On the Origin of the Above-Room-Temperature Magnetism in the 2D van der Waals Ferromagnet Fe3GaTe2

2D magnetic materials have attracted growing interest driven by their unique properties and potential applications. However, the scarcity of systems exhibiting magnetism at room temperature has limited their practical implementation into functional devices. Here we focus on the van der Waals ferromagnet Fe3GaTe2, which exhibits above-room-temperature magnetism (Tc = 350–380 K) and strong perpendicular anisotropy. Through first-principles calculations, we examine the magnetic properties of Fe3GaTe2 and compare them with those of Fe3GeTe2. Our calculations unveil the microscopic mechanisms governing their magnetic behavior, emphasizing the pivotal role of ferromagnetic in-plane couplings in the stabilization of the elevated Tc in Fe3GaTe2. Additionally, we predict the stability, substantial perpendicular anisotropy, and high Tc of the single-layer Fe3GaTe2. We also demonstrate the potential of strain engineering and electrostatic doping to modulate its magnetic properties. Our results incentivize the isolation of the monolayer and pave the way for the future optimization of Fe3GaTe2 in magnetic and spintronic nanodevices.


INTRODUCTION
The recent breakthrough of long-range magnetic order in two-dimensional (2D) van der Waals (vdW) magnetic materials represents a particularly exciting research avenue that opens new pathways for exploring physical phenomena and emerging applications.[1][2][3] The discovery of ferromagnetic materials at the 2D limit, such as the semiconductors CrI3 and Cr2GeTe6, has attracted a lot of attention since 2017 [1,2].However, practical applications are drastically hindered by their low critical temperatures below liquid nitrogen temperature [4,5].In this regard, materials such as CrSBr or the metallic Fe3GeTe2 have come to the forefront, retaining magnetic properties down to monolayer thickness with higher Curie temperatures (TC) of 146 and 130 K, respectively [6][7][8].In this context, external fields, strain engineering and electrostatic doping have proven to be powerful strategies to improve the magnetic behaviour of these systems [4,[9][10][11][12][13].
More recently, bulk Fe3GaTe2 has emerged as a magnetic material of interest due to its above-roomtemperature magnetism (350-380 K) and strong perpendicular magnetic anisotropy [28].Its electronic structure has been deeply investigated [29], it has been incorporated into spin-valve devices [30][31][32] and has shown potential for hosting magnetic skyrmions at room-temperature [33][34][35].Nevertheless, a comprehensive magnetic analysis to elucidate the main mechanisms governing the high critical temperature of Fe3GaTe2 and the possibility of maintaining its outstanding magnetic properties at the monolayer limit are still lacking.
In this work, through first-principles calculations combined with a Wannier-based tight-binding model, we investigate the fundamental mechanism responsible for the room-temperature magnetism in bulk Fe3GaTe2.Furthermore, we present a comparative analysis with Fe3GeTe2, unambiguously demonstrating the pivotal role of the competing ferromagnetic-antiferromagnetic in-plane couplings on the critical temperatures of both systems.In addition, we prove the dynamical stability of Fe3GaTe2 monolayer and explore the impact of strain engineering and electrostatic doping in the modification of the magnetic and electronic properties of this system.

RESULTS AND DISCUSSION
Fe3GaTe2 is a van der Waals (vdW) magnetic material, isostructural to the widely known Fe3GeTe2, that crystallizes within a hexagonal layered structure in the space group P63 /mmc (N0.194).Each unit cell contains two layers of the material stacked along the c direction (with an AB stacking pattern), which are separated by an interlayer spacing of ~7.8 Å.The lattice parameters are a = b = 3.986 Å and c = 16.229Å [28].Within each layer the system is formed by five atomic sublayers containing two external surfaces of Te atoms, a pair of internal layers formed by two equivalent Fe atoms (Fe1 and Fe2) and a central layer consisting in Ga and inequivalent Fe3 atoms (Figure 1(a)).
Firstly, we perform first-principles calculations on bulk Fe3GaTe2.The results reveal magnetic moments of 2.05 and 1.35 µB/Fe for the equivalent Fe1,2 and the inequivalent Fe3 atoms, respectively.Ga and Te showed negligible polarization.The calculated average magnetic moment is 1.82 µB/Fe atom, consistent with experimental observations at 3 K and previous theoretical studies [28,36].Furthermore, we extract the magnetic anisotropy energy (MAE) by means of the energy difference between in-plane and out-ofplane magnetic configurations (where a positive value indicates an out-of-plane magnetization easy axis).Our results yield a value of MAE of 0.31 meV/Fe, being in good agreement with previous studies [29,36] and confirming that the material exhibits a strong perpendicular anisotropy with spins aligned along the c axis.
To explore the reasons behind the different Tc of Fe3GaTe2 and Fe3GeTe2, which are 380 and 240 K, respectively, we determine the magnetic exchange couplings in both materials and perform atomistic simulations (see Computational details).The exchange interactions are determined by first formulating a tight-binding Hamiltonian in the basis of maximally localized Wannier Functions (MLWFs) [37], and then employing the TB2J package [38], which is based in the use of Green's functions.The spin Hamiltonian has the following form: where Jij represent the isotropic exchange interactions, Si and Sj are the magnetic moments of the different sites and A accounts for the magnetic anisotropy of the system.
Our calculations reveal that the magnetic interaction picture of Fe3GaTe2 and Fe3GeTe2 is quite intricate, where the emergent global ferromagnetism arises from a competition between different couplings.We categorize these interactions into two groups, namely (i) inter-plane interactions, which encompass J12 (interactions between atoms Fe1-Fe2) and J13 (Fe1-Fe3), as well as (ii) in-plane exchanges, comprising J11 (Fe1-Fe1) and J33 (Fe3-Fe3) (Figure 1(a)).In constructing the spin Hamiltonian, we included both interaction groups, taking into account all neighbour interactions and their evolution with distance.As one can observe in Figure 1(b), both systems display FM characteristics, with Fe3GaTe2 exhibiting lower values compared to Fe3GeTe2 when considering solely the inter-plane couplings.However, more significant differences appear upon examination of in-plane exchanges.In Fe3GeTe2, the J11 and J33 interactions are antiferromagnetic (AF), resulting in a geometrically frustrated spin lattice [39,40].(blue), as well as their evolution with distance to a maximum of 16 Å.
Considering the chemical and structural similarity between Fe3GaTe2 and Fe3GeTe2, which retains ferromagnetism and strong perpendicular anisotropy down to the monolayer limit [6,14], we investigate the electronic and magnetic properties of single-layer Fe3GaTe2.According to our first-principles calculations the optimized lattice parameters of the monolayer are a = b = 3.947 Å.These values are close to, but slightly smaller than, the abovementioned lattice constants of the bulk structure.We observe that single-layer maintains a FM ground state, with magnetic moments of 2.05 and 1.31 µB/Fe for Fe1,2 and Fe3 atoms, respectively.This yields an average magnetic moment for Fe atoms of 1.80 µB, which is nearly identical to the bulk and higher than the calculated for Fe3GeTe2 monolayer (1.61µB).
Then, we explore the dynamic stability of Fe3GaTe2 monolayer by performing phonon calculations.As observed in Figure 2(a) the phonon dispersion presents no imaginary frequencies and thus the structure is stable, closely resembling the one of Fe3GeTe2 monolayer [41].Furthermore, we study the energetic stability of Fe3GaTe2 monolayer calculating its formation energy relative to the bulk structure.This parameter is defined as the energy difference (per atom) between the monolayer and bulk forms, providing an estimation of the energy required to synthesize a single layer of material from its bulk counterpart.Using this definition, we obtain a value of 36.85 meV/atom, being close to the reported for Fe3GeTe2 monolayer (48.9 meV/atom) [41], and comparable to other vdW materials successfully exfoliated to the monolayer limit [42].Regarding the electronic band structure and projected density of  Furthermore, we estimate the magnetic exchange interactions of Fe3GaTe2 monolayer.A direct comparison with the bulk results reveals minor variations, implying that the monolayer is likely to preserve a robust FM character (Figure S1).This observation is further supported by the calculation of (a) ( MAE, which yields a value of 0.36 meV/Fe, indicating a substantial perpendicular anisotropy with spin orientations aligned along the c axis.Our atomistic simulations predict a high Tc value of 594 K, which is considerably higher than the one calculated for monolayer Fe3GeTe2 (100 K).
In Figure 3(a) we directly compare the exchange interactions in monolayers Fe3GaTe2 and Fe3GeTe2.
Our analysis indicates that as in bulk, the in-plane couplings J11 and J33 are the origin of the observed differences in Tc between these compounds.The distinct contributions originate from specific orbitals involved in the stabilization of long-range magnetic ordering in each system, as confirmed by our orbital-resolved analysis of the exchange parameters (Figure 3(b) and (c)).The ferromagnetic J12 interaction is predominantly governed by dz2-dz2, dxz-dxz, and dyz-dyz orbitals, with a small antiferromagnetic contribution arising from in-plane dxy-dxy and dx2-y2-dx2-y2.The overall reduction in ferromagnetism within J12 for Fe3GaTe2 can be attributed to a diminished FM contribution from dxz-dxz and dyz-dyz orbitals compared to the case of Fe3GeTe2 (Figure S2).This orbital-resolved description contrasts with the findings of Lee et al. [29], who associated the increase in Tc in Fe3GaTe2 with a higher value of J12 relative to Fe3GeTe2.Additionally, the FM nature of J11 is attributed to a predominant FM dyz-dxz superexchange pathway mediated by py and px orbitals of Ga and Te, respectively.This mechanism is highly diminished in Fe3GeTe2, eventually turning to become AF (Figure 3(b)).In addition, the J33 in Fe3GeTe2 is primarily sourced from an AF dz2-px-dz2 mechanism mediated by Ge, which is nearly supressed in Fe3GaTe2 (Figure 3(c)).Our findings discard the possibility that structural differences between both systems are the determinant factor to explain the variations in magnetic exchange couplings and the differences in Tc (Figure S3).
In order to assess the impact of J11 and J33 on the FM stabilization, we carry out calculations considering interactions until 16 Å (Figures S4, S5).In the case of only taking into account interactions up to 3 Å (thus only accounting for J12 and J13) we obtain values of Tc of 251 K for Fe3GaTe2 and 220 K for Fe3GeTe2.This slight variation in Tc between both systems suggests that a model restricted to nearestneighbours inter-plane couplings is not adequate to describe the magnetic behaviour.Extending to 4 Å, where J11 and J33 are also considered, the Tc of Fe3GaTe2 is enhanced up to 434 K, whereas the Tc of Additionally, we investigate the evolution of the magnetic properties of Fe3GaTe2 monolayer upon mechanical deformation and electrostatic doping.4(a), (b) and (c) illustrate the variation of the magnetic moments for both Fe and ligands, along with the dependence of MAE and exchange parameters under biaxial strain.Notably, under strain (-4% to 4%), the average Fe magnetic moments steadily increase, while they remain almost null in the ligands.The MAE results reveal that the magnetization easy axis remains off-plane in the entire range of study.Its value is almost null at -4% and reaches a maximum at 1%, suffering a decrease at values of e > 1%.According to our simulations, compression values slightly exceeding e = -4% would induce in-plane magnetization within the system.
On the other hand, a more substantial tensile strain would be required to modify the off-plane magnetization easy axis.Besides the evolution of MAE reflects the trend observed in the magnetic moments of Fe3, the change in the latter is considerably smaller than the variation in MAE.This strongly suggests that the drastic change in MAE is not primarily due to spin moment variations.Prior studies have attributed this phenomenon with band shifts involving significant spin-orbit coupling [43].This is further supported by our band structure calculations at various strain levels, demonstrating an effective modulation of the electronic properties around the Fermi level (Figure S6).In contrast, the evolution of the MAE diverges from that reported for Fe3GeTe2, where it parallels the evolution of magnetic moments and increases continuously upon tensile strain [41].In addition, we can observe that magnetic couplings are highly sensitive to mechanical deformation.In Figure 4(c) we show that the FM exchange couplings exhibit an increase (decrease) upon compression (elongation) over the studied range.Notably, at e < -2%, J12 significantly increases while J11 decreases, the latter transitioning towards an AF state.
We attribute these anomalous evolutions to the different trend of magnetic moments for e < -2% compared to the rest of the map.Note that Fe1 and Fe2 atoms are situated within same xy plane and consequently the distance Fe1-Fe2 remains intact upon applied strain.However, the angle governing the Fe1-Ga-Fe2 superexchange pathway is reduced from 57.1º (-4%) to 53.3º (+4%) upon tensile strain, thereby having an impact in the coupling J12 (see Figure S7   The analysis of the impact of electrostatic doping on the magnetic properties shows that the magnetic moments of both equivalent and inequivalent Fe atoms are similarly affected, increasing and decreasing under electron and hole doping, respectively (Figure 5(a)).In contrast, the magnetic moments of Ga remain almost unchanged, whereas those of Te experience a pronounced increase under electron doping.
Regarding the evolution of MAE reported in Figure 5(b), one can observe that the magnetization easy axis changes from off-plane (positive sign) to in-plane (negative sign) magnetization with a doping level around 0.15 h/f.u.This result agrees well with the theoretical observations of in-plane magnetism in bulk Fe3GaTe2 upon hole doping [29].The evolution of magnetic exchange couplings is reported in Figure 5(c).We observe that electron doping leads to an increase of the FM interactions J12 and J13, suggesting a potential rise in Tc.However, there is a softening of the FM character of J11 and J22 that results in a reduced Tc of 501 K (Figure 5(d)).This contrasts with the increase of the critical temperature reported for Fe3GeTe2 upon electron gating [14].On the other hand, hole doping leads to a slight reduction in J12 and J13 compared to the undoped system, while J11 and J33 become more FM, resulting in an almost unchanged critical temperature.This aligns with recent experimental findings in bulk Fe2.84GaTe2, where Fe deficiency is linked to a slightly lower Tc with respect to Fe3GaTe2 [33].Finally, we examine the evolution of the electronic band structure with dopings of 0.2h/f.u. and 0.2e/f.u., observing that there is an effective shift of the bands around the high symmetry points G and K (Figure 6).These particular bands are postulated to significantly influence the MAE [29,43].At zero doping, we note the presence of hole (around G) and electron (around K) pockets, which are mostly formed by p orbitals of Te and d orbitals of Fe atoms, respectively.Both pockets are shifted upwards (downwards) upon hole (electron) doping.Specifically, we note the presence of two electron pockets at K that contribute to an enhanced positive MAE.The first one (situated in the vicinity of K) suffers an effective downwards shift with increasing electron density, leading to heightened perpendicular anisotropy up to a value of 0.73 meV/Fe atom at 0.2e/f.u.In contrast, upon hole doping, the electron pockets rise in energy, with one band no longer intersecting the Fermi level.This results in a weakened perpendicular anisotropy and a preferential in-plane spin direction (-0.13 meV/Fe atom).

CONCLUSIONS
In summary, we have investigated the magnetic and electronic properties of the above-roomtemperature van der Waals ferromagnet Fe3GaTe2 via first principles calculations.By conducting a direct comparison between Fe3GaTe2 and Fe3GeTe2, we uncover the intricate microscopic mechanisms underlying their magnetic behaviour.Our analysis highlights the critical role of antiferromagneticferromagnetic in-plane exchange interactions and their contribution to the higher Tc observed in Fe3GaTe2 when contrasted with Fe3GeTe2.Moreover, our phonon calculations demonstrate the dynamical stability of single-layer Fe3GaTe2, while we predict strong perpendicular anisotropy and a high Tc, thus incentivising its isolation.Finally, we show that the exchange interactions of Fe3GaTe2 monolayer can be modified by strain engineering and electrostatic doping and prove their potential to tune the anisotropy of the system.Our findings lay a foundation for comprehending the origin of the critical temperature Tc in both Fe3GaTe2 and Fe3GeTe2 and their future manipulation in magnetic and spintronic devices.

COMPUTATIONAL DETAILS
We carried out spin polarized density-functional theory (DFT) using the Quantum ESPRESSO package [44].Spin-polarized local density approximation (LDA) [45] was used to approximate the exchangecorrelation functional given that it has been proved to properly describe properties of Fe3GaTe2 and Fe3GeTe2 [14,36,46], avoiding an overestimation of magnetic moments [41].For the bulk structures we relaxed the atomic coordinates while for the monolayers both atomic coordinates and lattice parameters were optimized.In both cases, the optimizations were carried out until forces on each atom were smaller than 1•10 −3 Ry/au and the energy difference between two consecutive relaxation steps was less than 1•10 −4 Ry.The electronic wave functions were expanded with well-converged kinetic energy cut-offs for the wave functions (charge density) of 75 (850) Ry.To properly describe the monolayers, a vacuum spacing of 18 Å was set along c direction to avoid unphysical interactions between layers.For the bulk(monolayer) structures the Brillouin zone was sampled by a fine Г-centered 10×10×3 (10×10×1) k-point Monkhorst-Pack, that was expanded to 15×15×3(15×15×1) for the calculations of MAE.The phonon spectrum was computed using a 3x3x1 supercell by means of the Phonopy code [47].A tightbinding model based was constructed based on maximally localized Wannier function as implemented in the Wannier90 code [48].Our reduced basis set is formed by the d orbitals of Fe and p orbitals of Ga, Ge and Te.Magnetic exchange interactions were determined using Green's function method as implemented in TB2J code [38] employing a 30×30×5 (30×30×1) supercell for the bulk (monolayer) structures.The TC was obtained by performing atomistic simulations as implemented in the VAMPIRE code [49].
To validate the obtained magnetic exchange couplings determined by the plane wave method, we double checked our results using a localized atomic orbital approach as implemented in the SIESTA package [50] (Figures S12 and S13), in which the local density approximation (LDA) was used to describe the exchange correlation energy [51,52].We used a double-ζ basis set for all atoms and core electrons were described using norm-conserving Troullier−Martins pseudopotentials.A real-space mesh cutoff of 500 Ry and a 64x64x10(64x64x1) Monkhorst−Pack k-point mesh was used for the bulk (monolayer) calculations.
Conversely, for Fe3GaTe2, J11 and J33 display values of 1.1 and -0.04 meV, respectively, indicating that the AF contribution of the in-plane couplings is almost suppressed, which in turn amplifies the net ferromagnetism.The values of Tc derived from the calculated exchange couplings and MAE are 644 K for Fe3GaTe2 and 135 K for Fe3GeTe2.While our results predict an overestimated Tc for Fe3GaTe2, they accurately reflect the comparatively higher critical temperature of Fe3GaTe2 over Fe3GeTe2.

Figure 1 .
Figure 1.(a) Lateral view of a unit cell of bulk Fe3GaTe2, which includes two single-layers.Colour code: Fe1,2 (red), Fe3 (pink), Ga (green) and Te (yellow).(b) Inter-plane exchange interactions J12, J13 (top panel) and in-plane couplings J11 and J33 (bottom panel) for bulk Fe3GaTe2 (red) and Fe3GeTe2 states (Figure 2(b)), one can observe that the system is metallic, with the d orbitals of the transition metal playing an important role around the Fermi level.In contrast, the contributions Ga and Te are almost negligible, predominantly from the p orbitals of these atoms.

Figure 2 .
Figure 2. (a) Phonon spectrum, (b) electronic band structure (left) and orbital-resolved density of states (right) of Fe3GaTe2 monolayer.Blue (red) colour in the band structure indicates spin up (down) states.
for further details).Based on the results of MAE and exchange parameters we computed the Tc at different levels of strain (Figure4(d)).For e = -4% we observe that besides J12, J13 and J33 are enhanced with respect the undistorted structure, the rapid decrease of J11 results in a drop of the Tc to a value of 320 K. Conversely, a 4% elongation yields a Tc reduction by 10%.

Figure 4 .
Figure 4. Evolution of (a) magnetic moments of metals and ligands (left and right panels, respectively), (b) MAE, (c) inter-(left) and in-plane (right) exchange parameters and (d) Tc of Fe3GaTe2 monolayer upon applied strain.

Figure 5 .
Figure 5. Evolution upon hole and electron doping of (a) magnetic moments of Fe, Ga and Te (left and right panels, respectively), (b) MAE, (c) inter-(left) and in-plane (right) exchange parameters and (d) Tc of Fe3GaTe2 monolayer.