 (2016).
Fisher, M. E., Ma, S.K. & Nickel, B. Critical exponents for longrange interactions. Phys. Rev. Lett. 29, 917–920 (1972).
Fisher, S., Kaul, S. N. & Kronmüller, H. Critical magnetic properties of disordered polycrystalline Cr_{75}Fe_{25} and Cr_{70}Fe_{30} alloys. Phys. Rev. B 65, 064443 (2002).
Carteaux, V., Moussa, F. & Spiesser, M. 2D Isinglike ferromagnetic behaviour for the lamellar Cr_{2}Si_{2}Te_{6} compound: A neutron scattering investigation. Europhys. Lett. 29, 251–256 (1995).
Lin, G. T. et al. Tricritical behavior of the twodimensional intrinsically ferromagnetic semiconductor CrGeTe_{3}. Phys. Rev. B 95, 245212 (2017).
Ouvrard, G., Sandre, E. & Brec, R. Synthesis and crystal structure of a new layered phase: The chromium hexatellurosilicate Cr_{2}Si_{2}Te_{6}. J. Solid State Chem. 73, 27–32 (1988).
May, A. F., Calder, S., Cantoni, C., Cao, H. & McGuire, M. A. Magnetic structure and phase stability of the van der Waals bonded ferromagnet Fe_{3−}_{x}GeTe_{2}. Phys. Rev. B 93, 014411 (2016).
Wildes, A. R. et al. Magnetic structure of the quasitwodimensional antiferromagnet NiPS_{3}. Phys. Rev. B 92, 224408 (2015).
Wildes, A. R. et al. Static and dynamic critical properties of the quasitwodimensional antiferromagnet MnPS_{3}. Phys. Rev. B 74, 094422 (2006).
Joe, M., Yang, U. & Lee, C. Firstprinciples study of ferromagnetic metal Fe_{5}GeTe_{2}. Nano Mater. Sci. 1, 299–303 (2019).
Tian, C. K. et al. Tunable magnetic properties in van der Waals crystals (Fe_{1−}_{x}Co_{x})_{5}GeTe_{2}. Appl. Phys. Lett. 116, 202402 (2020).
Chen, D. X., Pardo, E. & Sanchez, A. Demagnetizing factors of rectangular prisms and ellipsoids. IEEE Trans. Magn. 38, 1742 (2002).
Acknowledgements
The authors acknowledge the support by the National Natural Science Foundation of China (Grant No. 11874264). Y.F.G. acknowledges the starting grant of ShanghaiTech University and the Program for Professor of Special Appointment (Shanghai Eastern Scholar). L.M.C. is supported by the Key Scientific Research Projects of Higher Institutions in Henan Province (19A140018). The authors also thank the support from the Analytical Instrumentation Center (#SPSTAIC10112914), SPST, ShanghaiTech University.
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Y.F.G. conceived and planned the experimental project. Z.X.L. grew the crystals with the help from W. X. and Y.P.F. Z.X.L. measured the magnetization assisted by H.S., X.W. and Z.Q.Z. Z.X.L., Z.H.Y. and N.Y. contributed to single crystal xray diffraction chracterizations on the structure and quality of the crystals. Z.X.L. analyzed the data with help from L.M.C. Y.F.G., Z.X.L. and L.M.C. wrote the paper with inputs from all coauthors. All authors discussed the results and commented on the manuscript.
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Li, Z., Xia, W., Su, H. et al. Magnetic critical behavior of the van der Waals Fe_{5}GeTe_{2} crystal with near room temperature ferromagnetism. Sci Rep 10, 15345 (2020). https://doi.org/10.1038/s41598020722033
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DOI: https://doi.org/10.1038/s41598020722033
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