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## Layer-dependent magnetic phase diagram in FenGeTe2 (3 ≤ n ≤ 7) ultrathin films

14:38 10 junio in Artículos por Website

### 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.

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 La1−xSrxMO3 (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 ($${d}_{{x}^{2}-{y}^{2}}$$/dxy) and e2 (dxz/dyz). 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 $${d}_{{x}^{2}-{y}^{2}}$$, dxy, dxz, and dyz 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 $${d}_{{z}^{2}}$$, $${d}_{{x}^{2}-{y}^{2}}$$, dxy, dxz, and dyz orbitals of FenGeTe 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 dxz/dyz 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 dxz/dyz state of the Fe3+ ion will shift to lower energy. The corresponding number of occupied minority dxz/dyz states increases from 0.35 to 0.54. Therefore, the resulting calculated net magnetic moments are 3.0 μB and 2.6 μB for Fe3+ and Fe2+ ions, respectively. Similar to Fe ion in Li3FeN3, (PMe3)2FeCl3 and FePc/Ti3C2Tx compounds32,33,34, 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 μB and 1.6 μB 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 TC25. 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 La1−xSrxMnO3 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 dxz/dyz and $${d}_{{x}^{2}-{y}^{2}}$$/dxy 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.

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.

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 dxz and dyz orbitals, we consider mainly the coupling between $${d}_{{z}^{2}}$$ and dxz/dyz orbitals. Roughly speaking, the positive contributions to the total MAE originate mainly from unoccupied $${d}_{{z}^{2}}$$ orbitals and half-occupied dxz/dyz orbitals of Fe3+ ions, while the coupling of occupied $${d}_{{z}^{2}}$$ and unoccupied dxz/dyz 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 TC 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, TC is drastically reduced. In particular, the gate-tunable electrons sequentially fill the sub-band origin from the Fe-$${d}_{{z}^{2}}$$, dxz, and dyz orbitals, inducing room-temperature ferromagnetism in Fe3GeTe214. The value of TC depends mainly on the interaction between $${d}_{{z}^{2}}$$, dxz, and dyz orbitals, consistent with our previous discussion. Moreover, the transition from itinerant to localized magnetism increases TC, indicating that the FM coupling in Fe3GeTe2 comes mainly from localized double exchange (P1 path).

### Structure and magnetic behavior of FenGeTe2 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, TC 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 Fe3−xGeTe246, indicating that TC 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 FenGeTe2 system. To satisfy these requirements, we have designed a series of Fe-rich FenGeTe2 (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 FenGeTe2 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 FenGeTe2 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 FenGeTe2, 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(FenGeTe2) is the total energy of the 2D FenGeTe2 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 FenGeTe2 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 FenGeTe2 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 experiments11,12,49. Therefore, we have proposed that our predicted FenGeTe2 films could be grown on the surface of hexagonal Si phase. The calculated lattice mismatches between Si(001) and (5 × 5) FenGeTe2 superlattices are 0.5%, 3.7%, 0.7%, and 1.1% for Fe4GeTe2, Fe5GeTe2, Fe6GeTe2, and Fe7GeTe2, respectively. The optimized structures of FenGeTe2/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 FenGeTe2 superlattices.

We further discuss the electronic and magnetic properties of the proposed FenGeTe2 ultrathin films. Similar to 2D Fe3GeTe2, all the FenGeTe2 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 FenGeTe2 (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 FenGeTe2 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 FenGeTe2 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 FenGeTe2 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 μB for x = 0 to 2.73 μB for x = 1. This observation can be easily understood on the basis that Fe3+ ions contribute a larger magnetic moment than Fe2+ ions.

Because the TC in FenGeTe2 systems is determined mainly by localized double exchange, we consider the exchange parameters J of FenGeTe2 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 FenGeTe2 (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 FenGeTe2 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 dxz/dyz 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 dxz/dyz orbitals (I2) are stronger than the intralayer interactions (I1 path) in FenGeTe2 systems. Second, for all the FenGeTe2 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 dxz/dyz 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 FenGeTe2 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 FenGeTe2 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 dxz/dyz 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 dxz/dyz pairs by Eq. (4) (see Supplementary Fig. 10). When the Fe2+/Fe3+ ratio is 0 (Fe5GeTe2), there exist only positive contributions of occupied dxz/dyz 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 dxz/dyz and unoccupied $${d}_{{z}^{2}}$$ pairs and unoccupied dxz/dyz and occupied $${d}_{{z}^{2}}$$ pairs become 3.79 and 4.65 eV, respectively. Therefore, for the occupied dxz/dyz 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 FenGeTe2 is estimated using the 2D Heisenberg model, as shown in Fig. 4a. We have also simulated the MT curves for every Fe sublattice (see Supplementary Fig. 11). The obtained TC value of 138 K for 2D Fe3GeTe2 coincides well with previous experimental values of about 68–130 K13,14. 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 TC for bulk Fe3GeTe2 crystal is 280 K (see Supplementary Note 5), which is also comparable to the experimental value of 230 K27.

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 TC value of 68 K. One should note that this TC 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 TC (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 TC = 320 K. Such complicated magnetic behavior of Fe5GeTe2 has also been discussed in previous papers. For example, Ramesh et al.51 found that the Fe5−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 Fe5−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 TC of 570 K. The Néel’s R- and P-type magnetization profiles seen here have also been reported in mixed-valence complex alloys (Mn1.5FeV0.5Al52 and Mn2V0.5C0.5Z53), complex oxides (NiCo2O454), layered materials (AFeIIFeIII(C2O4)355) and core–shell nanoparticles56. The competition between interlayer AFM and FM coupling resulting in the transition from FM to FiM states in FenGeTe2 is derived from the coexistence of different electronic states.

The simulated values of TC 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, TC 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 TC of the FiM FenGeTe2 film increases with n, mainly owing to the higher MAE and stronger double exchange. A similar trend has also been observed in FM Fe3−xCrxGe and Fe3−xVxGe alloys57,58. Extrapolating to even thicker films, FenGeTe2 with n = 9 and an effective thickness of 14 Å yields a TC = 1006 K, which is comparable to the TC = 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 TC for various reported Fe-based compounds (Fig. 4b). In the three-dimensional (3D) compounds, the TC 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, TC is determined by a combination of composition and dimensional effects. Generally speaking, incorporation of Fe atoms into the system will increase TC. For example, Fe doping generates long-range spin ordering in GeTe films, and the TC of Fe0.18Ge0.82Te films is 100 K61. The TC 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 (TC = 220 K)12, even with the same Fe content. Moreover, the TC of Fe6GeTe2 (450 K) is slightly lower than that of the bulk Fe2Ge phase (TC = 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 TC.