Large anomalous transverse transport properties in atomically thin 2D Fe₃GaTe₂

Abstract

Anomalous transverse conductivities, such as anomalous Hall conductivity (AHC), anomalous Nernst conductivity (ANC), and anomalous thermal Hall conductivity (ATHC), play a crucial role in the emerging field of spintronics. Motivated by the recent fabrication of two-dimensional (2D) ferromagnetic thin film Fe₃GaTe₂, we investigate the thickness-dependent anomalous transverse conductivities of the 2D Fe₃GaTe₂ system (from one to four layers). The atomically ultrathin 2D Fe₃GaTe₂ system shows above-room-temperature ferromagnetism with a large perpendicular magnetic anisotropy energy. Furthermore, we obtain a large AHC of −485 S/cm in the four-layer thickness, and this is further enhanced to −550 S/cm with small electron doping. This AHC is seven times larger than the measured AHC in thicker 2D Fe₃GaTe₂ (178 nm). The ANC also reaches 0.55 A/K.m in the four-layer structure. Along with these, the four-layer system exhibits a large ATHC (−0.105 ~ −0.135 W/K.m). This ATHC is comparable to the large ATHC found in Weyl semimetal Co₃Sn₂S₂. Based on our results, the atomically ultrathin 2D Fe₃GaTe₂ system shows outstanding anomalous transverse conductivities and can be utilized as a potential platform for future spintronics and spin caloritronic device applications.

Introduction

Owing to their configurations and exceptional features for basic research and prospective industrial applications, two-dimensional (2D) materials have attracted extensive research interest and have triggered the curiosity of many researchers. To date, the fabrication of 2D magnetic materials are possible using advanced atomic manipulation techniques. Therefore, 2D ferromagnets and their magnetic ground states have caused intriguing issues: fabrication of atomically thin 2D magnetic materials and comprehension of magnetism in their 2D limit^1,2,3. For example, layer-dependent ferromagnetism with out-of-plane anisotropy was found in CrI₃ and Cr₂Ge₂Te₆ flakes with semiconducting behavior^4,5. However, these systems have low critical temperatures (<45 K). On the other hand, 2D VSe₂ was successfully grown from a bulk crystal with a 1 T phase^6,7. Although the stability and phase issues of 2D VSe₂ are still under debate, room-temperature ferromagnetism was observed in the 1T-VSe₂ monolayer. Moreover, the 2D Fe₃GeTe₂ monolayer was also successfully synthesized with a critical temperature of ~130 K⁸. Notably, an ionic gating approach was used to create gate-tunable room-temperature ferromagnetism in four-layer Fe₃GeTe₂ flakes⁹. Therefore, 2D ferromagnet materials have potential in various multifunctional devices for potential spintronics applications, especially at room temperature.

In this regard, the exploration of 2D magnetic materials with a high Curie temperature (T_C) has been ongoing. For instance, based on the success of the 2D Fe₃GeTe₂, the 2D ferromagnetic crystal Fe₃GaTe₂ was also recently fabricated in thickness ranges from bulk down to 9.5 nm (~14 layer thickness). This 2D Fe₃GaTe₂ system showed a T_C of 350 ~ 380 K with large perpendicular magnetic anisotropy. This T_C was potentially record-high for known intrinsic 2D vdW ferromagnetic crystals¹⁰. However, anomalous Hall measurements and direct magnetic domain imaging were performed in the Fe₃GaTe₂ nanosheet thickness (178 nm) and not in the ultrathin film structure. The maximum anomalous Hall conductivity in the nanosheet thickness (178 nm) was approximately 73 S/cm at low temperatures (~3 K) and was inversely proportional to the temperature. Moreover, the aforementioned anomalous Hall measurement was performed in the film thickness. Hence, the anomalous Hall properties in the ultrathin region or atomically thin (less than a few layer thickness) structures remain to be studied.

Moreover, the Berry phase and Berry curvature are key concepts in the study of condensed matter physics and are used to describe the geometric properties of a band structure in a periodic solid. This geometric information is related to anomalous transport properties such as the anomalous Hall effect (AHE), anomalous Nernst effect (ANE), and anomalous thermal Hall effect (ATHE) through anomalous off-diagonal coefficients. The interplay between the electron properties and the magnetic orders in magnetic materials is a central theme in the field of condensed matter physics due to its potential applications for high-density magnetic recording technologies: recording media, magnetic random access memory (MRAM), and magnetic sensors^11,12. Berry-curvature-related quantities, such as the anomalous Hall conductivity (AHC), anomalous Nernst conductivity (ANC), and anomalous thermal Hall conductivity (ATHC), are strongly sensitive to the thickness in the 2D limit¹³. However, no studies on the anomalous off-diagonal response (AHE, ANE, and ATHE) in atomically thin (few layer thickness) 2D Fe₃GaTe₂ systems have been performed. Hence, in this study, we investigate the thickness-dependent magnetic properties (monolayer to four-layer thickness), anomalous transverse transport properties, magnetic anisotropy, and Curie temperature (T_C). Then, we show that 2D Fe₃GaTe₂ have better transverse transport properties than bulk or thicker systems.

Results

Thickness-dependent magnetic and electronic properties

Figure 1a–d shows the top and side views from the monolayer to the four-layer Fe₃GaTe₂ systems. The optimized lattice parameters are 4.04, 4.02, 4.00, and 4.02 Å for the monolayer, bilayer, trilayer, and four-layer systems, respectively. Note that the 2D Fe₃GaTe₂ system is experimentally fabricated. Therefore, we follow the same stacking order reported in the experimental study. These lattice parameters are similar to the reported value in the bulk system (3.98 Å)¹⁰. The interlayer distance in the bilayer, trilayer, and four-layer systems is approximately 2.90 Å, and the distance between the two central Fe atoms is 7.90 Å in each system. This result is also in good agreement with a previously reported value in the bulk system¹⁰. Then, we calculate the magnetic ground state by considering the total energy difference between the antiferromagnetic (AFM) and ferromagnetic (FM) states (E_AFM-E_FM). In the monolayer system, the total energy difference (ΔE_ex) is calculated by considering intralayer exchange coupling, and we find an energy difference ΔE_ex of 272 meV/unit-cell. Therefore, the monolayer structure shows the FM ground state. In the bilayer, trilayer, and four-layer systems, we consider both the intralayer and interlayer spin configurations. However, the intralayer AFM spin configuration is the most unstable state. Thus, we consider the layer-to-layer exchange coupling. In the bilayer system, the total energy difference, ΔE_ex, between the layer-to-layer FM and AFM states is 9 meV/unit-cell, while ΔE_ex becomes 20 meV/unit-cell and 34 meV/unit-cell in trilayer and four-layer structures. Overall, all the systems have an FM ground state. The monolayer system has a total magnetic moment of 5.87 μ_B/unit-cell, and this magnetic moment increased to 11.67 μ_B/unit-cell, 17.37 μ_B/unit-cell, and 23.25 μ_B/unit-cell in the bilayer, trilayer, and four-layer systems, respectively. Here, the Fe atom in the central sublayer (Fe_central) has a magnetic moment of 1.4 μ_B, while those of the top and bottom sublayers (Fe_top and Fe_bottom) have a magnetic moment of 2.3 μ_B. However, the magnetic moments of the Te and Ga atoms are approximately −0.1 μ_B, which are very small compared to that of the Fe atoms.

Fig. 1: Structural information and electronic band structure.

Top and side views of a monolayer, b bilayer, c trilayer, and d four-layer Fe₃GaTe₂ systems. Electronic band structure including SOC of e monolayer, f bilayer, g trilayer, and h four-layer Fe₃GaTe₂ systems.

To further describe the magnetic properties of the 2D Fe₃GaTe₂ systems, we consider the Heisenberg model for the spins of the Fe atoms. Here, we use Green’s function method, which treats the local rigid spin rotation as a perturbation¹⁴. In this case, the Heisenberg Hamiltonian can be expressed as follows:

$$H=-\sum _{{ij}}{J}_{{ij}}({m}_{i}\,\cdot \,{m}_{j})$$

(1)

where J_ij is the Heisenberg isotropic exchange, and m_i and m_j are the normalized magnetic moments. Here, we consider the pairs I, j with an extent up to the 3rd nearest neighbor (NN). Owing to the symmetry, we select four different intralayer spin pairs: Fe_bottom-Fe_bottom (equivalently Fe_top-Fe_top), Fe_central-Fe_central, Fe_bottom-Fe_central (equivalently Fe_top-Fe_central), and Fe_bottom-Fe_top in the monolayer system. However, the bilayer, trilayer, and four-layer systems have both intralayer and interlayer exchange interactions. We select the layer 1 to layer 2 interaction in the bilayer and trilayer systems. Note that layer 1 and layer 3 have the same exchange interaction with layer 2 by symmetry in the trilayer system. Moreover, two interlayer interactions occur in the four-layer system owing to its symmetry: layer 1(4) to layer 2(3) and layer 3 to layer 2 interactions. We present the Heisenberg exchange constants J_ij up to the 3rd NN of the monolayer and bilayer systems in Tables 1 and 2. For the trilayer and four-layer systems, we present the results in the Supplementary Information (SI) Tables S1 and S2 with further discussion in Supplementary Note 1. Here, the positive exchange J values favor a ferromagnetic (FM) configuration, while the negative exchange J values indicate an antiferromagnetic (AFM) configuration.

Table 1 Calculated Heisenberg exchange (J) in meV using Green’s function method of the Fe₃GaTe₂ monolayer.

Table 2 Calculated Heisenberg exchange (J) in meV using Green’s function method of the Fe₃GaTe₂ bilayer.

In the monolayer system, the 1st NN Fe_bottom-Fe_bottom interaction has J ~ 2 meV, favoring the FM configuration, and then decreases to J ~ −0.25 meV and −0.42 meV in the 2nd and 3rd NN interactions, respectively. The 1st NN Fe_central - Fe_central exchange interaction favors AFM coupling with a moderate strength of ~−2 meV, and this decreases to ~0.34 meV and −1.28 meV in the 2nd and 3rd NN interactions, respectively. The 1st NN Fe_bottom-Fe_top interaction has the largest exchange coupling of ~75 meV. This exchange coupling quickly drops in the 2nd and 3rd NN interactions, as indicated in Table 1. The 1st NN Fe_bottom-Fe_central interaction has ~26 meV, with decreased values in the 2nd and 3rd NN interactions. Overall, the main exchange coupling is dominated by the 1st NN Fe_bottom-Fe_top interaction followed by the 1st NN Fe_bottom-Fe_central interaction.

In the bilayer system, we find similar intralayer interactions as in the monolayer system. The main contribution is dominated by the 1st NN Fe_bottom-Fe_top exchange interactions with J ~74 meV and the 1st NN Fe_bottom-Fe_central with J ~27 meV, favoring the FM configuration. Note that the exchange couplings quickly decrease in the 2nd and 3rd NN interactions, as displayed in Table 2. However, the interlayer exchange interaction is rather weak compared with the intralayer interaction. The 1st NN Fe_top (layer 1) - Fe_bottom (layer 2) exchange interaction is only ~1.90 meV. Note that the trilayer and four-layer systems also favor the FM configuration (see Supplementary Note 1). In addition, we also calculate the exchange coupling between the Te and Fe atoms in the 2D Fe₃GaTe₂ system. The Fe-Te exchange coupling is very small compared with the Fe-Fe exchange coupling. For instance, the 1st NN exchange coupling strength in the monolayer system is ~1 meV, and this decreases to J~ 0.04 meV in the 2nd NN interaction. This result indicates that the influence of the Fe-Te exchange coupling on ferromagnetism is negligible. Figure 1e–h shows the thickness-dependent band structures in the FM state. In all systems, conventional metallic band structures are observed because both majority and minority spin bands cross the Fermi level. Note that the 2D thin Fe₃GaTe₂ system was reported as being metallic in a previous experimental report¹⁰.

We also calculate the magnetic anisotropy, including the spin-orbit coupling (SOC). Here, the magnetic anisotropy energy (MAE) is obtained based on the total energy difference between the in-plane [001] and out-of-plane [100] magnetization directions. The monolayer system has an MAE of 0.59 meV/unit-cell, indicating perpendicular magnetic anisotropy. The perpendicular MAE is further increased to 1.89 meV/unit-cell, 3.28 meV/unit-cell, and 3.99 meV/unit-cell in the bilayer, trilayer, and four-layer systems, respectively. To understand this large perpendicular magnetic anisotropy, we further analyze the SOC matrix elements. Figure 2a, b shows the contributions to the magnetic anisotropy from the Fe and Te atoms of the monolayer and bilayer systems. For the trilayer and four-layer systems, we present the calculated results in Supplementary Information (SI) in Fig. S1(a)-(b). Note that no substantial contribution to the magnetic anisotropy from the Ga atoms is found. Here, a positive value indicates perpendicular magnetic anisotropy, while a negative value represents the in-plane contribution. The main contributions to the MAE originate from the Te element, followed by a relatively small contribution from the Fe element. In addition, the interface Te and Fe atoms are slightly more sensitive than the surface atoms. Hence, we present only the interface Te and Fe atoms as an illustration. As shown, the largest contribution to the perpendicular magnetic anisotropy originates from the SOC between p_x –p_y orbitals, whereas the Fe atom has a rather small magnitude of the in-plane (negative) magnetic anisotropy through the SOC between d_x2-y2 - d_xy orbitals and d_xz - d_yz orbitals. The MAE from the Te atom is ~0.6 meV in the monolayer and increases to ~0.80 meV in the bilayer system. However, the d_x2-y2 - d_xy orbitals and d_xz - d_yz orbitals of the Fe atom have MAE contributions of −0.17 meV and −0.14 meV in the monolayer, respectively, and these values become −0.16 meV in the bilayer system. Similar trends are found in both the trilayer and four-layer systems. Overall, the net magnetic anisotropy in all four systems is dominated by the Te atom and particularly the p_x - p_y orbitals. Moreover, the magnetic anisotropy is strongly sensitive to the orbital characteristics of the unoccupied and occupied states near the Fermi level because these states are correlated through SOC. In addition, the SOC strength is usually proportional to the atomic mass. Since the Te atom has a larger atomic mass than the Fe atom, the major contributions to the MAE originate from the Te atoms, although the Fe atoms dominate the magnetic moment. We also present the layer-dependent MAE to determine the evolution of the MAE on each layer. Figure 2c shows the MAE layer contribution of each system. In the bilayer system, each layer has the same contribution because both layers are in the same environment. However, the central layers have larger contributions to the MAE compared with the top and bottom layers in trilayer and four-layer systems.

Fig. 2: SOC-resolved MAE.

SOC-resolved MAE of the interface Te and Fe atoms in the a monolayer and b bilayer. c Layer-dependent magnetic anisotropy energy (MAE) of the monolayer, bilayer, trilayer, and four-layer Fe₃GaTe₂ systems.

The Curie temperature (T_C) is a key factor in determining the suitability of a ferromagnetic material for spintronic applications. Particularly, a material that has a T_C above room temperature is highly desirable. Thus, we calculate the thickness-dependent T_C of the 2D Fe₃GaTe₂ systems. We apply the Metropolis Monte Carlo (MC) simulation using the Heisenberg-type Hamiltonian. The Hamiltonian operator in the Heisenberg model with the magnetic anisotropy factor can be written as follows:

$$\hat{H}=-\sum _{i,j}J{\hat{m}}_{i}\,\cdot \,{\hat{m}}_{j}-{k}_{2}\sum _{i}{m}_{z}^{2}$$

(2)

where ${\hat{m}}_{i(j)}$ is the magnetic moment at sites i (j), and ${m}_{z}$ is the spin aligned along a single easy axis. J and ${k}_{2}$ represent the exchange coupling parameter and the anisotropy constant, respectively. Note that the J values are evaluated based on Green’s function method¹⁴. The anisotropy constant is estimated based on our calculated magnetic anisotropy energy (MAE). To calculate the temperature-dependent magnetization curve, we use a supercell of 50 × 50 for the monolayer system and 30 × 30 for the bilayer, trilayer, and four-layer systems using the VAMPIRE software package^15,16. We employ 10000 equilibration steps and 10000 averaging steps. Figure 3a–d shows the temperature-dependent magnetization curves of the monolayer, bilayer, trilayer, and four-layer Fe₃GaTe₂ systems. Here, we fit the temperature-dependent magnetic curve using the Curie-Bloch equation as given below:

$$m\left(T\right)={\left[1-\frac{T}{{T}_{C}}\right]}^{\beta }$$

(3)

Fig. 3: Monte Carlo simulation for the magnetization.

Temperature-dependent magnetization curve of the a monolayer, b bilayer, c trilayer, and d four-layer Fe₃GaTe₂ systems. Here, the calculated T_C is 300 K, 325 K, 340 K, and 352 K in the monolayer, bilayer, trilayer, and four-layer systems, respectively.

The monolayer system has a T_C of 310 K, which is further increased to 340 K, 358 K, and 374 K in the bilayer, trilayer, and four-layer systems, respectively. This enhancement of the T_C as a function of thickness is due to the introduction and enhancement of the interlayer exchange along with the intralayer exchange from the monolayer to the four-layer systems. Note that the layer-to-layer exchange is enhanced with increasing thickness from the bilayer to the four-layer thickness in the 2D Fe₃GaTe₂ systems, and this leads to the enhancement of the T_C up to the four-layer system. Overall, the calculated Curie temperatures of the atomically thin 2D Fe₃GaTe₂ systems are similar to those of the T_C in relatively thicker 2D Fe₃GaTe₂ (350 ~ 380 K) and bulk systems (365 K)¹⁰. The 2D Fe₃GaTe₂ shows a much larger T_C than the 2D Fe₃GeTe₂ monolayer (~130 K)⁸. Moreover, we investigate the effect of magnetic shape anisotropy on T_C. Thus, we calculate the critical temperature assuming in-plane anisotropy, and the calculated T_C is presented in SI Fig. S2. The calculated T_C with in-plane anisotropy slightly decreases by only 1–5 K in all 2D Fe₃GaTe₂ systems (from mono- to four-layer) in comparison to the T_C calculated with out-of-plane anisotropy. This result indicates the negligible effect of magnetic shape anisotropy on the critical temperature.

Thickness-dependent anomalous transport coefficients

Onsager’s generalized transport equations (Onsager’s reciprocal relations) relate the transport coefficients to the thermodynamic forces and fluxes in a system. For the off-diagonal transport coefficients, Onsager’s generalized transport equations can be expressed as follows:

$$\left(\begin{array}{c}J\\ Q\end{array}\right)=\left(\begin{array}{cc}{l}_{11} & {l}_{12}\\ {l}_{21} & {l}_{22}\end{array}\right)\left(\begin{array}{c}E\\ \nabla T\end{array}\right)$$

(4)

where ${J}$ and ${Q}$ are the charge current density and the heat current density, respectively, while $E$ and $\nabla T$ are the electric field and applied temperature gradient, respectively. Note that we consider the intrinsic anomalous transport coefficients to originate from an anomalous velocity that produces the intrinsic (Karplus–Luttinger) mechanism in terms of Berry curvature. Thus, the ${l}_{11}$ component represents the AHC. At zero temperature, the AHC (${\sigma }_{{xy}}$) can be obtained from the Kubo formula and Berry phase theory by integrating the Berry curvature over the whole Brillouin zone (BZ)^17,18:

$${{l}_{11}=\sigma }_{{xy}}=\frac{{e}^{2}}{{{\hslash }}}{\int }_{{BZ}}\frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}\varOmega \left(k\right)$$

(5)

where the Berry curvature $\varOmega \left(k\right)$ is given by the following:

$$\varOmega \left(k\right)=\sum _{n}{f}_{n}{\varOmega }_{n}\left(k\right)\,$$

(6)

$${\varOmega }_{n}\left(k\right)=-2{Im}\sum _{m\ne n}\frac{\left\langle {u}_{{nk}}\left|{v}_{x}\right|{u}_{{mk}}\right\rangle \left\langle {u}_{{mk}}\left|{v}_{y}\right|{u}_{{nk}}\right\rangle }{{({E}_{{mk}}-{E}_{{nk}})}^{2}}\,$$

(7)

where f_n, v_x,y, and u_nk represent the Fermi-Dirac distribution function, velocity operator, and periodic part of the Bloch wave function with eigenvalue E_nk, respectively. Based on the well-constructed maximize localized Wannier functions (MLWFs), the projected Bloch wave functions (density functional theory Hamiltonian) onto high symmetry Wannier functions are used to calculate the Fourier transformed Wannier Hamiltonian. Figure 4 shows the calculated thickness-dependent AHC as a function of the chemical potential of the monolayer, bilayer, trilayer, and four-layer systems. The magnitude of the AHC increases with film thickness. For instance, the four-layer thickness structure has an AHC of -485 S/cm at zero chemical potential (μ). By varying the chemical potential, the AHC increases with electron doping, while the reverse behavior is obtained with hole doping. In particular, the maximum magnitude of the AHC in the four-layer system is approximately −550 S/cm at a 40 meV chemical potential. Note that the chemical potential of 40 meV can be considered minor because it is almost half an order of magnitude smaller than that considered in 2D FeCl₂¹⁹, and the maximum AHC in the four-layer system is almost 1.5 times larger than that found in 2D FeCl₂. For comparison, we present both AHC (red line) and AHC per layer (black line) at zero chemical potential of the monolayer, bilayer, trilayer, and four-layer systems with an experimental measurement of AHC at 220-layer thickness Fe₃GaTe₂ (~178 nm)¹⁰ in Fig. 5a. To be consistent, we show the absolute AHC ($\left|{{\rm{\sigma }}}_{{xy}}\right|$) at zero carrier doping. The AHC in the monolayer system is approximately 165 S/cm and slightly increases to 185 S/cm in the bilayer. However, the AHC is substantially increased to 275 S/cm and 485 S/cm in the trilayer and four-layer systems, respectively. Note that the magnitude of the experimentally measured AHC of the 220-layer thickness system is approximately 73 S/cm at 3 K. Therefore, the ultrathin Fe₃GaTe₂ films with at least a four-layer thickness have larger values than those found in the thick film structure. In particular, the four-layer structure has an almost seven times larger AHC than the thick film structure. Additionally, the four-layer system shows a larger AHC than the thin film Fe₃GeTe₂ system (360 S/cm)²⁰. Note that the variation in the AHC is caused by its sensitivity to the Berry curvature, which will be discussed later. On the other hand, the AHC values per layer show different behaviors. The AHC of the monolayer is ~165 S/cm, as indicated by the black color in Fig. 5a. However, with increasing thickness, the contribution to the AHC per layer is in the range of ~80–120 S/cm. The AHC per layer in ultrathin 2D Fe₃GaTe₂ (mono- to four-layer) is larger than that measured in the thick Fe₃GaTe₂ system (AHC per layer ~0.33 S/cm)¹⁰. We attribute this behavior to the quantum confinement effects in low-dimensional systems.

Fig. 4: Thickness-dependent anomalous Hall conductivity.

Calculated anomalous Hall conductivity (σ_xy) as a function of the chemical potential of a monolayer, b bilayer, c trilayer, and d four-layer Fe₃GaTe₂ systems.

Fig. 5: Thickness-dependent anomalous transverse transport coefficients.

a Anomalous Hall conductivity (σ_xy) and AHC per layer (AHC/layer) at the Fermi energy of the 2D Fe₃GaTe₂ systems (from monolayer to four-layer and including experimental data). Thickness-dependent (b) anomalous Nernst conductivity (α_xy) and (c) anomalous thermal Hall conductivity (κ_xy) as a function of the chemical potential of the 2D Fe₃GaTe₂ systems. Here, the black, red, blue, and green lines represent the monolayer, bilayer, trilayer, and four-layer Fe₃GaTe₂ systems, respectively.

On the other hand, ANC (${\alpha }_{{xy}}$) can be expressed as follows²¹:

$${l}_{12}={\alpha }_{{xy}}=\frac{1}{{eT}}\int d\varepsilon \,\left(\varepsilon -\mu \right)\frac{\partial f}{\partial \varepsilon }{\sigma }_{{xy}}$$

(8)

where e, μ, and ${f}$ are the elementary positive charge, chemical potential, and Fermi-Dirac distribution function, respectively. Figure 5b shows the thickness-dependent ANC as a function of the chemical potential at 100 K by applying the Mott relation²¹. At zero chemical potential, the monolayer has an ANC of 0.23 A/K.m, with a slight decrease to 0.15 A/K.m in the bilayer system, and it is approximately 0.05 A/K.m in the trilayer structure. However, the ANC in the four-layer has 0.32 A/K.m. at zero chemical potential. Moreover, the ANC is not enhanced with either electron or hole doping in the monolayer and bilayer systems. However, the ANC can be manipulated by small carrier doping in the trilayer and four-layer systems. For instance, the trilayer system exhibits a maximum ANC of −0.35 A/K.m at a chemical potential of −35 meV, and the ANC is further increased to 0.55 A/K.m in the four-layer system at −45 meV. Furthermore, the ANC in the four-layer structure is slightly larger than those found in other 2D materials, such as 2D Fe₃GeTe₂ (0.3 A/K.m at 130 K)²⁰ and 2D FeCl₂ (0.3 ~ 0.4 A/K.m at 100 K)¹⁹. We also calculate the Nernst thermopower (S_xy) of the 2D Fe₃GaTe₂ systems, and they are shown in SI Fig. S3. Note that further details can be found in Supplementary Note 2. The Nernst thermopower is ~−0.32 μV/K to 1 μV/K at zero chemical potential, which is lower than that of ferromagnetic Co₃Sn₂S₂ (~3 μV/K) at 80 K²². However, these Nernst thermopower (S_xy) values can be largely increased by either hole or electron doping. In particular, the largest maximum Nernst thermopower of S_xy ~ 7.5 μV/K is obtained in the monolayer system with hole doping (μ ~ 0.34 meV).

We also calculate the anomalous thermal Hall conductivity (ATHC), which represents the transverse thermal current induced by a longitudinal gradient of temperature. The ATHC can be written according to the linear response relation¹¹:

$${l}_{22}={\kappa }_{{xy}}=\frac{1}{{{\hslash }}T}\int d\varepsilon \left(-{\left(\varepsilon -\mu \right)}^{2}\frac{\partial f}{\partial \varepsilon }\right){\sigma }_{{xy}}$$

(9)

Note that the AHC and ATHC can be correlated with each other using the Wiedemann-Franz (WF) law and based on the anomalous Lorenz ratio, which is defined as follows:

$${\kappa }_{{xy}}={L}_{{xy}}{\sigma }_{{xy}}T$$

(10)

where ${L}_{{xy}}$ is the anomalous Lorenz ratio, which is a constant ratio of ~2.44 × 10⁻⁸ Ω.W/K². Figure 5c shows the thickness-dependent ATHC as a function of the chemical potential at 100 K. Evidently, the spectral shape of the ATHC is similar to that of the AHC due to its correlation. At zero chemical potential, the ATHC is approximately −0.04 W/K.m, −0.045 W/K.m, −0.065 W/K.m, and −0.105 W/K.m in the monolayer, bilayer, trilayer, and four-layer systems, respectively. In particular, the ATHC of the four-layer system is similar to the reported large ATHC in the Weyl semimetal Co₃Sn₂S₂ (~0.14 W/K.m) at 80 K²³ and larger than that of the 2D Fe₃GeTe₂ system (0.04–0.11 W/K.m) at 130 K²⁴. Note that this can be further increased up to −0.135 W/K.m with small electron doping (chemical potential of 40 meV).

Thickness-dependent distribution of Berry curvature

All anomalous transverse transport coefficients are correlated with each other according to Onsager’s generalized transport equations. From a fundamental point of view, all transverse transport quantities are related to the AHC, and this AHC can be obtained from the Berry curvatures. For instance, the AHC is determined by the integration of the total Berry curvature of the occupied states at each k point in the whole Brillouin zone (BZ). Hence, as an illustration, we calculate the momentum resolved of the Berry curvature over the 2D plane (k_x, k_y) at the Fermi energy (zero chemical potential) to interpret the variation in the thickness dependent AHC behavior. Notably, the electronic structure is strongly sensitive to the film thickness in atomically thin structures, which affects the Berry curvature along with the structural symmetric feature. Figure 6a–d shows the distribution of the Berry curvature in the reciprocal space for monolayer, bilayer, trilayer, and four-layer systems. The boundary lines are indicated by black solid lines with Γ, M, K, and K^- points. Note that the blue color indicates a positive Berry contribution, while the red color represents a negative contribution. A positive value of the Berry curvature leads to a positive AHC magnitude and vice versa. Note that the odd number of layers in the 2D Fe₃GaTe₂ system belongs to the subgroup of D3h-1 (187 space group). It has both 3-fold symmetry and a broken inversion symmetry along with the lack of the additional three vertical mirror planes of the full D3h point group. In the monolayer system, due to the broken inversion symmetry, the Berry curvature contributions from K corners cancel each other and disappear. The small positive Berry curvatures originate near the Γ point, but more pronounced negative Berry curvatures are found by the many general k-points, and these contributions are accumulated. This results in a net negative AHC. Different from the odd number of layers structure, the even number of layers structure belongs to the subgroup of the D3d-3 point group (164 space group), which has a total of 6 symmetry operations, including the inversion symmetry. Therefore, in the bilayer structure, the K corners show the same Berry curvature contributions (with small positive contributions). Moreover, small positive Berry contributions are also present near the Γ point. However, the negative Berry curvature contributions are more dominant along the general k-points. These negative Berry curvature contributions accumulated, which results in a negative AHC in the bilayer system. In the trilayer system, we find similar behavior near the K corners as in the monolayer structure since it also has an odd number of layers. Additionally, small positive Berry contributions appear near the Γ point. However, greater dominance of the negative Berry contributions over the positive ones from the general k-points are observed. Thus, the negative AHC in the trilayer system is increased compared to those of the monolayer and bilayer systems. In the four-layer system, the Berry curvatures at the K corners show almost the same behavior as in the bilayer system due to the same symmetry. Additionally, small positive Berry curvatures are found around the Γ point. Nonetheless, the negative Berry curvatures along the general k-points are more dominant, and their magnitudes are larger than the positive ones. Consequently, this results in a larger negative AHC in the four-layer system. Overall, no single dominating contribution to the Berry curvature contribution appears from a particular k-point; however, all negative Berry contributions are accumulated from general k-points compared to the positive ones. Note that the behavior of ANC and ATHC properties are linked to the AHC features.

Fig. 6: Thickness-dependent Berry curvature.

Distribution of the Berry curvature (Ω) in the reciprocal space at the Fermi energy of the a monolayer, b bilayer, c trilayer, and d four-layer Fe₃GaTe₂ systems.

Discussion

In summary, we explore the thickness-dependent magnetic properties of the 2D Fe₃GaTe₂ system. The monolayer system has a T_C of 300 K, which is further increased to 325 K, 340 K, and 352 K in the bilayer, trilayer, and four-layer systems. We attain the perpendicular magnetic anisotropy from the monolayer to the four-layer thickness. The MAE energy increases with thickness. The perpendicular magnetic anisotropy originates from the SOC coupling of the Te atom, whereas the Fe contribution is quite weak. Similar to the magnetic anisotropy, the AHC also increases with increasing film thickness up to a four-layer thickness. For instance, the four-layer thickness structure has an AHC of −485 S/cm at zero chemical potential, and this can be enhanced to -550 S/cm with small electron doping. Based on the analysis of the momentum resolved from the Berry curvature, the AHC magnitude is mainly related to the accumulation of negative Berry curvature from general k-points and not from a single special k-point. The ANC of the four-layer system is 0.32 A/K.m at zero chemical potential. The magnitude of ANC can be further increased to 0.55 A/K.m with small hole doping. The Nernst thermopower (S_xy) is also calculated, and the largest maximum Nernst thermopower of S_xy ~ 7.5 μV/K can be obtained in the monolayer system with hole doping. In addition, the ATHC reaches approximately −0.105 W/K.m in the four-layer system and can be further enhanced to −0.135 W/K.m with electron doping. Overall, the 2D ultrathin Fe₃GaTe₂ system shows room-temperature ferromagnetism incorporating outstanding transverse transport properties. In particular, the four-layer system shows superior transverse transport properties. Our findings potentially indicate that the atomically thin 2D Fe₃GaTe₂ can be utilized for potential spintronics and spin caloritronic device applications.

Methods

First-principles calculations

We perform first-principles calculations using the Vienna ab initio simulation package (VASP)^25,26 with a projector augmented wave (PAW) set^26,27 and the Perdew-Burke-Ernzerhof (PBE) exchange correlational functional with generalized gradient approximation (GGA)²⁸. Note that the PBE-GGA is demonstrated to appropriately address the exchange and correlation effects, and the results are in agreement with experimental measurements¹⁰. The energy cutoff of the plane wave is set to 650 eV, and the energy criterion is set to 10⁻⁶ eV with the forces less than 0.02 eV/Å on all relaxed structures. The Brillouin zone is sampled on a k-mesh of 19 × 19 × 1, and a vacuum distance of 20 Å is used to prevent artificial interactions along the z-direction. The van der Waals (vdW) interaction is included using the Grimme DFT-D3 method²⁹, and spin-orbit coupling (SOC) is also considered in the noncollinear calculations.

Magnetic anisotropy, magnetic interactions, and Curie temperature calculations

The magnetic anisotropy energy is calculated as the total energy difference between the in-plane [001] and out-of-plane [100] magnetization directions. Since the magnetic anisotropy requires more accurate k-mesh points, we carefully check the energy convergence up to 21 × 21 k-mesh. The exchange magnetic interaction parameters are calculated using Green’s function method based on magnetic force theory (MFT) as implemented in the TB2J code¹⁴. The Curie temperature is estimated using Metropolis Monte Carlo simulations implemented in the VAMPIRE software package^15,16.

Wannier function (WF) calculations

The calculation of the Berry curvature and anomalous Hall conductivity is handled based on the Wannier functions, which are extracted from the band structure using the maximally localized Wannier function method³⁰. Here, a finer k-mesh of 300 × 300 × 1 is used to accurately construct real-space Wannier functions for the calculation of the transverse electric response properties using a tight-binding Hamiltonian.