Multiple spin couplings and layer−valley interactions in room-temperature ferromagnetic Fe₃GaTe₂

Abstract

Fe₃GaTe₂ has attracted significant interest due to its intrinsic room-temperature ferromagnetism, yet its magnetic interactions remain debated. We thoroughly investigate the magnetism of Fe₃GaTe₂ using critical analysis, nitrogen−vacancy (NV) center magnetometry, and Density Function Theory (DFT). Our critical phenomenon analysis with exponents [\beta = 0.3706(9), \gamma = 1.32(6), \delta = 4.7(2)] and DFT calculations reveal competition between itinerant and localized spins driven by anisotropic coupling, which can be attributed to a net charge transfer of approximately 0.22 electrons from Fe³⁺ to surrounding Ge/Te atoms. As confirmed by NV center magnetometry, the ferromagnetism in Fe₃GaTe₂ remains robust even in thin-layered sheet of 16 nm (corresponding to approximately 20 layers). The out-of-plane ferromagnetism in thin Fe₃GaTe₂ sheets is stabilized due to the distinct spin interaction energies between intralayers (J_1 ~ 66.74 meV and J_2 ~ 17.33 meV) and interlayers (J_z ~ 3.78 meV). In addition, the constant energy contour profiles near the Fermi surface of Fe₃GaTe₂ suggest the presence of both hole and electron pockets with a distorted contour around the \mathrm{K}/{\mathrm{K}}^{\mathrm{\prime }} point, indicating hexagonal trigonal warping effects. Furthermore, the layer-resolved electronic band structure uncovers a layer−valley coupling near the Fermi surface, with bands at valleys \mathrm{K} and {\mathrm{K}}^{\mathrm{\prime }} associated with different layers. These findings pave way for advanced electronic applications operating above-room-temperature.

Graphical abstract

Keywords

couplings / layer / interactions / above-room-temperature intrinsic ferromagnetism / strong magnetic anisotropy / critical phenomenon analysis / nitrogen−vacancy (NV) center magnetometry / Density Function Theory (DFT)

1 Introduction

Two-dimensional (2D) van der Waals (vdW) materials have garnered extensive interest due to their highly tunable physical properties and significant applications in spintronics. In particular, 2D ferromagnetic (FM) ordering at monolayer level has been a subject of intense study. Examples of such materials include Cr₂Ge₂Te₆, Cr₂Si₂Te₆, Fe₃GeTe₂, and CrI₃, which are known for their ability to sustain long-range ferromagnetism in two dimensions despite challenges posed by thermal fluctuation [1–6]. The presence of strong magnetic anisotropy, which is suggested to overcome the thermal fluctuation, is crucial in stabilizing the low-dimensional magnetic ordering [6]. Although advancements have been made in enhancing the Curie temperature (T_{\mathrm{C}}), such as rising up T_{\mathrm{C}} to 300 K in Fe₃GeTe₂ through liquid ion gating, these materials are still limited in their applicability for next-generation magnetoelectronic devices due to their low spin ordering temperatures and chemical unstability [6, 7]. Therefore, it has become increasingly important to explore intrinsic 2D-layered FM materials with higher T_{\mathrm{C}} and strong magnetic anisotropy for room-temperature operation [1, 8]. Fe₃GaTe₂ is a prominent 2D-vdW FM crystal that exhibits robust ferromagnetism up to 380 K, strong magnetic anisotropy, and significant anomalous Hall angle [9, 10]. These attributes make it highly promising for next-generation magnetoelectronic and spintronic devices [11–15]. However, there are ongoing debates among researchers regarding the nature of its magnetism, which is crucial for its various applications [16–20].

A recent study reported a critical temperature of 358 K for Fe₃GaTe₂ [17], which is significantly higher than previously reported values [16]. This discrepancy raises concerns about the quality of the single crystals used and the reliability of their critical exponents analysis. The enhanced T_{\mathrm{C}} in their study could be due to the presence of a Fe-intercalation layer, which induces local antiferromagnetic (AFM) coupling. This not only increases the critical temperature but also strengthens the AFM interaction. A similar phenomenon has also been observed in the related compound Fe₃GeTe₂ [21]. In a study by Chen et al. [16], it was claimed that the spins in Fe₃GaTe₂ follow a 3D Ising model by investigating the critical behavior up to a magnetic field of 5 T. However, Algaidi et al. [17] suggested a magnetic coupling of a 3D Heisenberg type. This assertion contradicts the established findings that Fe₃GaTe₂ exhibits strong anisotropic exchange interactions [18–20]. In addition, the anisotropic interactions extend beyond nearest neighbors and vary with orientation, the characteristics of which are not accommodated by the isotropic and nearest-neighbor limited assumptions of the 3D-Ising model. The critical behavior analysis restricted to a magnetic field of 5 T may not fully capture the magnetism of Fe₃GaTe₂, especially considering the contributions from itinerant spins. Higher magnetic fields are essential for accurate characterization as they suppress thermal fluctuations, enhance the Zeeman effect, and reduce the non-universal effects. Therefore, to reconcile these findings, further studies under higher magnetic fields are necessary to better elucidate the complex anisotropic spin interactions in Fe₃GaTe₂.

In this work, the magnetism of Fe₃GaTe₂ was studied through various methods such as critical analysis, nitrogen−vacancy (NV) center magnetometry, and Density Function Theory (DFT). We analyzed the critical behavior of Fe₃GaTe₂ under magnetic field up to 9 T, yielding reliable critical exponents such as \beta = 0.3706(9), \gamma = 1.32(6), and \delta = 4.7(2). Our critical phenomenon analysis and DFT calculations revealed multiple spin couplings with highly anisotropic exchange in Fe₃GaTe₂, which uncover competition between itinerant and localized spins driven by anisotropic coupling. This result was suggested to be attributed to a net charge transfer of approximately 0.22 electrons from Fe{}^{3+} to surrounding Ge/Te atoms by the DFT calculations. Using NV center magnetometry, we found that the ferromagnetism in Fe₃GaTe₂ remains robust even in thin-layered sheet with thicknesses of 16 nm corresponding to approximately 20 layers [22]. Theoretically, our calculations reveal the interlayer exchange coupling energies (J_1\sim 66.74 meV and J_2\sim 17.33 meV) and interlayer spin interaction energies (J_z\sim 3.78 meV). The stabilization of the out-of-plane ferromagnetism in Fe₃GaTe₂ is due to the distinct intralayer and interlayer spin interaction energies. This robust perpendicular magnetic anisotropy of Fe₃GaTe₂ is benefit to applications in spintronic devices operating above room temperature. Additionally, the constant energy contour profiles near the Fermi surface unveiled the presence of both hole and electron pockets in Fe₃GaTe₂ with a distorted contour around the \mathrm{K}/{\mathrm{K}}^{\prime } point, indicating hexagonal trigonal warping effects which can significantly impact transport properties, topological characteristics, and optoelectronic properties. Furthermore, our analysis of the layer-resolved band structure and Berry curvature uncovered promising features such as valley−valley coupling, hexagonal trigonal effects, and valley-locked Berry curvature. These findings are highly desirable for enhancing the spintronic functionalities of this material.

2 Results and discussion

The detailed synthesis and characterization of Fe₃GaTe₂ single crystals are included in the Supplementary Material. The Fe₃GaTe₂ has a hexagonal structure, with alternating layers separated by vdW gaps within each unit cell, as illustrated in Fig.1(a). This structure belongs to the P{6}_3/mmc space group. Chemical composition analysis via Energy Dispersive X-ray (EDX) spectrometry given in Fig.1(b) indicates an average composition of Fe{}_{2.98}Ga{}_{1.06}Te{}_{1.96}, closely matching the optimal Fe₃GaTe₂. X-ray diffraction (XRD) of the Fe₃GaTe₂ single crystal [Fig.1(c)] presents a set of (00l) peaks, indicating that the crystal surface is perpendicular to the c-axis and parallel to the ab-plane. The interlayer separation distance along the c-axis is approximately 0.79 nm [22]. The XRD pattern aligns with previous findings [9], affirming the layered nature and high crystalline quality of Fe₃GaTe₂.

The temperature-dependent magnetization [M(T)] of Fe₃GaTe₂ single crystal exhibits distinct behaviors in both in-plane (H//a b) and out-of-plane (H//c) orientations, as shown in Fig.1(d). At room temperature, the M(T) curve for H//ab shows significantly lower magnetization compared to that for H//c. The M(T) curve under zero field cooling (ZFC) for H//c reveals several noteworthy features, including a sudden increase indicative of a paramagnetic to ferromagnetic (PM−FM) transition around 344 K and a kink-like behavior at T_1. The observed kink-like behavior is similar to that in chiral layered magnets, which may stem from the magnetic skyrmions in this system [23–25]. Despite Fe₃GaTe₂ belonging to a centrosymmetric space group that lacks the Dzyaloshinskii−Moriya interaction (DMI), a characteristic associated with chiral spin-texture, large dipolar interactions detected in this material [23–25]. This may potentially stabilize such chiral spin-texture, similar to its sibling compound Fe₃GeTe₂ known for hosting skyrmion bubbles. However, recent studies suggest additional mechanisms for skyrmion formation in Fe₃GaTe₂. For instance, observed Néel-type skyrmions have been attributed to Fe vacancy-induced spatial symmetry breaking, which may introduce local non-centrosymmetry [26]. Another study reports the coexistence of Bloch-type and hybrid Bloch−Néel skyrmions, indicating that both DMI and dipole−dipole interactions could contribute to stabilizing diverse skyrmion types [25]. These findings imply that skyrmion formation in Fe₃GaTe₂ results from a complex interplay of dipolar interactions and structural factors. Variations in experimental conditions, such as temperature and magnetic field, may explain discrepancies between studies. Further work is needed to clarify the impact of these interactions and structural imperfections on skyrmion stability.

The hump-like peak observed at T_2 in the ZFC M(T) curve disappears under field cooling (FC), indicating variable magnetic states due to non-unidirectional alignment of magnetic moments between monolayers. A transition around T_3\approx 100 K is identified as the spin reorientation transition (T_{\mathrm{S}\mathrm{R}}) [27], which influences structural parameters and magnetic anisotropy energy through spin-lattice relaxation. The temperature derivatives of the ZFC (\mathrm{d}M/\mathrm{d}T@ZFC) and FC (\mathrm{d}M/\mathrm{d}T@FC) curves, shown in Fig.1(e), mirror the characteristics of the M(T) curves. The inset of Fig.1(e) shows the field-dependent magnetization [M(H)] curve at 300 K. At this temperature, the saturation magnetization (M_{\mathrm{S}}) reaches approximately 40 emu/g at 300 K, which closely aligns with the values reported for room temperature [9].

The low-temperature M(T) profile of Fe₃GaTe₂ is similar to that of Fe{}_{3-x}GeTe₂ and Fe{}_{5-x}GeTe₂, indicating a potential ferrimagnetic ground state with a change in magnetic anisotropy energy (MAE) from out-of-plane to in-plane [28]. Different magnetic states including non-magnetic, FM, and four AFM configurations are explored by the DFT calculations, as shown in Fig. S1 in the Supplementary Material. The results demonstrate an FM ground state for bulk Fe₃GaTe₂. The experimental magnetic moments of approximately m_{{\mathrm{F}\mathrm{e}}_{\mathrm{I}}}\sim 2.53 {\mu }_{\mathrm{B}} for {\mathrm{F}\mathrm{e}}_{\mathrm{I}} and m_{{\mathrm{F}\mathrm{e}}_{\mathrm{I}\mathrm{I}}}\sim 1.49 {\mu }_{\mathrm{B}} for {\mathrm{F}\mathrm{e}}_{\mathrm{I}\mathrm{I}} mathch the theory, suggesting an additional valence electron at the {\mathrm{F}\mathrm{e}}_{\mathrm{I}\mathrm{I}} site. The interlayer exchange coupling (J_z) of ~ 3.78 meV indicates interlayer ferromagnetism (see Supplementary Material). Furthermore, the non-collinear calculations using the DFT+U method generate MAE shown in Fig.1(f), indicating an out-of-plane preference of −3.42 meV per unit cell. The MAE of Fe₃GaTe₂ is significantly greater than that of Cr₂Ge₂Te₆ (0.5 meV/f.u.) or Fe₃GeTe₂ (1.5–2 meV/f.u.), highlighting its substantial potential for high-density data storage applications.

The PM−FM magnetic transition in Fe₃GaTe₂ is investigated by measuring magnetization isotherms around T_{\mathrm{C}} up to 9 T, as shown in Fig.1(g). As plotted in Fig.1(h), we construct the Arrott plot using assumed critical exponents \beta = 0.5 and \gamma = 1 from mean-field theory. However, we observe nonlinear behavior and downward curvature in each curve, indicating the invalidation of the traditional Landau mean-field model. The positive slope of the curve confirms a second-order PM−FM phase transition [29, 30]. We explore alternative models including 3D Heisenberg (\beta = 0.365, \gamma = 1.386), 3D-XY (\beta = 0.345, \gamma = 1.316), and 3D Ising (\beta = 0.325, \gamma = 1.24) [30–32], as shown in Figs. S2(a)−(c). The Normalized Slope (NS) approach is used to determine which model best describes the critical behavior of Fe₃GaTe₂. The NS values for the 3D Heisenberg and 3D-XY models, depicted in Fig.2(a), closely remain around “1”, suggesting that Fe₃GaTe₂ exhibits spin couplings with characteristics of these models.

The magnetization analysis was refined using the modified Arrott plot (MAP) method [33]. To determine critical exponents and critical temperature T_{\mathrm{C}}, the Arrott−Noakes equation of state (H/M{)}^{1/\gamma }=A\left[(T-T_{\mathrm{C}})/T_{\mathrm{C}}\right]+B{M}^{1/\beta } is adopted, as shown in Fig.2(b). Equations (S2) and (S3) are employed to refine the critical exponents \beta and \gamma. Initial values of spontaneous magnetization [M_{\mathrm{S}}(T)] and inverse initial susceptibility [{\chi }_0^{-1}(T)] are derived as plotted in Fig.2(c). The fitting yields critical exponents \beta = 0.3706 and \gamma = 1.326. As shown in Fig.2(b), all lines align parallel and that at T_{\mathrm{C}} = 344 K intersects the origin in the high-field region. Finally, critical exponents \beta = 0.3706(9) with T_{\mathrm{C}} = 344.1(1) K and \gamma = 1.32(6) with T_{\mathrm{C}} = 343.7(9) K are determined. Fig.2(d) illustrates the isothermal magnetization M(H) at critical temperature T_{\mathrm{C}} = 344 K, with the inset shown on a log−log scale. A third critical exponent \delta = 4.7(2) is estimated according to Eq. (S4). These obtained critical exponents are examined using the Widom scaling relation \delta =1+\frac{\gamma }{\beta }, which yields \delta = 4.57(7) based on MAP values. These results confirm the reliability and self-consistency of the obtained critical exponents. A detailed description of the fitting procedure is referred to the Supplementary Material.

A scaling analysis verifies the accuracy of the calculated critical exponents and T_{\mathrm{C}} for Fe₃GaTe₂. The renormalized magnetization is defined as m\equiv {\epsilon }^{-\beta }M(H,\epsilon), while the renormalized field is defined as h\equiv H{\epsilon }^{-(\beta +\gamma )}. As shown in Fig.2(e), scaled m versus scaled h is plotted according to Eq. (S6), with a log−log representation in the inset. The data bifurcated into distinct branches below and above T_{\mathrm{C}}, indicating correct renormalization in the critical regime. Tab.1 summarizes the critical exponents for Fe₃GaTe₂, along with related materials and theoretical predictions for comparison. The critical exponents of Fe₃GaTe₂ do not precisely align with any single theoretical model. However, \beta is close to that of the 3D Heisenberg model, and \gamma approaches that of the 3D-XY model. Deviations from theoretical values could stem from significant interlayer couplings or extended exchange interactions due to anisotropic exchange interactions involving itinerant electrons [35, 36].

In homogeneous magnets, the exchange distance of spin is expressed as J(r)\approx r^{-(d+\sigma )}, where d is spatial-dimensionality and \sigma represents the range of interaction [29]. This exchange distance of spin determines the universality class of the magnetic phase transition. The range of the spin interaction can be classified as either long or short based on \sigma: indicates a long-range interaction, while \sigma >2 suggests a short-range one. Furthermore, the range of interaction predicts the susceptibility exponent \gamma [37]:

(1)\begin{array}{rl}\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}\\ & \times \left[1+\frac{2G\left({\displaystyle \frac{d}{2}}\right)(7n+20)}{(n-4)(n+8)}\right]\mathrm{\Delta }{\sigma }^2,\end{array}

where \mathrm{\Delta }\sigma = \sigma -\frac{d}{2}, G\left(\frac{d}{2}\right)=3-\frac{1}{4}{\left(\frac{d}{2}\right)}^2 and n is the spin-dimensionality. Fitting to Eq. (1), we determined the \sigma, d, and n in Fe₃GaTe₂. By adjusting the parameters to achieve \gamma = 1.32(6), we derived other exponents, namely \beta = 0.37(4) and \delta = 4.52(9), which closely match the experimental values. This analysis suggests that the spin-dimensionality of Fe₃GaTe₂ cannot be simply treated as n=2 (3D-XY) or n=3 (3D Heisenberg). The exponent \gamma indicates 3D-XY spin interactions (d:n = 3:2) with an extended long-range interaction [\sigma = 1.92(9)]. On the other hand, the calculated \beta =0.37(4) aligns closely with the 3D Heisenberg model (d:n = 3:3), suggest a short-range interaction. These results imply a combination of localized and itinerant spins contributing to the magnetism of Fe₃GaTe₂. Therefore, the critical behaviors of Fe₃GaTe₂ confirm a complex magnetism with highly anisotropy that arise from the interplays between itinerant and local moments [20, 38].

As is known, 3D-XY type spin interactions exhibit strong anisotropic spin exchange primarily within the plane. This means that when Fe₃GaTe₂ is thinned to a few layers, the interlayer interaction in the out-of-plane direction has minimal impact on magnetism, while the strong in-plane interaction stabilizes magnetism in thin sheets. Therefore, we examined the magnetism of thin-layered Fe₃GaTe₂ sheet using NV center magnetometry, as illustrated in Fig.3(a). The magnetic properties under B_{\mathrm{a}\mathrm{p}} //c and B_{\mathrm{a}\mathrm{p}} //ab orientations are quantitatively imaged with sub-micron resolution, as shown in Fig.3(b). Fig.3(c) depicts a thin-layered Fe₃GaTe₂ configuration achieved by reducing the sample thickness to approximately 16 nm, corresponding to approximately 20 layers (0.79 nm between layers) [22]. Details regarding the methodology used and the orientations of the magnetic field are included in the Supplementary Material. This technique provides enhanced precision in measuring both the magnetic strength and orientation in a small sample. As illustrated in Fig.3(b) and (d), an electromagnet was used to induce magnetic fields both parallel (B_{\mathrm{a}\mathrm{p}}//ab) and perpendicular (B_{\mathrm{a}\mathrm{p}}//c) to the Fe₃GaTe₂ flake for magnetic imaging. Various magnetic fields B_{\mathrm{a}\mathrm{p}} were incrementally applied, as presented in Fig.3(d). The magnetic field detected by the NV centers, {\mathit{B}}_{\mathrm{N}\mathrm{V}}, is the sum of the applied field {\mathit{B}}_{\mathrm{a}\mathrm{p}} and the stray field {\mathit{B}}_{\mathrm{s}}, \left|B_{\text{ap}}+B_{\mathrm{s}}\right| generated by the magnetization of the sample. The stray field {\mathit{B}}_{\mathrm{s}}, indicative of the magnetization of the sample, is evaluated. When B_{\mathrm{a}\mathrm{p}}//c, a blue shading around the sample indicates a magnetized state opposing {\mathit{B}}_{\mathrm{a}\mathrm{p}}, which intensifies with increasing field strength, suggesting out-of-plane easy magnetization in thin-layered Fe₃GaTe₂ sheet [upper part of Fig.3(d)]. In contrast, for B_{\mathrm{a}\mathrm{p}}//ab, the absence of shading indicates weaker magnetization [lower parts of Fig.3(d)]. Two selected spots (red and purple) on the magnetic image show curves of their stray fields in relation to the applied magnetic field, highlighting that H//c is more easily magnetized than H//ab. Marfoua etal. [22] have systematically investigated the temperature-dependent magnetic properties of 2D tri-layered and four-layered Fe₃GaTe₂ based on first principles calculations. They found that both tri-layered and four-layered systems exhibit perpendicular magnetic anisotropy, supporting the conclusions of our present work. Our studies suggest that the magnetic interactions in both bulk and thin-layered Fe₃GaTe₂ sheets are primarily governed by intrinsic out-of-plane anisotropy, which remains robust even when the material is thinned to a few layers. The easy magnetization along the c-axis, crucial for the observed magnetic behavior, is not significantly altered by the reduction in thickness [22]. This indicates that the magnetic interactions, including the nature of the anisotropy, do not drastically change between the bulk and few-layered forms of Fe₃GaTe₂.

Our first-principle spin-polarized DFT simulation reveals that Fe₃GaTe₂ exhibits a stable AFM state, as shown in Fig.4(a) and (b). Sizable magnetic moments on the Fe atoms (m_{{\mathrm{F}\mathrm{e}}_{\mathrm{I}}} ~ 2.53 {\mu }_{\mathrm{B}} for {\mathrm{F}\mathrm{e}}_{\mathrm{I}} and m_{{\mathrm{F}\mathrm{e}}_{\mathrm{I}\mathrm{I}}} ~ 1.49 {\mu }_{\mathrm{B}} for {\mathrm{F}\mathrm{e}}_{\mathrm{I}\mathrm{I}}) are observed, indicating a local moment nature. However, the non-magnetic density of states shown in Fig.4(c) suggests instability at the Fermi surface, indicating a phase transition to a stable magnetic state. Additionally, charge calculations indicate a net charge transfer of approximately 0.22 electrons from each {\mathrm{F}\mathrm{e}}_{\mathrm{I}} ion to the surrounding Te atoms, suggesting itinerant behavior in Fe₃GaTe₂. Although slight Fe vacancies and excess Te could influence the local charge distribution, the insights derived from our DFT calculations provide a strong baseline for understanding the itinerant magnetism of this material. Any effects arising from structural imperfections can be regarded as perturbations to this underlying behavior. This dual itinerant and local moment nature is similar to that observed in Fe₃GeTe₂ [39], whereas Fe₃GaTe₂ exhibits stronger local behavior compared to Fe₃GeTe₂. This difference may contribute to Fe₃GaTe₂ maintaining magnetism at high temperatures. Our critical analysis also reveals strong anisotropic spin couplings in Fe₃GaTe₂, predominantly in the ab-plane, which indicates 2D spin interactions. Evaluating the interlayer exchange coupling, we estimate J_z to be approximate 3.78 meV, suggesting an interlayer ferromagnetic ground state. In terms of intralayer near-neighbor exchange couplings, i.e., the obtained J_1\sim 66.74 and J_2\sim 17.33 meV, both of them are significantly stronger than J_z\sim 3.78 meV (see the Supplementary Material) [39]. These results strongly support our findings of a 3D-XY type magnetic coupling, confirming the presence of a nearly 2D FM state in bulk Fe₃GaTe₂.

The observed out-of-plane anisotropy in the thin-layered Fe₃GaTe₂ sheet aligns cohesively with that in its bulk form, which is critical for a wide range of significantly anisotropic out-of-plane ferromagnets that could be employed in vdW heterostructures and spintronics devices. Recent studies further emphasize the potential of Fe₃GaTe₂ for high-density data storage and spintronics applications. In particular, Fe₃GaTe₂, a room-temperature ferromagnet, has been shown to host topologically protected magnetic skyrmions, which can be engineered using magnetic force microscopy. These skyrmions, characterized by distinct topological charges, can be controlled through field-cooling processes and tip stray fields, making Fe₃GaTe₂ a strong candidate for future spintronic devices and data storage technologies [40, 41].

On the other hand, Fig.4(d) shows the layer-resolved electronic band structure of Fe₃GaTe₂, which reveals a layer−valley coupling near the Fermi surface with bands at valleys \mathrm{K} and {\mathrm{K}}^{\prime } belonging to different layers. This promising characteristic in Fe₃GaTe₂ can be utilized to explore various phenomena related to layer degrees of freedom, such as the layer Hall effect and other valley-related phenomena [39, 42–45]. As shown in Fig.4(d), the absence of a global band gap confirms that Fe₃GaTe₂ is metallic. Further analysis of the spin-resolved band structure indicates a predominance of spin-up bands crossing the Fermi level and the \mathrm{K}/{\mathrm{K}}^{\prime } pseudo-gap, suggesting a highly spin-polarized Fermi surface (refer to the Supplementary Materials for more details). This, combined with the out-of-plane MAE, highlights the potential of Fe₃GaTe₂ for spin manipulation applications. Additionally, near the Fermi surface, constant energy contour profiles exhibit both hole and electron pockets and a distorted contour around the \mathrm{K}/{\mathrm{K}}^{\prime } point, indicating the presence of hexagonal trigonal warping effects, as depicted in Fig.4(g) and (h). This warping, which distorts the hexagonal shape of the electronic bands, can have an impact on transport properties, topological hall marks, and optoelectronic properties. By utilizing the maximally localized Wannier function approach, calculations show that the anomalous hall conductivity (AHC) displays a plateau-like behavior within the \mathrm{K}/{\mathrm{K}}^{\prime } pseudo-gap, suggesting the presence of topological characteristics, as shown in Fig.4(e). The contributions to the Berry curvature primarily come from the \mathrm{K} and {\mathrm{K}}^{\prime } valley points, as depicted in Fig.4(f). An additional peak in the Berry curvature near the Q point in the Brillouin zone influences the AHC, causing it to switch from positive to negative below the Fermi level. However, the AHC is predominantly influenced by contributions from the \mathrm{K}/{\mathrm{K}}^{\prime } valleys, indicating potential topological features in Fe₃GaTe₂ when the Fermi level resides within the pseudo-gap.

3 Conclusions

In conclusion, the magnetism of Fe₃GaTe₂ single crystals has been thoroughly studied through critical exponent analysis, NV center magnetometry, and DFT calculations. Analysis of critical behavior at the PM−FM transition reveals asymptotic exponents \beta = 0.3706(9), \gamma = 1.32(6), and \delta = 4.7(2) for H//c, which indicates pronounced out-of-plane MAE and the coexistence of itinerant and localized spins. The DFT calculations suggest that the itinerant magnetic behavior observed can be attributed to a net charge transfer of approximately 0.22 electrons from Fe³⁺ to surrounding Ge/Te atoms. Remarkably, as confirmed by NV center magnetometry, the ferromagnetism in Fe₃GaTe₂ remains robust even in thin-layered sheet of 16 nm (~ 20 layers), suggesting potential applications in spintronics devices operating above room temperature. The stability of the out-of-plane FM state in thin Fe₃GaTe₂ sheets is attributed to the anisotropic exchange interactions, i.e., the distinct intralayer spin interaction energy (J_1\sim 66.74 meV and J_2\sim 17.33 meV) and interlayer spin interaction energy (J_z\sim 3.78 meV). Additionally, the constant energy contour profiles near the Fermi surface of Fe₃GaTe₂ uncover the presence of both hole and electron pockets with a distorted contour around the \mathrm{K}/{\mathrm{K}}^{\prime } point, indicating hexagonal trigonal warping effects. This warping can significantly impact transport properties, topological characteristics, and optoelectronic properties. Furthermore, the layer-resolved electronic band structure demonstrates layer−valley coupling near the Fermi surface, with bands at valleys \mathrm{K} and {\mathrm{K}}^{\prime } associated with different layers. This functionality can be utilized to explore various phenomena, such as the layer Hall effect and other valley-related properties, thereby enhancing spintronics and optoelectronic applications. This study provides crucial insights into the unique properties of layered Fe₃GaTe₂, paving the way for advanced electronic applications.

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