Layer-dependent magnetic phase diagram in Fe_nGeTe₂ (3 ≤ n ≤ 7) ultrathin films

Abstract

Two-dimensional (2D) ferromagnets with high Curie temperature T_C are desirable for spintronics applications. However, they are rarely obtained in experiments mainly due to the challenge of synthesizing high-quality 2D crystals, and their T_C values are below room temperature. Using first-principles calculations, we design a family of stable 2D Fe_nGeTe₂ (4 ≤ n ≤ 7) ultrathin films with coexisting itinerant and localized magnetism. Among them, 2D Fe₃GeTe₂ and Fe₄GeTe₂ are ferromagnetic metals with T_C = 138 and 68 K; 2D Fe₅GeTe₂, Fe₆GeTe₂, and Fe₇GeTe₂ are Néel’s P-, R-, and R-type ferrimagnetic metals with T_C = 320, 450, and 570 K. A thickness-induced magnetic phase transition originates from competition between itinerant and localized states, and also correlates with Fe³⁺ and Fe²⁺ content. A valence/orbital-dependent magnetic exchange model is proposed for these effects. Our results reveal a universal mechanism for magnetic coupling in complex magnetic systems.

Introduction

Since the first successful exfoliation of monolayer CrI₃ and bilayer CrGeTe₃ sheets, the family of 2D magnetic materials has undergone tremendous growth during the past few years. At present, the range of 2D magnets covers insulators, semiconductors, half-metals, and metals. Among these, those of most interest are ferromagnetic (FM) semiconductors, such as CrX₃, NiX₃, CrGeTe₃, and RuX₃ (X = Cl, Br and I)^1,2,3,4,5,6. Magnetism in these compounds originates from localized d electrons, and the magnetic ordering is usually mediated by superexchange interaction between the magnetic ions through the nonmetal atoms. The semi-empirical Goodenough–Kanamori–Anderson (GKA) rules provide a valuable picture to describe the magnetic interactions in these 2D compounds^7,8,9. According to the GKA rules, the ferromagnetism in 2D semiconductors is mainly derived from 90° d–p–d superexchange. In this situation, the occupied orbitals overlap with different orthogonal p orbitals of a ligand. It is thus not surprising that weak ferromagnetism is usually found in these systems. As representatives, the observed Curie temperatures T_C of the 2D compounds CrI₃ and CrGeTe₃ are 45 K and 30 K, respectively, which are far below room temperature^1,4.

As well as these FM semiconductors, metallic ferromagnets are another important class of 2D ferromagnets. A significant advantage of metallic ferromagnets is that their metallic nature enables an interplay between spin and charge degrees of freedom, which are the main concern in spintronics¹⁰. The reported metallic ferromagnets, such as CrTe_x, Cr₂BC, FeSe₂, FeTe, MnSe, and Fe_nGeTe₂, exhibit robust ferromagnetism with high T_C (130–846 K)^{11,12,13,14,15,16,17,18,19,20,21}. In particular, 2D metallic Fe–Ge–Te ternary (FGT) compounds with high T_C and huge magnetic anisotropy energy (MAE) along the c axis have attracted attention. Among FGT thin films, 2D Fe₃GeTe₂ was first obtained by cleaving Fe₃GeTe₂ bulk crystal onto a gold film evaporated on top of an SiO₂/Si substrate. The polar reflective magnetic circular dichroism measurement confirmed that the T_C was preserved at 68–130 K, with an MAE value of ~2.0 meV at the monolayer limit^13,14. Subsequently, Kim et al.¹⁵ successfully synthesized and exfoliated seven-layer Fe₄GeTe₂ flakes with 7 nm thickness and determined that the T_C was about 270 K. However, the observed MAE was reduced from 1.03 to 0.23 J cm⁻³ when the composition changed from Fe₃GeTe₂ to Fe₄GeTe₂¹⁵. Another important member of the FGT family is Fe₅GeTe₂, and mechanically exfoliated Fe_5−xGeTe₂ nanoflakes have been found to be metallic ferromagnets with a high T_C of 270–332 K^16,22.

Spontaneous magnetization in most 2D metallic ferromagnets is generally accepted to be due to itinerant electrons, which can be understood in terms of the well-known Stoner model²³. The electrons behave ferromagnetic just because of their repulsive Coulomb interaction, while the contributions from lattice and band structure are totally ignored. Beneficial from the delocalized electrons, most of the reported metallic ferromagnets have higher T_C than FM semiconductors. However, there is a considerable amount of evidence suggesting that metallic FGT systems are not conventional Stoner ferromagnets^14,24,25. Itinerant magnetism cannot fully explain the variation of T_C in FGT systems. For example, Dai et al.²⁴ found that increased hydrostatic pressure led to enhanced electron itinerancy but decreased T_C in thin Fe₃GeTe₂ flakes. Yang et al.²⁵ noted that the band dispersions of Fe₃GeTe₂ barely changed upon heating towards the ferromagnetic transition near 225 K, which also represents a strong deviation from the itinerant Stoner model. Deng et al.¹⁴ used the localized Heisenberg model to estimate the magnetic properties of 2D Fe₃GeTe₂, which were consistent with experimental results. All these results suggest that local magnetic moments may play a crucial role in the FM ordering of Fe–Ge–Te systems. From the above experimental reports, we can conclude that 2D metallic FGT compounds are prospective candidates for room-temperature ferromagnets. However, there remain several unresolved issues, such as the physical origin of the localized magnetism in these metallic systems, the validity of the Stoner model and the Heisenberg model, the effects of composition and thickness, and the influence of magnetic anisotropy.

To resolve these issues, the electronic and magnetic properties of 2D Fe₃GeTe₂ are systemically investigated in this work. We find that the five 3d orbitals of 2D Fe₃GeTe₂ can be divided into two parts: there are a₁ (${d}_{{z}^{2}}$) states that are mostly localized on the Fe sites and give rise to local spin moments, whereas the other e₁ (d_xy/${d}_{{x}^{2}-{y}^{2}}$) and e₂ (d_xz/d_yz) states are itinerant. According to the orbital occupation behavior of the localized a₁ (${d}_{{z}^{2}}$) states and the different coordination environments, we infer the valence states of Fe atoms in Fe₃GeTe₂ are +2 and +3, respectively. For localized spins on Fe atoms, we propose a valence-dependent multipath magnetic coupling mechanism to describe the competition between the interlayer ferromagnetism and antiferromagnetism, while the itinerant e₁ and e₂ states always favor intra- and interlayer ferromagnetism in 2D Fe₃GeTe₂. Furthermore, the MAE also depends on the valence state of the Fe ions and originates from the coupling between a₁ and e₂ states. Based on these findings, we construct a series of 2D Fe_nGeTe₂ ultrathin films (4 ≤ n ≤ 7) with different Fe contents and thicknesses. The combined effects of these differences on magnetic moment, magnetic exchange parameters, and MAE are discussed. We find that the T_C value of Fe_nGeTe₂ ultrathin films does indeed depend on the competition between localization and itinerant magnetism. Interesting thickness-induced magnetic phase transformations from ferromagnetism to Néel’s P-type ferrimagnetism and then to R-type ferrimagnetism are observed in 2D Fe_nGeTe₂ films, whose T_C are in the range of 68–570 K. Our results not only reveal a route for the design of 2D intrinsic magnets with high Curie temperature, but also provide a universal theoretical model for analyzing the itinerant and localized magnetism in complex materials.

Results

Coexistence of localized and itinerant magnetism in 2D Fe₃GeTe₂

The atomic configuration of 2D Fe₃GeTe₂ is shown in Fig. 1. Each Fe₃GeTe₂ unit has a thickness of five atomic layers. Clearly, there are two types of Fe atoms with different coordination environments, namely, trivalent iron (Fe³⁺) and divalent iron (Fe²⁺). The middle of the 2D Fe₃GeTe₂ is a Fe²⁺Ge layer, sandwiched by bottom and top Fe³⁺ layers. The entire surface of each Fe³⁺ layer is then covered by an atomic layer of Te. The corresponding ratio of number of Fe³⁺ and Fe²⁺ is 2:1. The 2D Fe₃GeTe₂ is metallic¹³, as can be seen from both the electronic band structure and the total density of states (TDOS) in Supplementary Note 1. Nonmagnetic (NM), FM, and antiferromagnetic (AFM) states are all considered, to determine their ground spin configurations. The corresponding FM and various AFM configurations are shown in Supplementary Note 2. Our results indicate that 2D Fe₃GeTe₂ has an FM ground state.

Fig. 1: Crystal structure of Fe_nGeTe₂ (4 ≤ n ≤ 7).

a Schematic illustration of Te-substituted Fe₇Ge₄ crystal and of five structures in the series Fe_nGeTe₂. The green, purple and gold spheres correspond to Te, Ge, and Fe atoms, respectively. b Stacked plane views along the [001] direction of Fe_nGeTe₂ multilayer films with different Fe²⁺/Fe³⁺ ratios.

In addition to their coordination environment, the Fe³⁺ and Fe²⁺ ions are more accurately distinguished in 2D Fe₃GeTe₂ by their different electronic behaviors, which is confirmed by the charge density distributions and the Bader charge on Fe atoms²⁶ (see Supplementary Notes 1 and 4) Electrons are more strongly localized around the Fe²⁺ ions than around the Fe³⁺ sites. The electrons are localized between Fe²⁺ and Ge/Te ions, indicating covalent bonding characteristics. By comparison, more delocalized ionic bonding takes place between Fe³⁺ and Ge/Te ions. This difference between localized and delocalized electron distributions around the Fe³⁺ and Fe²⁺ ions is also supported by deformation charge density analysis and Bader charge (see Supplementary Fig. 2 and Note 4). There is a net charge transfer of about 0.4 electron from each Fe³⁺ ion to its surrounding Ge/Te ions. However, there is no evident charge transfer in the case of Fe²⁺ ions. Following this picture, we find that the Fe²⁺ ions are localized relative to the Fe³⁺ ions in 2D Fe₃GeTe₂, which is consistent with a previous report by ref. ²⁷. The coexistence of localized and itinerant magnetism has also been found in iron-based superconductors and double perovskite materials, such as LaOFeAs, Sr₂FeMoO₆, and La_1−xSr_xMO₃ (M = Mn and Co)^28,29,30,31, all of which are polyvalent materials.

The partial density of states (PDOS) can provide further clarification of the origin of different electronic and magnetic features in Fe³⁺ and Fe²⁺ ions of 2D Fe₃GeTe₂, which are shown in Supplementary Note 1. Under a hexagonal crystal field, the five 3d orbitals of the Fe atom split into a single state a₁ (${d}_{{z}^{2}}$), two twofold-degenerate states e₁ (${d}_{{x}^{2}-{y}^{2}}$/d_xy) and e₂ (d_xz/d_yz). From the PDOS of Fe atoms, one can see that the ${d}_{{z}^{2}}$ orbital is clearly narrower and sharper than the other 3d orbitals, suggesting a localized feature. However, it is also obvious that the ${d}_{{x}^{2}-{y}^{2}}$, d_xy, d_xz, and d_yz orbitals in the minority-spin channels are obviously wide and are hybridized with Ge/Te-p states, indicating a delocalized feature. Similar to LaOFeAs, the localized d electrons differ from the itinerant electrons in coming from more isolated ${d}_{{z}^{2}}$ orbitals²⁸. Moreover, the 3d bands of the majority spin for both Fe³⁺ and Fe²⁺ ions are fully occupied, while those of the minority spin are partially occupied.

Based on the occupation matrix, the electron occupation numbers of the ${d}_{{z}^{2}}$, ${d}_{{x}^{2}-{y}^{2}}$, d_xy, d_xz, and d_yz orbitals of Fe_nGeTe ultrathin films are listed in Supplementary Note 4, which further confirms the valance states of Fe ions. The main difference between Fe³⁺ and Fe²⁺ ions in terms of the PDOS is due to the ${d}_{{z}^{2}}$ and d_xz/d_yz states in the minority-spin channels. Specifically, the electron occupation numbers in the minority ${d}_{{z}^{2}}$ orbitals are 0.04 and 0.30 for Fe³⁺ and Fe²⁺ ions, respectively. That is to say, the Fe²⁺ ion has ~0.3 more electrons than Fe³⁺ ion to occupy the minority ${d}_{{z}^{2}}$ orbital. The occupied minority ${d}_{{z}^{2}}$ orbital results in the Fe²⁺ ion being more localized than the Fe³⁺ ion. In addition, compared with the Fe²⁺ ion, the energy level of the minority d_xz/d_yz state of the Fe³⁺ ion will shift to lower energy. The corresponding number of occupied minority d_xz/d_yz states increases from 0.35 to 0.54. Therefore, the resulting calculated net magnetic moments are 3.0 μ_B and 2.6 μ_B for Fe³⁺ and Fe²⁺ ions, respectively. Similar to Fe ion in Li₃FeN₃, (PMe₃)₂FeCl₃ and FePc/Ti₃C₂T_x compounds^32,33,34, the magnetic moments values may suggest that both Fe³⁺ and Fe²⁺ ions are probably in their intermediate spin states. The intermediate-spin state can become the ground states of the system due to the relative stability of the ligand hole states that it hybridizes with ref. ³⁵ A density functional theory (DFT) calculation by Zhu et al.³⁶ gave similar values of 2.5 μ_B and 1.6 μ_B for the magnetic moments of Fe³⁺ and Fe²⁺ ions, respectively, in bulk Fe₃GeTe₂. The localized a₁ and delocalized e₁/e₂ states result in the unique magnetic properties of 2D Fe₃GeTe₂, with coexistence of local and itinerant magnetism, consistent with a previous report by ref. ²⁵

Magnetic coupling mechanism of 2D Fe₃GeTe₂

Because of the coexistence of localized and itinerant magnetism, the magnetic behavior of metallic ferromagnetic Fe₃GeTe₂ will deviate from the itinerant Stoner model. The recent experiment by Yang et al.²⁵ indeed confirmed that metallic Fe₃GeTe₂ exhibits non-Stoner ferromagnetism. Yang et al. did not observe any considerable change in electronic structure with temperature, which is not consistent with expectations. According to the itinerant Stoner model, a ferromagnetic metal will exhibit a temperature-dependent exchange splitting that disappears above T_C²⁵. Moreover, Tovar et al.²⁹ used a corrected Stoner parameter to describe the magnetic behaver in polyvalent Sr₂FeMoO₆ and found evidence for the coexistence of localized and itinerant magnetism in this material too. The corrections for Landau diamagnetism to the Stoner parameter need to derived from experimental measurements²⁹. Therefore, we need a new model to describe the complicated ferromagnetism in Fe₃GeTe₂ systems.

In this paper, two main magnetic exchange mechanisms have been introduced in 2D Fe₃GeTe₂, namely, the itinerant magnetism between e₁ and e₂ electrons and the localized magnetism in a₁ spins. Therefore, we propose a multipath magnetic interaction mechanism to understand the localized magnetic exchange in 2D Fe₃GeTe₂. According to the splitting of Fe²⁺ and Fe³⁺ orbitals in the crystal field and the multilayer structure of 2D Fe₃GeTe₂, three possible exchange paths are considered. Figure 2 shows the exchange paths between unoccupied ${d}_{{z}^{2}}$ orbitals (Fe³⁺–Fe³⁺), between occupied ${d}_{{z}^{2}}$ orbitals (Fe²⁺–Fe²⁺), and from an unoccupied to an occupied ${d}_{{z}^{2}}$ orbital (Fe²⁺–Fe³⁺), respectively. The hopping from an occupied Fe-${d}_{{z}^{2}}$ orbital to an unoccupied Fe-${d}_{{z}^{2}}$ orbitals induces extremely strong FM coupling (path P₁), which occurs between magnetic ions in different oxidation states, i.e., double exchange³⁷. Double exchange plays an essential role in polyvalent ferromagnetic materials such as La_1−xSr_xMnO₃ that also exhibit both localized and itinerant magnetism³⁸. However, spin crossover between both unpaired Fe-${d}_{{z}^{2}}$ orbitals (path P₂) and paired Fe-${d}_{{z}^{2}}$ orbitals (path P₃) gives rise to an AFM interaction according to the Pauli exclusion principle. On the other hand, the non-spin-polarized PDOS (see Supplementary Note 1) shows that d_xz/d_yz and ${d}_{{x}^{2}-{y}^{2}}$/d_xy are mainly contributed at the Fermi level, and their lower kinetic energy makes them contribute to the itinerant ferromagnetism in 2D Fe₃GeTe₂. Therefore, the interaction between itinerant electrons in e₁ states favors intralayer FM (I₁), while the coupling between electrons in e₂ states favors interlayer FM (I₂). Of these, the intralayer FM is contributed only by itinerant electrons (I₁), but there is competition between interlayer FM (P₁ and I₂) and AFM (P₂ and P₃) coupling. This explains why there is some debate regarding Fe atoms behaving ferromagnetically or antiferromagnetically in Fe₃GeTe₂³⁹. Moreover, Fu et al.⁴⁰ have also found that the coexistence of localized and itinerant 3d electrons in BiFeO₃/SrTiO₃ superlattices and itinerant Fe-3d electrons tends to cause ferromagnetism.

Fig. 2: Magnetic exchange interaction in 2D Fe₃GeTe₂.

a Schematic representation of the splitting of d orbitals in Fe²⁺ and Fe³⁺ ions, respectively. b Three possible processes for the exchange between localized ${d}_{{z}^{2}}$ orbitals. c The corresponding three possible localized Fe–Fe exchange paths in the Fe₃GeTe₂ crystal. d Side view showing the magnetic exchange parameters J₁, J₂, and J₃ for the Fe–Fe coupling in Fe₃GeTe₂ crystal, together with the Fe–Fe exchange interaction paths for these parameters in monolayer Fe₃GeTe₂. e–g Schematic representations of the exchange parameters J₁, J₂, and J₃, respectively.

A local Heisenberg model can provide a good description of the FM ordering in the Fe₃GeTe₂ system¹⁴. In 2D Fe₃GeTe₂, there are three types of exchange interaction between Fe ions, corresponding to the first, second, and third nearest neighbor magnetic exchange constants J₁, J₂, and J₃, as shown in Fig. 2. The values of J₁, J₂, and J₃ for 2D Fe₃GeTe₂ can be extracted from the total energy difference between different spin orderings. As summarized in Table 1, the derived exchange interaction parameters are J₁ = −0.44 meV, J₂ = 3.27 meV, and J₃ = 0.47 meV. It is known that a positive J value favors FM ordering, while a negative J value favors AFM coupling. Therefore, the calculated J₁ of −0.44 meV yields weak AFM coupling, which occurs mainly through the path P₂. The path P₁ corresponds to strong FM coupling, with J₂ having a value of 3.27 meV. Moreover, itinerant magnetism (I₁ and I₂) gives a value of 0.47 meV for J₃, corresponding to long-range intralayer FM coupling. The coincidence between the magnetic interaction parameters and the effect of coexisting localized and itinerant magnetism suggests that our proposed magnetic interaction mechanism is valid for understanding the magnetic ground state of 2D Fe₃GeTe₂. Its validity is also verified by other theoretical results. For example, first-principles calculations by Hu et al.⁴¹ have shown that the stability of ferromagnetism can be greatly enhanced by tensile strain in Fe₃GeTe₂ monolayer. According to our picture, tensile strain will shorten the Fe³⁺–Fe²⁺ distance (P₁ path) but lengthen the other interatomic distances, which in turn will enhance FM double exchange between Fe³⁺ and Fe²⁺ ions.

Table 1 Structure and magnetic parameters in 2D Fe_nGeTe₂ (4 ≤ n ≤ 7).

MAE as an important parameter of ferromagnets counteracts thermal fluctuations and preserves long-range FM ordering⁴². From noncollinear calculations with inclusion of the spin–orbit coupling (SOC) effect, the MAE of 2D Fe₃GeTe₂ has been determined as 0.94 meV/Fe, favoring perpendicular anisotropy, whereas the previously reported value was 0.67 meV/Fe¹⁴. For comparison, the MAE of 2.5 meV/Fe in bulk Fe₃GeTe₂ is slightly higher. The physical origin of a positive MAE can be ascribed to the matrix element differences between the occupied and unoccupied spin-down d orbitals of the Fe atom⁴³. For the contributions from d electrons, all nonvanishing matrix elements will make nonnegligible contributions to the MAE. In a simple analysis, the matrix elements that are near the Fermi level in spin-down states are most important to the MAE. According to Eq. (4) in the Methods section, the contribution to MAE is dominated by the coupling of $\left\langle {xz},|,{L}_{z},|,{yz}\right\rangle$ and $\left\langle {xz},{yz},|,{L}_{x},|,{z}^{2}\right\rangle$. Owing to the degeneracy of the d_xz and d_yz orbitals, we consider mainly the coupling between ${d}_{{z}^{2}}$ and d_xz/d_yz orbitals. Roughly speaking, the positive contributions to the total MAE originate mainly from unoccupied ${d}_{{z}^{2}}$ orbitals and half-occupied d_xz/d_yz orbitals of Fe³⁺ ions, while the coupling of occupied ${d}_{{z}^{2}}$ and unoccupied d_xz/d_yz orbitals of Fe²⁺ ions make a negative contribution to the MAE. Such a mechanism also accounts for the variation in MAE for Fe₃GeTe₂ monolayer when the Fe³⁺ content is decreased by hole doping, as observed by Park et al.⁴⁴.

An experimental study by Hwang et al.⁴⁵ found AFM coupling between pristine Fe₃GeTe₂ layer and oxidized Fe₃GeTe₂ layers. Their DFT calculations further revealed that such AFM coupling mainly originates from the oxygen atoms located at the bilayer interface, while bilayer Fe₃GeTe₂ with oxygen atoms adsorbed on the top or bottom sites still preferentially exhibit an FM state. According to our localized Fe–Fe exchange model, the intermediate oxygen atoms could provide an oxygen-mediated P₂ path between two Fe₃GeTe₂ layers, thereby inducing AFM coupling. Dai et al.²⁴ reported a pressure-dependent phase diagram of Fe₃GeTe₂ thin flakes, with a magnetic transformation temperature from ferromagnetic to paramagnetic states of 203 K at 3.7 GPa and 163 K at 7.3 GPa. Moreover, the T_C showed a clear decreasing trend from 4.0 GPa to 7.3 GPa because of the reduced local magnetic moment and increased electronic itinerancy. On the one hand, the increased electronic itinerancy could weaken the localized double exchange (P₁ path). On the other hand, by analyzing structural characteristics, we found that the Fe³⁺–Te distance clearly decreases at pressures below 7 GPa. The corresponding Fe³⁺–Fe³⁺ exchange though the Te-mediated P₂ path is stronger. As a consequence of the weakened FM coupling and enhanced AFM coupling, T_C is drastically reduced. In particular, the gate-tunable electrons sequentially fill the sub-band origin from the Fe-${d}_{{z}^{2}}$, d_xz, and d_yz orbitals, inducing room-temperature ferromagnetism in Fe₃GeTe₂¹⁴. The value of T_C depends mainly on the interaction between ${d}_{{z}^{2}}$, d_xz, and d_yz orbitals, consistent with our previous discussion. Moreover, the transition from itinerant to localized magnetism increases T_C, indicating that the FM coupling in Fe₃GeTe₂ comes mainly from localized double exchange (P₁ path).

Structure and magnetic behavior of Fe_nGeTe₂ ultrathin films

The above discussions on the one hand again demonstrate the coexistence of itinerant and localized magnetism in the Fe₃GeTe₂ system. On the other hand, the interlayer competition between localized exchange coupling (paths P₁, P₂, and P₃) and itinerant electrons (I₁ and I₂) is also crucial in determining the nature of the magnetic ground states and the values of the Curie temperature and MAE of 2D Fe₃GeTe₂. Moreover, T_C has been found to increase from 143 K to 226 K when the Fe content is increased from 2.75 to 3.10 in bulk Fe_3−xGeTe₂⁴⁶, indicating that T_C is very sensitive to Fe content. These findings motivate us to explore new high-temperature Fe–Ge–Te systems with optimal Fe²⁺/Fe³⁺ ratio and thickness, in which the valences of Fe ions are related to the direction of MAE and the competition between localized and itinerant magnetism in the Fe_nGeTe₂ system. To satisfy these requirements, we have designed a series of Fe-rich Fe_nGeTe₂ (4 ≤ n ≤ 7) ultrathin films with various thicknesses (Fig. 1), which could exhibit abundant magnetism through more complicated competition between itinerant and localized magnetism in a multilayer structure. Similar to 2D Fe₃GeTe₂, these Fe_nGeTe₂ ultrathin films also belong to the P-3m1 space group. The effective thicknesses (Table 1) of Fe₄GeTe₂, Fe₅GeTe₂, Fe₆GeTe₂, and Fe₇GeTe₂ ultrathin films are 5.63 Å, 6.79 Å, 7.56 Å, and 8.73 Å, respectively, which are moderately larger than that of Fe₃GeTe₂ (5.14 Å). The atomic arrangements of Fe_nGeTe₂ ultrathin films can be regarded as six, seven, eight, and nine atomic layered thickness (001) surfaces of a Te-substituted Fe₇Ge₄ crystal⁴⁷. Fortunately, the atomic arrangement of a five atomic layered thickness Te-substituted Fe₇Ge₄ crystal is the same as that of the experimentally reported Fe₃GeTe₂ phase.

To further check the experimentally feasibility of Fe_nGeTe₂, we have calculated their formation energies, defined as

$${E}_{{{{{{\rm{f}}}}}}}=[E({{{{{{\rm{Fe}}}}}}}_{n}{{{{{{\rm{GeTe}}}}}}}_{2})-E({{{{{{\rm{Fe}}}}}}}_{2}{{{{{\rm{Ge}}}}}})-E({{{{{{\rm{Te}}}}}}}_{2})-({{{{{\rm{n}}}}}}-2)E({{{{{\rm{Fe}}}}}})]/n,$$

(1)

where E(Fe_nGeTe₂) is the total energy of the 2D Fe_nGeTe₂ compound, and E(Fe₂Ge), E(Te₂), and E(Fe) are the total energies of Fe₂Ge, Te, and Fe in their most stable bulk phases⁴⁸. The formation energies of four Fe_nGeTe₂ ultrathin films from our theoretical design are −0.03 eV/atom (n = 4), −0.11 eV/atom (n = 5), −0.05 eV/atom (n = 6), and −0.01 eV/atom (n = 7), which are comparable to the formation energy of −0.08 eV/atom for Fe₃GeTe₂. All these negative values indicate that the formation processes are exothermic. More importantly, we find that the total energy of our proposed Fe₅GeTe₂ ultrathin film is 0.24 eV per atom lower than that of the experimentally reported layered phase with the same stoichiometry¹⁶. It should be noted, however, that our DFT simulation results only mean that our proposed Fe₅GeTe₂ ultrathin film is energetically favorable than the experimentally reported one at 0 K. Anyway, the satisfactory stability of these Fe_nGeTe₂ ultrathin film implies that they are feasible from a theoretical point of view.

It is noteworthy that ultrathin films of Cr₂S₃, CrSe, and FeTe in a FM state have been synthesized by chemical vapor deposition and molecular beam epitaxy methods in previous experiments^11,12,49. Therefore, we have proposed that our predicted Fe_nGeTe₂ films could be grown on the surface of hexagonal Si phase. The calculated lattice mismatches between Si(001) and (5 × 5) Fe_nGeTe₂ superlattices are 0.5%, 3.7%, 0.7%, and 1.1% for Fe₄GeTe₂, Fe₅GeTe₂, Fe₆GeTe₂, and Fe₇GeTe₂, respectively. The optimized structures of Fe_nGeTe₂/Si (001) heterostructures are shown in Supplementary Note 3. Evidently, the lattice misfit due to the Si substrate does not cause noticeable structural distortion in 2D Fe_nGeTe₂ superlattices.

We further discuss the electronic and magnetic properties of the proposed Fe_nGeTe₂ ultrathin films. Similar to 2D Fe₃GeTe₂, all the Fe_nGeTe₂ systems are metallic, as can be seen from the electronic band structures in Supplementary Fig. 6. The orbital projected densities of states in Supplementary Fig. 7 demonstrate that the metallicity still originates from d orbitals of Fe atoms. The coexistence of itinerant and localized d electrons in Fe_nGeTe₂ (3 ≤ n ≤ 7) can be revealed by the Bader charge (see Supplementary Note 4) and the PDOS. The distributions of Fe²⁺ and Fe³⁺ ions vary with the thickness and composition of the 2D Fe_nGeTe₂ ultrathin films. With increasing Fe content, the Fe²⁺/Fe³⁺ (x) ratio is 0.5, 1.0, 0, 0.2, and 0.75 for n = 3, 4, 5, 6, and 7, respectively, which correspond to a progressive change in magnetic behavior from itinerant to localized. To investigate the ground states of Fe_nGeTe₂ ultrathin films, we consider FM and various AFM configurations (see Supplementary Fig. 8). Owing to the multilayer structure, the considered AFM configurations increase with increasing Fe content. From our DFT calculations, FM ordering in all Fe_nGeTe₂ systems is more favored than its AFM or NM counterparts. The magnetic moment as a function of x is plotted in Fig. 3a. With increasing Fe²⁺/Fe³⁺ ratio, the average magnetic moment per Fe atom decreases slightly from 3.18 μ_B for x = 0 to 2.73 μ_B for x = 1. This observation can be easily understood on the basis that Fe³⁺ ions contribute a larger magnetic moment than Fe²⁺ ions.

Fig. 3: Magnetic properties of multilayer Fe_nGeTe₂.

a Calculated magnetic anisotropy energy (MAE) and magnetic moment per atom for various Fe²⁺/Fe³⁺ ratios (x). The detailed data are listed in Table 1. b Exchange parameters for first, second, and third nearest neighbors (see Supplementary Fig. 9). The green, gold, gray, blue, and purple regions correspond to x = 0, 0.2, 0.5, 0.75, and 1.0, respectively.

Because the T_C in Fe_nGeTe₂ systems is determined mainly by localized double exchange, we consider the exchange parameters J of Fe_nGeTe₂ ultrathin films that are presented in Table 1 and Supplementary Note 5. Meanwhile, the long-range magnetic coupling with exchange parameter J = 5 instead of J = 3 is considered for Fe_nGeTe₂ (n = 5–7) with increasing Fe content (see Supplementary Note 5). For Fe²⁺/Fe³⁺ ratios up to 0.5, the magnitude and sign of the coupling are insensitive to the distance between magnetic ion pairs, and there obviously exists competition between localized and itinerant magnetism. As the Fe²⁺/Fe³⁺ ratio is increased further, localized magnetic exchange becomes dominant, and the magnitude decreases with the distance between Fe ions. Meanwhile, the variations in J₁, J₂, and J₃ can also be interpreted in terms of the magnetic interaction mechanism, as has earlier been established for 2D Fe₃GeTe₂.

To further clarify the magnetic ground states of 2D Fe_nGeTe₂ ultrathin films, the relationship between the exchange-path-dependent parameters J₁, J₂, and J₃ and the Fe²⁺/Fe³⁺ ratio x is displayed in Fig. 3b, from which we can deduce several arguments. First, because the exchanges through ${d}_{{z}^{2}}$ and d_xz/d_yz orbitals are dominant in multilayer structures, the interlayer localized ${d}_{{z}^{2}}$ orbital interactions (P₁, P₂, and P₃ paths) and the itinerant electron coupling of d_xz/d_yz orbitals (I₂) are stronger than the intralayer interactions (I₁ path) in Fe_nGeTe₂ systems. Second, for all the Fe_nGeTe₂ systems considered here, the dominant J parameter for FM coupling comes mainly from double exchange of localized ${d}_{{z}^{2}}$ orbitals (P₁ path) and coupling of itinerant electrons in d_xz/d_yz orbitals (I₂). However, the major J parameter for AFM ordering comes mainly from coupling between localized ${d}_{{z}^{2}}$ orbitals in Fe³⁺–Fe³⁺ and Fe²⁺–Fe²⁺ exchange (P₂ and P₃ paths). Therefore, the competition between interlayer AFM and FM coupling results from that between itinerant and localized magnetism in Fe²⁺–Fe²⁺ or Fe³⁺–Fe³⁺ coupling. In an Fe₅GeTe₂ ultrathin film, when the distance between interlayer Fe layers is shorter, the localized magnetic exchange through the P₂ path can compete with itinerant e₂ electrons. However, the itinerant e₂ electrons become dominant as the Fe–Fe distance increases, such that the value of J₄ becomes 5.9 meV. Subsequently, the itinerant magnetism weakens as the Fe–Fe distance continues to increase, with the value of J₅ becoming 0.1 meV. Two competing ferromagnetisms of localized and itinerant are responsible for these complicated behaviors of the magnetic exchange parameters. As the Fe²⁺/Fe³⁺ ratio increases, the itinerant behavior of d orbitals is weakened. That is to say, more localized P₁/P₃ paths (FM/AFM) and fewer I₂ (FM) appear. Therefore, there is no simple trend of variation of the J parameters.

The MAE values for all the Fe_nGeTe₂ ultrathin films with different Fe²⁺/Fe³⁺ ratios are also shown in Fig. 3a. One can see that MAE first increases from 0.91 meV/Fe atom for x = 0 (Fe₅GeTe₂) to 1.05 meV/Fe atom for x = 0.2 (Fe₆GeTe₂). Then, it decreases almost monotonically with increasing Fe²⁺/Fe³⁺ ratio in the mixed-valence Fe_nGeTe₂ compounds. As x further increases to 1, the easy axis flips from a perpendicular into an in-plane orientation. The amplitude and direction of magnetic anisotropy are affected by two competing factors simultaneously. One is the Fe²⁺/Fe³⁺ ratio. As we have discussed with regard to 2D Fe₃GeTe₂, the Fe³⁺ and Fe²⁺ ions contribute to positive and negative MAE, respectively. Another important factor is the interaction between ${d}_{{z}^{2}}$ and d_xz/d_yz orbitals, since the electronic band structures reveal that the spin-minority components of these orbitals are affected by SOC associated with the inserted Fe layers. To further unveil the origin of MAE enhancement from Fe₅GeTe₂ to Fe₆GeTe₂, we decompose the MAE into the coupling of ${d}_{{z}^{2}}$ and d_xz/d_yz pairs by Eq. (4) (see Supplementary Fig. 10). When the Fe²⁺/Fe³⁺ ratio is 0 (Fe₅GeTe₂), there exist only positive contributions of occupied d_xz/d_yz and unoccupied ${d}_{{z}^{2}}$ pairs, with a difference in orbital energy levels of about 4.08 eV, leading to an out-of-plane MAE of 0.91 meV/Fe atom. When the Fe²⁺/Fe³⁺ ratio is increased from 0 (Fe₅GeTe₂) to 0.2 (Fe₆GeTe₂), the energy level differences between occupied d_xz/d_yz and unoccupied ${d}_{{z}^{2}}$ pairs and unoccupied d_xz/d_yz and occupied ${d}_{{z}^{2}}$ pairs become 3.79 and 4.65 eV, respectively. Therefore, for the occupied d_xz/d_yz and unoccupied ${d}_{{z}^{2}}$ pairs, the positive contributions to MAE prevail over the negative contributions.

Based on the obtained magnetic exchange constants and MAE, the Curie temperature of Fe_nGeTe₂ is estimated using the 2D Heisenberg model, as shown in Fig. 4a. We have also simulated the M–T curves for every Fe sublattice (see Supplementary Fig. 11). The obtained T_C value of 138 K for 2D Fe₃GeTe₂ coincides well with previous experimental values of about 68–130 K^13,14. The temperature-dependent magnetic moments (i.e., M–T curves) for each type of Fe ion (Fe³⁺ and Fe²⁺) in Fe₃GeTe₂ compounds are also presented in Supplementary Fig. 11a. One can see that the magnetizations of both Fe³⁺ and Fe²⁺ sublattices indeed behave like ferromagnets. Additionally, the estimated T_C for bulk Fe₃GeTe₂ crystal is 280 K (see Supplementary Note 5), which is also comparable to the experimental value of 230 K²⁷.

Fig. 4: Thickness-dependent magnetic properties of multilayer Fe_nGeTe₂ and comparison with other Fe-rich ferromagnets.

a Calculated normalized magnetization of Fe atoms in Fe_nGeTe₂ as a function of temperature from Monte Carlo simulation. The M_max and T_C represent the maximum of magnetization and Curie temperature, respectively. b Ternary phase diagram of various Fe-rich compositions, together with the stoichiometric line of Fe:Ge:Te = n:1:2 (3 ≤ n ≤ 7). The color indicate the values of T_C. The squares and circles represent 3D and 2D structures, respectively.

For various stoichiometries, three kinds of M–T curves are observed. Similar to Fe₃GeTe₂, Fe₄GeTe₂ is also a true ferromagnet. The magnetic spin moments of all Fe atoms align in the same direction, and they decrease with increasing temperature, yielding a T_C value of 68 K. One should note that this T_C is significantly lower than the experimentally reported one (270 K), owing to differences in thickness and symmetry¹⁵. For Fe₅GeTe₂, the magnetic moment continues to increase with temperature, and full compensation is not observed anywhere in the entire temperature range. The maximum in spontaneous magnetization appears between 0 K and T_C (320 K). For each Fe sublattice, the Fe₁ and Fe₅ layers are thermally disturbed more easily, and their magnetic moments decrease almost linearly with increasing temperature, while the other three layers (Fe₂, Fe₃, and Fe₄) drastically decrease around 320 K. The correlation here between magnetization and temperature is characteristic of a Néel’s P-type ferrimagnet⁵⁰. The ferrimagnetic (FiM)-to-paramagnetic transition occurs at a critical temperature T_C = 320 K. Such complicated magnetic behavior of Fe₅GeTe₂ has also been discussed in previous papers. For example, Ramesh et al.⁵¹ found that the Fe_5−xGeTe₂ system exhibited a temperature-dependent FM-to-FiM phase transition, and existed glassy cluster behavior at low temperature. Li et al.²² performed spin dynamics simulations of Fe₅GeTe₂, the results of which support the existence of the magnetic transition but not that of a spin glass state. Compared with 2D Fe_5−xGeTe₂, Fe₆GeTe₂ and Fe₇GeTe₂ sheets exhibit a relatively rapid decline in magnetization within an intermediate range of temperatures, showing the characteristics of a Néel’s R-type ferrimagnet. From a careful analysis of the spin coupling strength, we speculate that the main feature distinguishing FiM Fe₅GeTe₂ from Fe₆GeTe₂ and Fe₇GeTe₂ is the existence of a frustration effect. In this situation, the magnetic moments of Fe₅GeTe₂ will be more sensitive to the thermal fluctuations induced by temperature.

The M–T curves exhibit monotonic decreases with increasing temperature. For the Fe sublattices of Fe₆GeTe₂, the spin moments of the Fe₁, Fe₃, Fe₄, Fe₅, and Fe₆ layers and those of the Fe₂ layers show opposite directions, although all the moments decrease with temperature. For the M–T curves of 2D Fe₇GeTe₂, the spin directions of the Fe₂, Fe₄, and Fe₅ layers and those of the Fe₁, Fe₃, Fe₆, and Fe₇ layers are opposite. The FiM-to-paramagnetic transition occurs at a T_C of 570 K. The Néel’s R- and P-type magnetization profiles seen here have also been reported in mixed-valence complex alloys (Mn_1.5FeV_0.5Al⁵² and Mn₂V_0.5C_0.5Z⁵³), complex oxides (NiCo₂O₄⁵⁴), layered materials (AFe^IIFe^III(C₂O₄)₃⁵⁵) and core–shell nanoparticles⁵⁶. The competition between interlayer AFM and FM coupling resulting in the transition from FM to FiM states in Fe_nGeTe₂ is derived from the coexistence of different electronic states.

The simulated values of T_C are 138 K for Fe₃GeTe₂, 68 K for Fe₄GeTe₂, 320 K for Fe₅GeTe₂, 450 K for Fe₆GeTe₂, and 570 K for Fe₇GeTe₂. For truly FM systems, T_C drops from 138 K for Fe₃GeTe₂ to 68 K for Fe₄GeTe₂ because of the flipping of out-of-plane MAE brought about by the increased ratio of Fe²⁺ ions. Further increases in Fe content lead to a transition of magnetic ordering from FM to FiM at Fe₅GeTe₂. For n ≥ 5, the T_C of the FiM Fe_nGeTe₂ film increases with n, mainly owing to the higher MAE and stronger double exchange. A similar trend has also been observed in FM Fe_3−xCr_xGe and Fe_3−xV_xGe alloys^57,58. Extrapolating to even thicker films, Fe_nGeTe₂ with n = 9 and an effective thickness of 14 Å yields a T_C = 1006 K, which is comparable to the T_C = 1043 K for pure Fe solid of bcc phase¹⁵ (see Supplementary Fig. 13).

To provide a more general view of the composition and dimensional effects on the magnetic behavior of Fe–Ge–Te systems, we plot a ternary phase diagram of T_C for various reported Fe-based compounds (Fig. 4b). In the three-dimensional (3D) compounds, the T_C of Fe-rich compositions increases monotonically from 279 K for FeGe⁵⁹ to 485 K for Fe₂Ge⁶⁰, and then to 1043 K for pure Fe¹⁵, revealing a prominent composition effect. In the 2D Fe–Ge–Te films, T_C is determined by a combination of composition and dimensional effects. Generally speaking, incorporation of Fe atoms into the system will increase T_C. For example, Fe doping generates long-range spin ordering in GeTe films, and the T_C of Fe_0.18Ge_0.82Te films is 100 K⁶¹. The T_C of our Fe₃GeTe₂ with an effective thickness of 0.86 nm (138 K) is lower than that of an FeTe ultrathin film with thickness 2.80 nm (T_C = 220 K)¹², even with the same Fe content. Moreover, the T_C of Fe₆GeTe₂ (450 K) is slightly lower than that of the bulk Fe₂Ge phase (T_C = 485 K). Both of them have an Fe content of 0.67. However, the role of nonmetal element (Ge and Te) inclusions in ultrathin Fe–Ge–Te films is very complicated. These inclusions can not only tune the chemical valence state and the electronic behavior of the variable element Fe, but also provide a crystal field to control the MAE, which will change the magnetic behavior and T_C.

Conclusion

We have designed a family of 2D Fe_nGeTe₂ ultrathin films with different Fe contents and thicknesses, which are experimentally more feasible than the reported 2D layered phase. By first-principles calculations, we have systemically studied their electronic and magnetic properties and have some important findings to obtain low-dimensional magnetic materials with high working temperature. All the 2D Fe_nGeTe₂ ultrathin films considered here are robust ferromagnetic/ferrimagnetic materials with magnetic transition temperatures of 68–570 K, which can be ascribed to the coexistence of itinerant and localized electronic states. The localized magnetism comes from the electrons in ${d}_{{z}^{2}}$ orbital, while the itinerant magnetism derives from electrons in d_xz/d_yz/d_xy/${d}_{{x}^{2}-{y}^{2}}$ orbitals. The coexistence of itinerant and localized electronic states is also correlated well with the values of many critical magnetic parameters, such as magnetic moment, exchange parameters, MAE, and T_C, all of which have been discussed in detail in this paper. Based on these results, we have proposed a localized Fe–Fe exchange Heisenberg model that provides a good description of the exchange between ${d}_{{z}^{2}}$ orbitals in Fe–Ge–Te systems. It may also be appropriate for application to variable-valence element-based magnetic compounds. Meanwhile, the itinerant magnetism introduced to our mechanism to explain the competitive intra- and interlayer ferromagnetism. Moreover, the established thickness-dependent magnetic order suggests the possibility of tuning the interlayer exchange energy of Fe–Ge–Te systems by changing the composition. The results of this study should prove very helpful in further understanding the modulating effect of thickness on 2D Fe_nGeTe₂ ultrathin films with variable-valence elements. They also indicate that 2D magnetic Fe_nGeTe₂ ultrathin films are promising candidates for future room-temperature spintronic applications.

Methods

Electronic band structure calculation

Our first-principles calculations were based on density functional theory (DFT) within the generalized gradient approximation (GGA)⁶², as implemented in the VASP code⁶³. The projected augmented wave (PAW) potential was used to describe ion–electron interaction⁶⁴. The energy cutoff of the plane-wave basis was set as 500 eV. A vacuum space of 20 Å thickness was added to avoid interaction between adjacent layers. During geometry optimization, a Monkhorst–Pack k-point mesh of 0.02 Å⁻¹ was chosen for sampling the 2D Brillouin zones. To remove self-interaction errors, the effective Hubbard U parameter (U = 4.3 eV) was included within the PBE+U framework, which is consistent with previous studies^65,66. Different U values at various Fe²⁺/Fe³⁺ ratios x were also tested, as shown in Supplementary Note 5. Our results shown that the best choice for all Fe_nGeTe₂ compounds is U = 4.3, although a different value would not affect our conclusions. Correction of van der Waals interactions using the DFT-D3 scheme⁶⁷ was included in the bulk Fe₃GeTe₂ calculations.

Magnetic parameters

To describe the magnetic properties of Fe_nGeTe₂ crystals, the magnetic anisotropy energy (MAE) is defined as

$${{{{{\rm{MAE}}}}}}={E}_{{{{{{\rm{tot}}}}}}}[100]-{E}_{{{{{{\rm{tot}}}}}}}[001],$$

(2)

where E_tot[100] and E_tot[001] are to the total energies of states whose magnetization direction is parallel and perpendicular to the basal plane, respectively⁶⁸. The MAE is determined by considering the SOC effect through noncollinear calculations.

In the present system, the minority spin states dominate the magnetic anisotropy, and so the MAE can be expressed as⁶⁹

$${{{{{\rm{MAE}}}}}}\approx \triangle {E}^{{{{{{\rm{dd}}}}}}}={\xi }^{2} \mathop{\sum} \limits _{{o}^{-}{u}^{-}}\frac{{\left|\left\langle {o}^{-},|,{L}_{z},|,{u}^{-}\right\rangle \right|}^{2}-{\left|\left\langle {o}^{-},|,{L}_{x},|,{u}^{-}\right\rangle \right|}^{2}}{{\varepsilon }_{{u}^{-}}{-\varepsilon }_{{o}^{-}}}.$$

(3)

Here, L_x and L_z are the x and z components of the angular momentum operator, and o and u denote the spin-down orbitals in the occupied and unoccupied states, respectively. From Eq. (3), we can see that $\triangle {E}^{{{{{{\rm{dd}}}}}}}$ is not only determined by the orbital character of the occupied states, but also depends on the coupling with the empty states and the splitting between them through the energy denominator. For a simple analysis, we decompose Eq. (3) into matrix elements with the Fe-d orbitals that predominant near the Fermi level in spin-down states, omitting the SOC constant⁷⁰. The MAE can then be expressed approximately as

$${{{{{{\rm{MAE}}}}}}}^{{{{{{\rm{dd}}}}}}}\approx \frac{{\left|\left\langle {xz},|,{L}_{z},|,{yz}\right\rangle \right|}^{2}}{{\varepsilon }_{{xz}}-{\varepsilon }_{{yz}}}-\frac{{\left|\left\langle {xz},{yz},|,{L}_{x},|,{z}^{2}\right\rangle \right|}^{2}}{{\varepsilon }_{{xz},{yz}}-{\varepsilon }_{{z}^{2}}}.$$

(4)

Monte Carlo simulations

Monte Carlo simulations were carried out to determine the magnetic transition temperatures. The Hamiltonian of the system was expressed as follows:

$${ {\mathcal H} }=-\mathop{\sum} \limits_{(ij)}{J}_{ij}{{{{{{\bf{S}}}}}}}_{i}\cdot {{{{{{\bf{S}}}}}}}_{j}-K \mathop{\sum} \limits_{i}{({{{{{{\bf{S}}}}}}}_{i}\cdot {{{{{{\bf{e}}}}}}}_{i})}^{2},$$

(5)

where S_i is the unit vector of the magnetic moment at site i, J_ij is the exchange-coupling constant between magnetic Fe atoms, K is the anisotropy constant, and e_i is the unit vector along the easy direction of the magnetic anisotropy. The parameters used in the MC simulations were obtained from first-principles calculations. To determine the Curie temperature, the magnetization M per atom and the specific heat C_m were calculated by

$$M=\frac{1}{N}\langle {[{(\mathop{\sum} \limits _{i}{S}_{i}^{x})}^{2}+{(\mathop{\sum} \limits _{i}{S}_{i}^{y})}^{2}+{(\mathop{\sum} \limits _{i}{S}_{i}^{z})}^{2}]}^{1/2}\rangle$$

(6)

and

$${C}_{{{{{{\rm{m}}}}}}}=\frac{\langle {E}^{2}\rangle -{\langle E\rangle }^{2}}{N{k}_{{{{{{\rm{B}}}}}}}{T}^{2}},$$

(7)

respectively. Here, N is the total number of magnetic Fe atoms, and k_B is the Boltzmann constant. The simulation supercells were constructed by 50 × 50 expansion of the unit cell. For each temperature, the first 10⁵ MC steps were discarded for thermal equilibration, and the successive 10⁵ MC steps were then used to collect data and determine the thermodynamic averages of given physical quantities. All thermodynamic properties were averaged over five different seed numbers.