Giant anomalous transverse transport properties of Co-doped two-dimensional Fe₃GaTe₂

Abstract

In spintronics, transverse anomalous transport properties have emerged as a highly promising avenue surpassing the conventional longitudinal transport behaviors. Here, we explore the transverse transport properties of monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. All the systems exhibit ferromagnetic ground states with metallic features and also have perpendicular magnetic anisotropy. Besides, the magnetic anisotropy is substantially enhanced with increasing Co-doping concentration. However, unlike magnetic anisotropy, the Curie temperature is suppressed by increasing the Co-doping concentration. For instance, the monolayer and bilayer Fe_2.917Co_0.083GaTe₂ hold a Curie temperature of 253 K and 269 K, which decreases to 163 K and 173 K in monolayer and bilayer Fe_2.67Co_0.33GaTe₂ systems, respectively. We find a giant anomalous Nernst conductivity (ANC) of 6.03 A/(K·m) in the monolayer Fe_2.917Co_0.083GaTe₂ at −30 meV, and this is further enhanced to 11.30 A/(K·m) in the bilayer Fe_2.917Co_0.083GaTe₂ at −20 meV. Moreover, the bilayer Fe_2.917Co_0.083GaTe₂ structure has a large anomalous thermal Hall conductivity (ATHC) of −0.14 W/(K·m) at 100 K. Overall, we find that the Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) structures have better anomalous transverse transport performance than the pristine Fe₃GaTe₂ system and can be used for potential spintronics and spin caloritronics applications.

Graphical abstract

Keywords

two-dimensional (2D) material / Fe₃GaTe₂ / ferromagnetism / magnetic anisotropy / Curie temperature / anomalous Hall conductivity / anomalous Nernst conductivity / anomalous thermal Hall conductivity

1 Introduction

Introducing ferromagnetism in two-dimensional (2D) materials through some external agencies, such as introducing defects [1, 2], doping [3], or making edges [1, 4] has been widely studied in the past. However, the ferromagnetism created in this way does not have long-range ordering. Thanks to the advancement in exfoliation and measurement techniques, it is possible to exfoliate van der Waals (vdW) layered magnetic materials from a few layers down to even monolayer limits and measure their magnetic properties experimentally [5, 6]. Similarly, with the development of growth techniques, it becomes possible to control the growth of atomically thin layers [7, 8]. These developments have brought extensive research interest in 2D magnetic materials. The real breakthrough started with the discovery of layer-dependent ferromagnetism in the monolayer CrI₃ and bilayer Cr₂Ge₂Te₆ [9, 10]. However, these materials possess relatively low Curie temperatures ~45 Kelvin (K). In this same family of the CrI₃ structure, monolayer CrCl_3, and CrBr₃ are also ferromagnetic semiconductors but with low Curie temperatures of ~17 K and ~31 K [11]. CrSe₂ is another experimentally synthesized material with metallic ferromagnetic (FM) order. This CrSe₂ shows a Curie temperature of 110 K in a 16-layer thick film. However, the Curie temperature decreases to 65 K in the monolayer thickness on the WSe₂ substrate [12]. For spintronics device applications, it is necessary to find a material that possesses a room-temperature ferromagnetism. In this aspect, Fe₃GeTe₂ has attracted extensive research interest because the bulk Fe₃GeTe₂ exhibited FM order with large out-of-plane magnetocrystalline anisotropy and a Curie temperature of ~220 K. It was also reported that the Curie temperature of 2D Fe₃GeTe₂ (130 K) could be enhanced to 230 K, and ~300 K through Fe-intercalation and ionic gating approach [13-15]. Recently, Fe₃GaTe₂ was also successfully synthesized down to a thickness of ~9.5 nm. A large Curie temperature of ~380 K and out-of-plane anisotropy were reported in the Fe₃GaTe₂ thin film [16].

Transport properties of magnetic materials play an important role in diverse spintronics applications and are widely studied on both experimental and theoretical sides. In the transport properties of magnetic materials, it is advantageous to utilize the transverse transport properties such as anomalous Hall conductivity (AHC), anomalous Nernst conductivity (ANC), and anomalous thermal Hall conductivity (ATHC) than the longitudinal transport properties [17, 18]. A key quantity among these transverse transport properties is the AHC because all the transverse transport properties are correlated to this AHC. The intrinsic AHC can be computed as a sum of the Berry curvature of the Bloch wave function over the occupied electronic states [19-21]. Extensive studies have been performed on the transport properties of bulk-type or 2D materials. For instance, layer-dependent ferromagnetism and intrinsic AHC were reported in 2D Fe₃GeTe₂ film down to the monolayer limit [22]. Besides, the strain-dependent room temperature ferromagnetism and AHC of ~400 S/cm was also found in monolayer Cr₂Ge₂Se₆ and Cr₂Ge₂Te₆ [23]. Furthermore, carrier-doping-induced ANC of −6.65 µV/K at 100 K was also theoretically reported in the FeCl₂ monolayer [24]. In addition, a large ANC of −14 A/(K·m) at 300 K was reported in the Co₂FeAl thin film of thickness ~10 nm, while the Co₂MnSi had a small ANC of 0.28 A/(K·m) at 300 K [25]. In a recent experimental study, an AHC of ~73 S/cm was reported in the Fe₃GaTe₃ thin film with a thickness of around 178 nm at 3 K. Note that the AHC of this relatively thick Fe₃GaTe₃ film shows a decreasing trend with increasing temperature [16]. Recently, we also calculated the thickness-dependent AHC of the Fe₃GaTe₂ in the thickness range from monolayer to four layers. A large AHC of around −490 S/cm was found in a four-layer Fe₃GaTe₂ system [26]. This implies that the AHC displays strong thickness-dependent characteristics. Since the experimental study shows that the magnetic properties of the Fe₅GeTe₂ flakes can be tuned by Co doping [27], we expect that the transverse transport properties of the Fe₃GaTe₂ system can also be affected by Co doping. Thus, in this report, we aim to systematically explore the anomalous transverse transport properties of Co doped Fe₃GaTe₂ by changing the Co doping concentration in both monolayer and bilayer structures, Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. Herein, we will propose that the anomalous transverse transport properties can be substantially enhanced in atomically thin bilayer Fe_3−xCo_xGaTe₂ layer systems compared with that found in the pristine four-layer Fe₃GaTe₂ system.

2 Numerical method

The concentration-dependent structural, electronic, and magnetic properties of monolayer and bilayer Co-doped Fe_3−xCo_xGaTe₂ systems are investigated using the Vienna ab initio simulation package (VASP) [28, 29]. We consider four different doping concentrations; for instance, x = 0.083, 0.167, 0.250, and 0.330 in monolayer and bilayer Fe_3−xCo_xGaTe₂ systems. We apply the generalized gradient approximation of Perdew−Burke−Ernzerhof (PBE) as exchange and correlation functional [30]. A slab geometry with a vacuum distance of more than 12 Å in the z-direction is applied to simulate all the layered structures. The structures are fully optimized after Co-doping without any constraints using the conjugate gradient method. A plane-wave basis set with an energy cutoff of 650 eV is used and the force and energy convergence criteria are set to 0.02 eV/Å and 10⁻⁶ eV for all the calculations. A Γ-centered k-points mesh of 11 × 11 × 1 is used for structure relaxation and electronic calculations. Furthermore, due to the weak van der Waals interaction between layers, we include the dispersion correction using the DFT-D3 method with Becke−Johnson damping function [31, 32]. To calculate the magnetocrystalline anisotropy energy (MAE), we utilize a non-collinear total energy method including the spin−orbit coupling (SOC) [33]. The MAE is determined by calculating the total energy difference between in-plane [001] and out-of-plane [100] magnetization directions. For these calculations, we employ a denser k-mesh of 15 × 15 × 1. The Metropolis Monte Carlo (MC) simulations are used to calculate the Curie temperature (T_C) [34, 35]. Furthermore, the Wannier interpolation technique is used to obtain the intrinsic Berry curvature contribution to the anomalous Hall conductivity (AHC). The Wannier functions are extracted from band structures using the maximally localized Wannier function method [36]. For calculating the AHC, we used the numerical tight-binding model Hamiltonian and employed the linear response Kubo formula approach [37]. A denser k-mesh of 150×150×1 is used to accurately construct real-space Wannier functions for calculating the AHC using a tight-binding Hamiltonian. The anomalous Nernst conductivity (ANC) is determined at 100 K using the Mott relation [38, 39].

3 Results and discussion

Herein, we investigate the magnetic properties of the Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems for monolayer and bilayer thickness. Fig.1(a) and (b) show the top and side views of the optimized crystal structure of the pristine Fe₃GaTe₂ monolayer (2 × 2 × 1 supercell), respectively. The orange, green, and purple spheres represent the Fe, Ga, and Te atoms, respectively. For x = 0.083 doping concentration, we replace one Fe with Co in this (2 × 2 × 1) supercell. Since the Fe₃GaTe₂ monolayer has three sublayers, we consider two doping positions represented by A and B in Fig.1(b); by replacing one Fe with one Co in the (i) central Fe layer and (ii) in the upper Fe layer (or bottom). According to our total energy calculations, the A-site doping is more stable than the B-site by an energy difference of 158 meV. For x = 0.167 doping concentration, we replace two Fe atoms with two Co atoms in the same supercell size as shown in Fig.1(c). Here, we consider four different combinations. For instance, one Fe is replaced with a Co atom at A and the other Fe with Co atoms at B (C, D, and E). We find that the (AC) configuration is more stable than the (AB), (AD), and (AE) by an energy difference of 2 meV, 189 meV, and 180 meV. For x = 0.25 doping concentration, we replace three Fe atoms with three Co atoms as shown in Fig.1(d). Here, we consider four possible combinations, (ABC), (ACD), (ACE), and (ACF), respectively. For instance, we replace the Fe at A, B, and C sites with Co in (ABC) configuration. The (ABC) combination becomes the most stable one with an energy difference of 11 meV, 197 meV, and 213 meV compared with the (ACD), (ACE), and (ACF) combinations. For x = 0.33 doping concentration, we consider a unit cell of the pristine Fe₃GaTe₂ and replace one Fe either from the central (A) or from the upper (B) layer as shown in Fig.1(e). The central layer doping is more stable than the B site doping by an energy difference of 144 meV. Thus, we conclude that the Co always prefers the central layer in all doping concentrations in the monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330). The optimized lattice constants for all these stable crystal structures are provided in Tab.1.

Fig.1 Crystal structures (a) top view of the monolayer Fe2.917Co0.083GaTe2, (b) side view of the monolayer Fe2.917Co0.083GaTe2, (c) side view of the monolayer Fe2.833Co0.167GaTe2, (d) side view of the monolayer Fe2.75Co0.25GaTe2, (e) side view of the monolayer Fe2.67Co0.33GaTe2, (f) AA stacking in the bilayer Fe2.917Co0.083GaTe2, (g) AB stacking in the bilayer Fe2.917Co0.083GaTe2, (h) AA stacking in the bilayer Fe2.833Co0.167GaTe2, (i) AB stacking in the bilayer Fe2.833Co0.167GaTe2, (j) AA stacking in the bilayer Fe2.75Co0.25GaTe2, (k) AB stacking in the bilayer Fe2.75Co0.25GaTe2, (l) AA stacking in the bilayer Fe2.67Co0.33GaTe2, and (m) AB stacking in the bilayer Fe2.67Co0.33GaTe2.

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Tab.1 Lattice constants of monolayer and bilayer Fe3−xCoxGaTe2 (x = 0.083, 0.167, 0.250, and 0.330) along with interlayer distance (d) in the bilayer. The formation energy (EF) of monolayer and exchange energy (ΔEex) of monolayer and bilayer.

System	x = 0.083	x = 0.167	x = 0.25	x = 0.33
a = b, Monolayer (in Å)	8.155	8.161	8.165	4.089
a = b, Bilayer (in Å)	8.058	8.064	8.072	4.040
d (in Å)	3.11	3.02	3.00	2.97
EF molayer (meV/atom)	−166	−171	−175	−171
ΔEex(meV/cell) - Intralayer monolayer	983	884	760	581
ΔEex(meV/cell) - Interlayer bilayer	19	16	16	6

Using these monolayers Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330), we create a bilayer in every doping concentration. In bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems, we consider two stacking orders; AA- and AB-type of stacking. As an illustration, we present the AA- and AB-type stacking with x = 0.083 in Fig.1(f) and (g), respectively. In the AB-type stacking, the second layer is shifted by half of the lattice constant in the xy plane with respect to the first layer. For all Co doping concentrations, we find that the AB stacking is more stable than the AA type stacking. We present the details in Table S1 of the Electronic Supplementary Materials (ESM). The AB stacking order in the bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems is the same as that reported in the experimentally fabricated Fe₃GaTe₂ system. In Tab.1, we present the optimized lattice constants for bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems along with the interlayer distances. The lattice constants of monolayer and bilayer Fe_3−xCo_xGaTe₂ are slightly increasing with the increasing doping concentration, while the interlayer distance in the bilayer decreases with increasing Co concentration. Note that we also consider random site dopings as shown in Figs. S1(a)−(j) of the Electronic Supplementary Materials (ESM) for bilayer Fe_3−xCo_xGaTe₂ with x = 0.083 and x = 0.33. However, all random site dopings are less stable compared with the regular site dopings with energy differences in the range of 163−368 meV/cell in x = 0.083 and 123−270 meV/cell in x = 0.33 doping concentrations. In the bilayer Fe_3−xCo_xGaTe₂ with x = 0.167 and x = 0.25, there are also several random site doping configurations. However, the energy differences in both the lowest (x = 0.083) and highest (x = 0.33) Co doping concentrations are very large. Therefore, we do not consider the random site doping in bilayer Fe_3−xCo_xGaTe₂ with x = 0.167 and x = 0.25.

Based on these optimized crystal structures, we calculate the formation energy using the following relation:

(1)$E_{\mathrm{F}}={\displaystyle \frac{E_{\mathrm{T}}-{nE}_{\mathrm{F}\mathrm{e}}-{4E}_{\mathrm{G}\mathrm{a}}-{8E}_{\mathrm{T}\mathrm{e}}-{mE}_{\mathrm{C}\mathrm{o}}}{N}}.$

Here, E_T, E_Fe, E_Ga, E_Te, E_Co, n(m), and N represent the total energy of monolayer Fe_3−xCo_xGaTe₂, the chemical potential of the Fe atom, the chemical potential of the Ga atom, the chemical potential of the Te atom, the chemical potential of Co atom, number of Fe (Co) atoms, and the total number of atoms in the cell. The formation energy of monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) is also provided in Tab.1. The negative values of formation energies in all systems confirm their thermodynamical stability. Furthermore, we also calculate the phonon dispersion curves for monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) to confirm the dynamical stabilities using the finite difference approach [40]. Here, we consider a 2 × 2 × 1 supercell, and the force criterion for the ionic step is set to 0.001 eV/Å. The phonon band structures for Co doping in monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) are presented in Fig. S2 of the ESM. No trace of imaginary frequencies over the whole Brillouin zone is found in all the systems, indicating that monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems are dynamically stable.

After confirming the stabilities, we now discuss the magnetic properties of the monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. To find the magnetic ground state, we calculate the total energy difference between the antiferromagnetic (AFM) and ferromagnetic (FM) states (ΔE_ex = E_AFM−E_FM). In the monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems, the total energy difference is calculated by considering an intralayer exchange coupling, and the energy differences are 983, 884, 760, and 581 meV/cell. All the monolayer structures show the FM ground state. However, the exchange energy decreases with increasing Co concentrations. In the bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems, we consider both intra and interlayer exchange interactions. The bilayer structures have also FM ground states. Here, we adopt the intra-layer exchange interaction from the monolayer, while the interlayer exchange energy (ΔE_ex) is provided in Tab.1 for all bilayer systems. In the monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems, the local magnetic moment of the Fe atom has position dependency. For instance, the central sublayer (Fe_center) has a local magnetic moment of 1.48 μ_B, 1.55 μ_B, and 1.62 μ_B in monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, and 0.25) systems, respectively. In Fe_3−xCo_xGaTe₂ (x = 0.333), all the Fe atoms in the central layer are replaced with Co atoms. Therefore, the magnetic moment of Fe_central atom in this system is not listed here. In contrast, the top (or bottom) Fe atom (Fe_top(bottom)) has a higher local magnetic moment of 2.46 μ_B, 2.52 μ_B, 2.57 μ_B, and 2.60 μ_B at x = 0.083, 0.167, 0.250, and 0.330, respectively. On the other hand, the Co atom has a local magnetic moment of 0.81 μ_B, 0.83 μ_B, 0.85 μ_B, and 0.86 μ_B in the monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. Besides, a small negative magnetic moment of ~ −0.06 μ_B is induced in Te and ~ −0.08 μ_B in Ga atoms in monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. In the bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems, the local magnetic moments of both Fe and Co atoms have a similar trend as found in the monolayer systems. The detailed descriptions are given in Table S2 of the ESM.

We also investigate the atom-to-atom exchange interaction as a function of an interatomic distance using Green’s function method [41]. In this method, the Heisenberg Hamiltonian can be written as

(2)$H=-{{\displaystyle \sum }}_{ij}{J}_{ij}(m_i\cdot m_j),$

where m_i(m_j) and J_ij are the normalized magnetic moments at site i(j) and the Heisenberg isotropic exchange. In all the studied systems, we consider the pairs (i, j) with up to the third nearest neighbor (NN). We select seven different intralayer spin pairs; Fe_bottom−Fe_bottom (equivalently Fe_top−Fe_top), Fe_central−Fe_central, Fe_bottom−Fe_top, Fe_bottom−Fe_central (equivalently Fe_top−Fe_central), Fe_central−Co_central, Co_central−Co_central, and Fe_bottom−Co_central (equivalently Fe_top−Co_central) in the monolayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. In the bilayer structure, we consider both intralayer and inter-layer exchange interactions. As an illustration, we present the J_ij up to the third NN of the monolayer and bilayer Fe_2.917Co_0.083GaTe₂ system in Tab.2 and Tab.3. The positive (negative) exchange J values favor a ferromagnetic (antiferromagnetic) interaction. In the monolayer Fe_2.917Co_0.083GaTe₂ system, the dominant exchange coupling originates from the first NN Fe_bottom−Fe_top interaction followed by the first NN Fe_bottom−Fe_central and Fe_bottom−Co_central interaction as shown in Tab.2.

Tab.2 Exchange parameter (J) in meV monolayer Fe2.917Co0.083GaTe2.

Exchange interactions	J (1st NN) (meV)	J (2nd NN) (meV)	J (3rd NN) (meV)
Febottom−Febottom (Fetop−Fetop)	2.91	0.46	0.41
Fecentral−Fecentral	−1.58	−0.42	1.44
Febottom−Fetop	81.79	−2.65	1.66
Febottom−Fecentral (Fetop−Fecentral)	27.74	2.60	−1.33
Fecentral−Cocentral	−1.69	−0.22	−0.12
Cocentral−Cocentral	0.19	−0.01	0
Febottom−Cocentral (Fetop−Cocentral)	15.76	2.66	−1.20

Tab.3 Exchange parameter (J) in meV for bilayer Fe2.917Co0.083GaTe2.

Exchange interactions (intra-layer)	J (1st NN) (meV)	J (2nd NN) (meV)	J (3rd NN) (meV)
Febottom−Febottom (Fetop−Fetop)	0.53	0.90	0.51
Fecentral−Fecentral	−0.96	−0.15	−0.07
Febottom−Fetop	77.38	−1.23	0.65
Febottom−Fecentral (Fetop−Fecentral)	26.06	1.77	−0.68
Fecentral−Cocentral	−1.34	0.04	0.0
Cocentral−Cocentral	0.07	0.01	−0.05
Febottom−Cocentral (Fetop−Cocentral)	15.89	−0.84	−0.17
Exchange interactions (inter-layer)	J (1st NN) (meV)	J (2nd NN) (meV)	J (3rd NN) (meV)
Fetop (layer 1)−Febottom (layer 2)	0.09	0.24	−0.13
Fetop (layer 1)−Fecentral (layer 2)	0.54	0.27	−0.02
Fetop (layer 1)−Fetop (layer 2)	−0.07	−0.11	−0.04
Fetop (layer 1)−Cocentral (layer 2)	0.15	−0.04	−0.03
Fecentral (layer 1)−Cocentral (layer 2)	−0.22	0.02	0.01
Cocentral (layer 1)−Cocentral (layer 2)	−0.13	0.02	0.01
Febottom (layer 1)−Cocentral (layer 2)	0.07	−0.01	−0.01

In the bilayer Fe_2.917Co_0.083GaTe₂ system, the intra-layer interactions show the same trend as in the monolayer Fe_2.917Co_0.083GaTe₂ system, although the magnitudes are slightly changed. For instance, the main contribution is dominated by the first NN Fe_bottom−Fe_top exchange interactions with J ~ 77.38 meV as shown in Tab.3. The inter-layer exchange interaction is quite weak compared with the intra-layer interaction. For instance, it is only 0.54 meV for the first NN Fe_top (layer 1)−Fe_central (layer 2). Note that the effective J values for monolayer (18.34 meV) and bilayer (18.03 meV) Fe_2.917Co_0.083GaTe₂ systems using the Green’s function are in close agreement with (20.19 meV) and (17.77 meV) obtained from VASP. A similar feature is found in other systems as well.

Next, we calculated the spin-projected electronic band structures including the spin−orbit coupling (SOC) for the monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. Fig.2(a) and (b) show spin-projected band structures including SOC for monolayer and bilayer Fe_2.917Co_0.083GaTe₂ systems. The band structures for all the other systems are given in Figs. S3(a)−(f) of the ESM. The horizontal zero line represents the Fermi level. We find conventional metallic band structures in monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. For comparison, we also plot the spin projected band structure including SOC for pristine monolayer and bilayer Fe₃GaTe₂ systems in Figs. S4(a) and (b) of the ESM. We also calculate the magnetocrystalline anisotropy energy (MAE) due to SOC. All the systems show perpendicular magnetic anisotropy. The calculated results are presented in Tab.4. Note that the MAE of the pristine Fe₃GaTe₂ monolayer is 0.20 meV/atom (per Fe atom), while it is increased to 0.32 meV/atom in bilayer Fe₃GaTe₂ [26]. From Tab.4, it is clear that the MAE is largely enhanced with increasing Co concentration in both monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems.

Fig.2 Spin-projected band structures including spin−orbit coupling (a) monolayer Fe2.917Co0.083GaTe2, (b) bilayer Fe2.917Co0.083GaTe2.

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Tab.4 Magnetocrystalline anisotropy (in meV/atom) and Curie temperatures of monolayer and bilayer.

System	x = 0.083	x = 0.167	x = 0.25	x = 0.33
Monolayer MAE	0.49	0.75	1.07	1.48
Bilayer MAE	0.53	0.76	1.04	1.37
Monolayer TC (K)	253	230	200	163
Bilayer TC (K)	269	240	210	173

To understand this out-of-plane anisotropy, we analyze the SOC matrix elements of the monolayer Fe_2.917Co_0.083GaTe₂. Fig.3(a)−(d) show the SOC matrix elements of the Te, Fe-top, Fe-center, and Co atoms in monolayer Fe_2.917Co_0.083GaTe₂ respectively. Although the magnetic moment mainly originates from Fe and Co atoms, a small negative magnetic moment is found in the Te atom. Nonetheless, the dominant contribution to the MAE originates from the Te atoms. The SOC through the (p_x−p_y) orbitals of the Te atom results in the out-of-plane anisotropy as shown in Fig.3(a), while the contributions to the MAE from the Fe-top and Fe-center (Co) atoms are rather weak as shown in Fig.3(b)−(d). For comparison, the SOC matrix elements of Te, Fe-top, and Fe-center atoms in pristine Fe₃GaTe₂ monolayer is presented in Fig. S5 of the ESM. In Fig. S6 of the ESM, we also present the contribution to the total MAE of every system from Fe, Te, and Co atoms in different layers in monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.25, and 0.33) systems. We now calculate the Curie temperatures using the Metropolis Monte Carlo simulation. Here, we use the Heisenberg model Hamiltonian with single-ion magnetic anisotropy,

Fig.3 Orbital resolved MAE of monolayer Fe2.917Co0.083GaTe2 (a) Te atom, (b) Fe-top atom, (c) Fe-center atom, (d) Co atom.

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(3)$\hat{H}=-{{\displaystyle \sum }}_{i,j}J{\hat{m}}_i\cdot {\hat{m}}_j-k_2{{\displaystyle \sum }}_i{m}_z^2.$

Here, J represents the exchange parameter, while ${\hat{m}}_{i\left(j\right)}$, $k_2$, and $m_z$ are the magnetic moments (in µ_B) at sites i(j), single-ion anisotropy constant, and spin pointing along the easy axis, respectively. The exchange parameter J is calculated using the relation $J=\frac{E_{\mathrm{e}\mathrm{x}}}{{nm}^2}$. Here E_ex is the exchange energy difference between FM and AFM, n represents the number of magnetic atoms and m denotes the magnetic moment. A supercell of (50 × 50 × 1) with periodic boundary conditions in xy-plane is used to calculate the temperature-dependent magnetization curve. All the systems are equilibrated for time steps of 10 000 followed by a statistical average over another 10 000 steps to evaluate the average magnetization [34]. Fig.4(a)−(h) show the temperature-dependent magnetization curves of monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. For comparison, we present the temperature-dependent magnetization curves for pristine monolayer and bilayer Fe₃GaTe₂ systems in Figs. S7(a) and (b) of the ESM. The temperature-dependent magnetization curves are fitted with the Curie−Bloch equation $m\left(T\right)={\left(1-T/T_{\mathrm{C}}\right)}^{\beta }$. Here, T_C and β are the Curie temperature and critical exponent, respectively. The value of the β is ranging from 0.28 to 0.34 in all the systems. Note that the Curie temperature (T_C) is primarily affected by the exchange energy; higher exchange energy correlates with a higher T_C for the system, whereas the influence of the MAE on the Curie temperature is rather weak. The Curie temperature of monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems are given in Tab.2. The Curie temperatures of monolayer and bilayer Fe_2.917Co_0.083GaTe₂ are 253 K and 269 K, respectively. The Curie temperature is decreasing in both monolayer and bilayer systems with increasing the Co doping ratio. The decrease in Curie temperatures is due to the decrease in exchange energy in the presence of Co.

Fig.4 Temperature-dependent magnetization curves for (a) monolayer Fe2.917Co0.083GaTe2, (b) monolayer Fe2.833Co0.167GaTe2, (c) monolayer Fe2.75Co0.25GaTe2, (d) monolayer Fe2.67Co0.33GaTe2, (e) bilayer Fe2.917Co0.083GaTe2, (f) bilayer Fe2.833Co0.167GaTe2, (g) bilayer Fe2.75Co0.25GaTe2, (h) bilayer Fe2.67Co0.33GaTe2.

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We also calculate the transverse transport properties; anomalous Hall conductivity (AHC), anomalous Nernst conductivity (ANC), and anomalous thermal Hall conductivity (ATHC). By integrating the Berry curvature throughout the whole Brillouin zone (BZ), the AHC (σ_xy) can be found using the Berry phase theory and the linear response Kubo formula as given below:

(4)${\sigma }_{xy}=-{\displaystyle \frac{e^2}{\hslash }}{{\displaystyle \sum }}_n{\displaystyle {\int }_{BZ}^{BZ}{\displaystyle \frac{\mathrm{d}k}{{\left(2\pi \right)}^3}}{f}_n\left(k\right){\mathrm{\Omega }}_{n,z}\left(k\right),}$

where e, f_n, and ${\mathrm{\Omega }}_{n,z}\left(k\right)$ are the elementary charge, Fermi−Dirac distribution function, and Berry curvature. The Berry curvature can be calculated by the following relations:

(5)${\mathrm{\Omega }}_{n,z}\left(k\right)=-{\displaystyle \sum _n{f}_n{\mathrm{\Omega }}_n\left(k\right),}$

(6)${\mathrm{\Omega }}_n\left(k\right)=-2Im{\displaystyle \sum _{m\ne n}{\displaystyle \frac{⟨u_{nk}|{\nu }_x|u_{mk}⟩⟨u_{mk}|{\nu }_y|u_{nk}⟩}{{\left(E_{mk}-E_{nk}\right)}^2}}.}$

Here, ${\nu }_{x,y}$, u_nk(u_mk), and E_nk(E_mk) are the velocity operators, the periodic part of the Bloch wave function with eigenvalue E_nk(E_mk). The Berry curvature is calculated using the Wannier interpolation technique [36], in which a set of maximally localized Wannier functions (MLWFs) are constructed from the Bloch wave functions. A Fourier-transformed Wannier Hamiltonian is calculated using the projected Bloch wave functions onto these MLWFs. Fig.5(a) and (b) show the Co concentration dependent AHC of the monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. For comparison, we also present the AHC of the pristine structure. Compared with the AHC of the pristine monolayer structure, the magnitude of AHC is increased with Co doping at zero chemical potential (no carrier doping) although no monotonic behavior is observed as shown in Fig.5(a) inset. Particularly, the largest AHC of −294 S/cm is obtained in monolayer Fe_2.75Co_0.25GaTe₂. In the bilayer structure, the AHC is also increased compared with that of the pristine bilayer system. Unlike in the monolayer case, the AHC of the Co doped structures has no substantial change with increasing the Co concentration at zero chemical potential as shown in Fig.5(b) inset. The most noticeable behavior is found in the hole-doped bilayer system. Particularly, the bilayer Fe_2.917Co_0.083GaTe₂ shows an AHC of −578 S/cm at a small chemical potential shifting of just −10 meV. This value is substantially larger than those found in Fe₃GeTe₂ (360 S/cm) and Co₃Sn₂S₂ (150 S/cm) [42, 43]. Besides, it is also almost three times larger than the pristine bilayer Fe₃GaTe₂ at the same chemical potential. We also calculate the longitudinal electrical conductivity using the Boltzmann approach as implemented in the BoltzTrap2 code [44]. Since our systems are metallic, we apply the typical relaxation time of metallic systems (~10⁻¹⁴ s) to calculate the longitudinal electrical conductivity σ_xx of monolayer and bilayer Fe_3−xCo_xGaTe₂ systems. The electrical conductivity σ_xx as a function of chemical potential for all the systems is presented in Figs. S8(a)−(h) of the ESM. We find that the σ_xx from 1.4 × 10⁵ to 10.7 × 10⁵ S/cm. Note that it has been identified that the ${\sigma }_{xy}^{\mathrm{A}\mathrm{H}}$ is determined by scattering-independent or purely intrinsic Berry curvature if the σ_xx is ranged from 10⁴ to 10⁶ S/cm [45]. Since the longitudinal conductivities for all our studied systems are within this critical range, we infer that the dominant contribution to ${\sigma }_{xy}^{\mathrm{A}\mathrm{H}}$ stems from the intrinsic Berry curvature.

Fig.5 Thickness-dependent (a, b) anomalous Hall conductivity as a function of chemical potential for monolayer and bilayer Fe3−xCoxGaTe2 systems, (c, d) anomalous Nernest conductivity as a function of chemical potential for monolayer and bilayer Fe3−xCoxGaTe2 systems, and (e, f) anomalous thermal Hall conductivity as a function of chemical potential for monolayer and bilayer Fe3−xCoxGaTe2 systems.

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Fig.5(c) and (d) show the ANC at 100 K for monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems using the Mott relation [38, 39]:

(7)${\alpha }_{xy}=-{{\displaystyle \frac{{\pi }^2{k}_{\mathrm{B}}^{2}T}{3e}}{\displaystyle \frac{\mathrm{d}{\sigma }_{xy}}{\mathrm{d}\epsilon }}|}_{\epsilon =\mu }.$

Note that the ANC is related to the first derivative of the AHC. Fig.5(c) shows the ANC in monolayer systems. The largest ANC of −1.63 A/(K·m) is obtained in the Fe_2.75Co_0.25GaTe₂ system at zero chemical potential. Moreover, the ANC of the Fe_2.917Co_0.083GaTe₂ system is largely enhanced to 6.03 A/(K·m) at a small chemical potential of −30 meV. Fig.5(d) shows the ANC of the bilayer systems. Like in the monolayer case, the highest ANC of 1.79 A/(K·m) is found in the bilayer Fe_2.75Co_0.25GaTe₂ system at zero chemical potential. However, the ANC of the bilayer Fe_2.917Co_0.083GaTe₂ system is greatly enhanced to −6.52 A/(K·m) at −5 meV and 11.30 A/(K·m) at a chemical potential of −20 meV. This is almost twenty times larger than that found in the four-layer pristine Fe₃GaTe₂ [0.55 A/(K·m)] at −45 meV [26]. Besides, the obtained ANC at 100 K is also very large compared to the experimentally reported 2D Fe₃GeTe₂ thin film [0.3 A/(K·m) at 140 K] and also theoretically reported half metallic FeCl₃ monolayer [−0.45 A/(K·m) at 100 K] [24, 43].

We also calculate the ATHC which is a measure of the transverse thermal current generated in a material due to a longitudinal temperature gradient. The ATHC can be calculated using the following linear response relation [46]:

(8)${\kappa }_{xy}={\displaystyle \frac{1}{\hslash T}}{\displaystyle \int \mathrm{d}\epsilon \left[-{\left(\epsilon -\mu \right)}^2{\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\epsilon }}\right]{\sigma }_{xy}.}$

The Wiedemann−Franz (WF) law correlates both AHC and ATHC in the following relation:

(9)${\kappa }_{xy}=L_{xy}{\sigma }_{xy}T,$

where $L_{xy}$ having a value of ~2.44 × 10⁻⁸ Ω·W/K² is known as the Lorenz number. Fig.5(e) shows the ATHC of the monolayer systems at 100 K. Since the AHC and ATHC are correlated to each other, both AHC and ATHC display the same spectral shapes. At zero chemical potential, the ATHC is rather small ~ −0.07 W/(K·m) as maximum in the monolayer Fe_2.75Co_0.25GaTe₂ and bilayer Fe_2.917Co_0.083GaTe₂ systems as shown in Fig.5(e) and (f). However, the ATHC of the bilayer Fe_2.917Co_0.083GaTe₂ is greatly enhanced to −0.14 W/(K·m) at a chemical potential of −10 meV. This is larger than that found in four-layer pristine Fe₃GaTe₂ [−0.105 W/(K·m)] [26] and close to the reported giant ATHC in Weyl semimetal Co₃Sn₂S₂ [~0.14 W/(K·m)] at 80 K [17]. The AHC is a key factor for governing all the transverse transport coefficients (AHC, ANC, and ATHC). Therefore, we analyze the AHC. As a representative illustration, we select monolayer Fe_2.75Co_0.25GaTe₂ and bilayer Fe_2.917Co_0.083GaTe₂ systems. Fig.6(a) and (b) show the Berry curvatures along the high symmetry directions for the monolayer Fe_2.75Co_0.25GaTe₂ and bilayer Fe_2.917Co_0.083GaTe₂ systems at zero chemical potential [i.e., at the Fermi level (FL)]. The Berry curvatures for all other systems at FL are provided in Fig. S9 of the ESM. The positive and negative Berry curvature leads to positive and negative AHC. The negative and positive Berry curvature can also originate from the general k-points in the BZ, but here for a qualitative analysis, we only consider along the high symmetry lines. In the monolayer Fe_2.75Co_0.25GaTe₂ at the FL, two negative peaks and positive peaks are found in the Γ−K path as shown in Fig.6(a). The negative and positive Berry curvatures compensate resulting in a suppression of the AHC. Similarly, in the bilayer Fe_2.917Co_0.083GaTe₂ system at FL, a large negative peak is found from Γ to K point along with a comparatively small positive Berry curvature peak as shown in Fig.6(b). Thus, the AHCs of both mono (−294 S/cm) and bilayer (−311 S/cm) structures are almost the same. However, the negative peaks are substantially enhanced at a chemical potential of −40 meV in the monolayer Fe_2.75Co_0.25GaTe₂ and −10 meV in the bilayer Fe_2.917Co_0.083GaTe₂ system. Particularly, the enhancement of the Berry curvature in the bilayer is more noticeable. Thus, we find that the bilayer structure shows the most outstanding transverse transport properties.

Fig.6 Berry curvature along high symmetry lines for (a) monolayer Fe2.75Co0.25GaTe2 at zero chemical potential, (b) bilayer Fe2.917Co0.083GaTe2 at zero chemical potential, (c) monolayer Fe2.75Co0.25GaTe2 at −40 meV and (d) bilayer Fe2.917Co0.083GaTe2 at −10 meV.

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4 Conclusion

In summary, we investigate the magnetic ground state, magnetocrystalline anisotropy, Curie temperature, and transverse transport properties of monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330) systems. Both mono and bilayer structures have FM ground states with ordinary metallic behavior. All the systems show out-of-plane magnetic anisotropy with large magnitudes in both monolayer and bilayer Fe_3−xCo_xGaTe₂ (x = 0.083, 0.167, 0.250, and 0.330). For instance, both monolayer and bilayer Fe_2.917Co_0.083GaTe₂ have MAEs of 0.49 and 0.53 meV/atom, respectively, and these are greatly enhanced with increasing Co concentration. We obtain the MAEs of 1.48 meV/atom and 1.37 meV/atom in the monolayer and bilayer Fe_2.67Co_0.33GaTe₂, respectively. These MAEs are more than twice larger than those found in the pristine monolayer and bilayer Fe₃GaTe₂. The monolayer and bilayer Fe_2.917Co_0.083GaTe₂ systems have Curie temperatures of 253 K and 269 K, but the Curie temperatures are decreased with increasing Co concentration. For instance, these Curie temperatures are decreased to 163 K and 173 K in monolayer and bilayer Fe_2.67Co_0.33GaTe₂ systems. We find an AHC of −294 S/cm in the monolayer Fe_2.75Co_0.25GaTe₂ at zero chemical potential, but this is further enhanced to −387 S/cm at a small chemical potential of −40 meV. Similarly, an AHC of −311 S/cm is found in the bilayer Fe_2.917Co_0.083GaTe₂ system at zero chemical potential, and this is increased to −578 S/cm at −10 meV chemical potential. Besides, the ANC is largely enhanced to 6.03 A/(K·m) at −30 meV and 11.30 A/(K·m) at −20 meV in monolayer and bilayer Fe_2.917Co_0.083GaTe₂ systems respectively at 100 K. Furthermore, the ATHC in the bilayer Fe_2.917Co_0.083GaTe₂ structure at 100 K reaches −0.14 W/(K·m) at a chemical potential of −10 meV which is comparable to the giant ATHC of Weyl semimetal Co₃Sn₂S₂ [~0.14 W/(K·m)] at 80 K. Overall, we propose that the Fe_3−xCo_xGaTe₂ structure shows better anomalous transverse transport performance than the pristine Fe₃GaTe₂. These findings may stimulate further experimental verifications.

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Author contributions

J. H. conceived the idea of this study; I. K. performed DFT calculations; J. H. and I. K. wrote and revised the manuscript.

Electronic supplementary materials

The online version contains supplementary material available at https://doi.org/10.1007/s11467-024-1424-5 and https://journal.hep.com.cn/fop/EN/10.1007/s11467-024-1424-5. Supporting information includes the information about different random site Co doping in bilayer Fe_3−xCo_xGaTe₂, phonon dispersion curves for monolayer Fe_3−xCo_xGaTe₂, average local magnetic moments in bilayer Fe_3−xCo_xGaTe₂, spin projected band structures including SOC for monolayer and bilayer systems, MAE contributions, temperature dependent magnetization curves, thickness-dependent longitudinal conductivity, and Barry curvature along high symmetry lines for monolayer and bilayer Fe_3−xCo_xGaTe₂ systems.

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2022R1A2C1004440) and the Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2023-CRE-0190).

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