Magnetic critical behavior of the van der Waals Fe₅GeTe₂ crystal with near room temperature ferromagnetism

Abstract

The van der Waals ferromagnet Fe₅GeTe₂ 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 Fe₅GeTe₂ 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 Fe₅GeTe₂ could be interpreted by a three-dimensional magnetic exchange with the exchange distance decaying as J(r) ≈ r^−4.916, close to that of a three-dimensional Heisenberg model with long-range magnetic coupling.

Introduction

A prominent virtue of the quasi-two-dimensional (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^1–6, etc. Known as the Merin-Wagner theorem⁷, intrinsic long-range 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 long-range 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 long-range ferromagnetic (FM) order in mono- or few-layer CrI₃⁸, Cr₂Ge₂Te₆⁹, Cr₂Si₂Te₆¹⁰, VSe₂¹¹, and MnSe₂¹², etc. The vdW magnets in the 2D limit host rich magneto-electrical, magneto-optical, or spin–lattice coupling effects that are capable of producing intriguing properties which are scarcely observed in bulk. Very recently, current-induced magnetic switch was observed in the few-layer Fe₃GeTe₂¹³, 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 spin-filter magnetic vdW CrI₃ heterostructures even approaches 1.9 × 10⁴%, remarkably superior to that constructed by using conventional magnetic thin films¹⁶. 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 CrI₃ is an Ising ferromagnet with very low Curie temperature (T_C) of about 45 K due to the weak superexchange interaction along the Cr-I-Cr pathway^8,17. The similar weak FM superexchange in the Heisenberg magnet bilayer Cr₂Ge₂Te₃ also results in a low T_C of ~ 30 K, and FM order is even not present in the monolayer⁹. As a contrast, the FM exchange with an itinerant character mediated by carriers in metallic Fe₃GeTe₂ monolayer is much stronger than the superexchange in CrI₃ and Cr₂Ge₂Te₆, 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 quasi-2D vdW Fe_3-xGeTe₂, which can even be enhanced up to room temperature¹⁸. Interestingly, similar as Fe_3-xGeTe₂, bulk Fe_5-xGeTe₂ 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 first-order transition near 100 K was detected by neutron powder diffraction²⁰. Considering the exotic physical properties in exfoliated Fe₃GeTe₂ nanoflakes and its heterostructures, such as the extremely large anomalous Hall effect²², planar topological Hall effect²³, Kondo lattice physics²⁴, anisotropic magnetostriction effect²⁵, spin filtered tunneling effect¹⁶, magnetic skyrmions²⁶, etc., Fe₅GeTe₂ would also be expected to provide extraordinary opportunities to explore intriguing physical properties. To well understand the physical properties of Fe₅GeTe₂, 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 CrI₃²⁷, VI₃²⁸, Fe₃GeTe₂^29,30, Co₂TiSe³¹, and Fe_0.26TaS₂³², etc. In this work, we have reported the investigation on the critical behavior of Fe₅GeTe₂, which finds that the obtained set of critical exponents are close to those calculated from the renormalization group approach for a long-range 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 Fe₅GeTe₂ measured with zero-field-cooling (ZFC) and field-cooling (FC) mode under the applied magnetic field H = 1 kOe along the ab-plane 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 Fe²⁺ with the values raging from 4.90 to 6.70 μ_B in various materials including sphalerite and monoclinic pyroxenes obtained from magnetic susceptibility analysis³³ 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 Fe₅GeTe₂, which is similar as that of Fe₃GeTe₂^29,30. The saturation magnetic moment along the c-axis 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//c-axis were shown in Fig. 1c and the Arrott plot³⁴, that is, M² vs. H//M, is shown in Fig. 1d. The positive slope of all M² vs. H/M curves, according to the Banerjee’s criterion³⁵, indicates that the PM to FM transition has a second-order in nature. The Arrott plot was initially tried to for the analysis of the measured magnetizations, so the mean Landau mean-field theory with the critical exponents β = 0.5 and γ = 1.0 is involved. If it works, the M² 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 M² vs. H//M curves are clearly nonlinear with a downward curvature, suggesting that the fit does not work for Fe₅GeTe₂. The failure of the Arrott plot within the framework of Landau mean-field theory lies in that the itinerant ferromagnetism in Fe₅GeTe₂ should have significant electronic correlations and spin fluctuations, which however are neglected in the Landau mean-field theory.

Figure 1.

(a) Temperature dependence of magnetization M(T) for Fe₅GeTe₂ under H = 1 kOe. The inset shows the inverse susceptibility plotted against temperature and the straight dotted line is Curie–Weiss law fitting. (b) Isothermal magnetization M(H) measured at 2 K. (c) Typical initial magnetization M(H) curves measured from 261 to 285 K with an interval of 1 K. (d) Arrott plots in the form of M² vs. H/M (mean field model) around T_C.

The second-order PM to FM phase transition in Fe₅GeTe₂ 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 second-order phase transition, the spontaneous magnetization M_S(T) below T_C, the inverse initial susceptibility χ₀^–1(T) above T_C and the magnetization M at T_C can be used to obtain β, γ and δ by using the equations³⁶:

$$M_S\left(T\right)=M_0{\left(-,\epsilon \right)}^{\beta },\epsilon <0,T<T_C,$$ 1

$${\chi }_0^{-1}\left(T\right)=\left(h_0,/,m_0\right){\epsilon }^{\gamma },\epsilon >0,T>T_C,$$ 2

$$\mathrm{and}M=D{H}^{1/\delta },\epsilon =0,T=T_C,$$ 3

where ε = (T—T_C)/T_C is the reduced temperature, and M₀, h₀/m₀, and D are the critical amplitudes. Though the Landau mean-field theory can not be used, the critical isothermal magnetizations, alternatively, can be analyzed with the Arrott-Noakes equation of state³⁷:

$${(H/M)}^{1/\gamma }=a\mathrm{\epsilon }+b{M}^{1/\beta },$$ 4

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 mean-field 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 mean-field 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 Fe₅GeTe₂ 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 Fe₅GeTe₂ may not belong to a single universality class. The fact also likely indicates that the magnetic character of Fe₅GeTe₂ is nearly isotropic above T_C and the enhancement of the anisotropic exchange below T_C.

Figure 2.

The isotherms of M^1/β versus (H/M)^1/γ with (a) 3D Heisenberg model, (b) 3D Ising model, (c) 3D XY model, (d) Tricritical mean-field model and (e) 2D Ising model. (f) Normalized slope versus temperature curves for six sets of critical exponents.

To achieve in-depth insights into the nature of the PM to FM transition in Fe₅GeTe₂, 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 χ₀^–1(T), respectively. The extracted M_S(T) and χ₀^–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 χ₀^–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 χ₀^–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.

Figure 3.

Modified Arrott plot of isotherms with β = 0.351(1) and γ = 1.413(5) for Fe₅GeTe₂.

Figure 4.

(a) Temperature dependence of the spontaneous magnetization M_S (left) and the inverse initial susceptibility ${\chi }_0^{-1}\left(T\right)$ (right) with solid fitting curves for Fe₅GeTe₂. (b) Kouvel-Fisher plots of M_S(T)/(dM_S(T)/dT) (left) and χ₀^–1(T)/(dχ₀^–1(T)/dT) (right) with solid fitting curves for Fe₅GeTe₂. (c) Isotherm M(H) collected at T_C = 274 K for Fe₅GeTe₂. Inset: the same plot in log–log scale with a solid fitting curve.

It is necessary to check the accuracy of above analysis. The Kouvel-Fisher (K-F) method can also be employed to fit the critical exponents and critical temperature, which is expressed as⁴⁰:

$$\frac{M_S(T)}{d{M}_S(T)/dT}=\frac{T-T_C}{\beta }$$ 5

$$\mathrm{and}\frac{{\chi }_0^{-1}\left(T\right)}{d{\chi }_0^{-1}\left(T\right)/dT}=\frac{T-T_C}{\gamma },$$ 6

where M_S(T)/(dM_S(T)/dT) and χ₀^–1(T)/(dχ₀^–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 relation⁴¹:

$$\delta =1+\frac{\gamma }{\beta },$$ 7

which gives δ = 5.02(6) and δ = 4.94(0) by using the β and γ obtained with modified Arrott plot and Kouvel-Fisher 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 Fe₅GeTe₂, 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 as⁴²:

$$M\left(H,,,\epsilon \right)={\epsilon }^{\beta }{f}_{\pm }(\frac{H}{{\epsilon }^{\beta +\gamma }})$$ 8

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:

$$m=f_{\pm }(h),$$ 9

where $m\equiv {\epsilon }^{-\beta }M(H,\epsilon )$ and $h\equiv {\epsilon }^{-(\beta +\gamma )}$. 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 m² 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 is⁴²:

$$\frac{H}{M^{\delta }}=h\left(\frac{\epsilon }{H^{1/\beta }}\right),$$ 10

where h(x) is a scaling function. From Eq. (10), the εH^−(βδ) vs. MH^−1/δshould fall on one universal curve⁴³, 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.

Figure 5.

(a) The $m\equiv {\epsilon }^{-\beta }M(H,\epsilon )$ as a function of the $h\equiv {\epsilon }^{-(\beta +\gamma )}$ below and above T_C for Fe₅GeTe₂. Inset is the same m(h) data in log–log scale. (b) Plot in the form of m²(h/m) for Fe₅GeTe₂. Inset shows the plot of εH^−(βδ) vs. MH^−1/δ below and above T_C.

It is valuable to compare the critical exponents of Fe₅GeTe₂ with those of other layered vdW magnets and those predicted by various models. The critical exponents of Fe₅GeTe₂ 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_3-xGeTe₂ (x = 0, 0.15, and 0.36), Cr₂Si₂Te₆, and Cr₂Ge₂Te₆. The previous comprehensive study reached a conclusion that the critical exponent β for a 2D magnets lies in the range of ~ 0.1 ≤ β ≤ 0.25⁴⁴. It is apparent that the β values of Cr₂Si₂Te₆ and Cr₂Ge₂Te₆, which were verified as 2D Ising magnets^45,46, are actually within the window, while those of Fe_3-xGeTe₂ and Fe₅GeTe₂ are apparently larger than 0.25, thus excluding the 2D Ising model for them^29,30. Moreover, the γ values of Fe_3-xGeTe₂ and Fe₅GeTe₂ are much larger than those for the tricritical mean-field and 3D Ising models^38,39, suggesting the two models are not appropriate. Combining the β and γ values, the magnetic critical behavior in Fe₅GeTe₂ should have a 3D nature, indicating that the interlayer magnetic exchange can not be neglected. It was suggested that Fe_3-xGeTe₂ has a smaller vdW gap and hence a stronger interlayer magnetic exchange than that in Cr₂(Si,Ge)₂Te₆¹⁷. It is therefore a natural hypothesis that the vdW gap in Fe₅GeTe₂ is also very small. To achieve more insights, the critical exponents of Fe₅GeTe₂ should be compared with the several 3D models more carefully. The β of Fe₅GeTe₂ is much closer to that of the 3D XY model³⁹ while the γ is closer to that of the 3D Heisenberg model³⁸, likely implying that the obtained critical exponents of Fe₅GeTe₂ can not be simply categorized into any conventional universality classes.

Table 1.

A summary of the critical exponents of Fe₅GeTe₂, Fe_3-xGeTe₂, Cr₂Si₂Te₆, Cr₂Ge₂Te₆ and those predicted by different models (MAP: Modified Arrott plot; KF: Kouvel-Fisher method; CI: critical isotherm analysis).

Composition	References	Technique	β	γ	δ	{d:n}	J(r)
Fe5GeTe2	This work	MAP	0.351 (1)	1.413 (5)	5.02 (6)	{3:3}	r–4.916
	This work	KF	0.346 (4)	1.364 (9)	4.94 (0)
	This work	CI			5.02 (1)
3D Heisenberg	4	Theory	0.365	1.386	4.8
3D XY	4	Theory	0.345	1.316	4.81
3D Ising	4	Theory	0.325	1.24	4.82
Tricritical mean field	5	Theory	0.25	1.0	5
Mean field	4	Theory	0.5	1.0	3
Fe2.64Ge0.87Te2	12	KF	0.372 (4)	1.265 (1)	4.401 (6)	{3:3}	r–4.89
Fe2.85GeTe2	13	KF	0.363	1.228	4.398		r–4.8
Fe3GeTe2	3	KF	0.322 (4)	1.063 (8)	4.301 (6)		r–4.6
Cr2Si2Te6	14	KF	0.175 (9)	1.562 (9)	9.925 (5)	{2:1}	r–3.63
Cr2Ge2Te6	15	KF	0.200 (3)	1.28 (3)	7.405	{2:1}	r–3.52

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 short-range magnetic exchange and J(r) ~ r^–(d+σ) for the long-range 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 as⁴⁷:

$$\gamma =1+\frac{4}{d}\left(\frac{n+2}{n+8}\right)\mathrm{\Delta }\sigma +\frac{8(n+2)(n-4)}{d^2{(n+8)}^2}\left[1,+,\frac{2G(\frac{d}{2})(7n+20)}{(n-4)(n+8)}\right]{\mathrm{\Delta }\sigma }^2,$$ 11

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 long-range 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 short-range magnetic exchange, while when σ ≤ 3/2, the mean-field model works and J(r) decreases slower than r^-4.547,48. To obtain the values of d, n, and σ for Fe₅GeTe₂, 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 long-range interaction decaying as J(r) ≈ r^–4.916 can account for the magnetic properties of Fe₅GeTe₂, which is consistent with our analysis presented above.

Table 2.

Critical exponents calculated by the renormalization group theory.

d	n	σ	β	γ	δ
3	3	1.9160	0.3851	1.3613	4.5351
2	1	1.3603	0.3168	1.370	5.3241
2	3	1.2740	0.3904	1.370	4.5096

The magnetic exchange in quasi-2D vdW magnets has been subjected to immense investigations. For Cr₂Si₂Te₆, 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 Cr₂Ge₂Te₆, its critical behavior shows a transition from the 2D Ising-type to a 3D tricritical mean-field type⁵⁰. It is useful to compare the magnetic critical behavior of Fe₅GeTe₂ with that of Fe₃GeTe₂. The mean distance between the two adjacent Te layers that across the vdW gap in Fe₃GeTe₂ is 0.423 nm⁵¹, which is rather close to that of CrGeTe₃, 0.377 nm⁴⁹, which presumably can account for the 3D Heisenberg characteristics of the critical behavior. Previous studies on Fe₅GeTe₂ indicate small magnetic anisotropy at high temperature²⁰, so the 3D magnetism for the critical behavior in Fe₅GeTe₂ is reasonable. Moreover, it is found that the magnetic anisotropy in Fe_3-xGeTe₂ strongly depends on the Fe deficiency⁵², which can be largely suppressed with increasing the deficiency content x. If we pay a close attention to the critical exponents of Fe₅GeTe₂, it is easily found that they are much closer to those of Fe deficient Fe_3-xGeTe₂²⁹, likely further demonstrating the weak magnetic anisotropy in Fe₅GeTe₂. 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 NiPS₃ and an anisotropic 2D to 3D magnetism below T_C in MnPS₃ were experimentally confirmed^53,54. Though such possibility has not been examined yet in Fe_3-xGeTe₂, considering that Fe_3-xGeTe₂ indeed shares similarities as MPS₃ (M = Mn, Fe, and Ni) in that they all have 2D antiferromagnetic ground state with the ferromagnetic layers in them order antiferromagnetically along the c-axis at low temperature, as well as the 3D critical behavior near T_C, the critical phase transition definitely need to be checked in Fe_3-xGeTe₂. For Fe₅GeTe₂, it is somewhat different from MPS₃ and Fe_3-xGeTe₂, which behaves as an easy-axis vdW ferromagnet with the magnetic moments preferring to align along the c-axis 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 Fe₅GeTe₂ 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_C²¹. Short-range 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 quasi-2D van der Waals ferromagnet Fe₅GeTe₂ 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 long-range exchange. The critical phenomena indicate weak magnetic anisotropy of Fe₅GeTe₂ at high temperature, possibly due to its small vdW gap. The very recent calculations indicate that monolayer formation energy of Fe₅GeTe₂ lies inside the energy range of other 2D materials⁵⁵, 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 Fe₅GeTe₂ would find extraordinary opportunities for applications in next-generation 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 X-ray 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 (MPMS-3) with the magnetic field applied parallel to c-axis 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 size⁵⁷.

Author contributions

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 x-ray 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.

Competing interests

The authors declare no competing interests.

Supplementary information

is available for this paper at 10.1038/s41598-020-72203-3.