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

Two-dimensional (2D) ferromagnets with high Curie temperature TC 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 TC values are below room temperature. Using first-principles calculations, we design a family of stable 2D FenGeTe2 (4 ≤ n ≤ 7) ultrathin films with coexisting itinerant and localized magnetism. Among them, 2D Fe3GeTe2 and Fe4GeTe2 are ferromagnetic metals with TC = 138 and 68 K; 2D Fe5GeTe2, Fe6GeTe2, and Fe7GeTe2 are Néel’s P-, R-, and R-type ferrimagnetic metals with TC = 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. 2D magnetism offers the prospect of ordered and controllable magnetic states at reduced dimensions and there is a desire to obtain materials which can realise these properties at room temperature. Here, the authors use first-principles calculations to demonstrate that a family of 2D FenGeTe-based ultrathin films with varying Fe content can exhibit a range of magnetic properties at elevated temperatures.

S ince the first successful exfoliation of monolayer CrI 3 and bilayer CrGeTe 3 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 CrX 3 , NiX 3 , CrGeTe 3 , and RuX 3 (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 CrI 3 and CrGeTe 3 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 spintronics 10 . The reported metallic ferromagnets, such as CrTe x , Cr 2 BC, FeSe 2 , FeTe, MnSe, and Fe n GeTe 2 , 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 Fe 3 GeTe 2 was first obtained by cleaving Fe 3 GeTe 2 bulk crystal onto a gold film evaporated on top of an SiO 2 /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 Fe 4 GeTe 2 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 Fe 3 GeTe 2 to Fe 4 GeTe 2 15 . Another important member of the FGT family is Fe 5 GeTe 2 , and mechanically exfoliated Fe 5−x GeTe 2 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 model 23 . 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 Fe 3 GeTe 2 flakes. Yang et al. 25 noted that the band dispersions of Fe 3 GeTe 2 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 Fe 3 GeTe 2 , 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 roomtemperature 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 Fe 3 GeTe 2 are systemically investigated in this work. We find that the five 3d orbitals of 2D Fe 3 GeTe 2 can be divided into two parts: there are a 1 (d z 2 ) states that are mostly localized on the Fe sites and give rise to local spin moments, whereas the other e 1 (d xy /d x 2 Ày 2 ) and e 2 (d xz /d yz ) states are itinerant. According to the orbital occupation behavior of the localized a 1 (d z 2 ) states and the different coordination environments, we infer the valence states of Fe atoms in Fe 3 GeTe 2 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 e 1 and e 2 states always favor intra-and interlayer ferromagnetism in 2D Fe 3 GeTe 2 . Furthermore, the MAE also depends on the valence state of the Fe ions and originates from the coupling between a 1 and e 2 states. Based on these findings, we construct a series of 2D Fe n GeTe 2 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 n GeTe 2 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 n GeTe 2 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 Fe 3 GeTe 2 . The atomic configuration of 2D Fe 3 GeTe 2 is shown in Fig. 1. Each Fe 3 GeTe 2 unit has a thickness of five atomic layers. Clearly, there are two types of Fe atoms with different coordination environments, namely, trivalent iron (Fe 3+ ) and divalent iron (Fe 2+ ). The middle of the 2D Fe 3 GeTe 2 is a Fe 2+ Ge layer, sandwiched by bottom and top Fe 3+ layers. The entire surface of each Fe 3+ layer is then covered by an atomic layer of Te. The corresponding ratio of number of Fe 3+ and Fe 2+ is 2:1. The 2D Fe 3 GeTe 2 is metallic 13 , 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 Fe 3 GeTe 2 has an FM ground state.
In addition to their coordination environment, the Fe 3+ and Fe 2+ ions are more accurately distinguished in 2D Fe 3 GeTe 2 by their different electronic behaviors, which is confirmed by the charge density distributions and the Bader charge on Fe atoms 26 (see Supplementary Notes 1 and 4) Electrons are more strongly localized around the Fe 2+ ions than around the Fe 3+ sites. The electrons are localized between Fe 2+ and Ge/Te ions, indicating covalent bonding characteristics. By comparison, more delocalized ionic bonding takes place between Fe 3+ and Ge/Te ions. This difference between localized and delocalized electron distributions around the Fe 3+ and Fe 2+ 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 Fe 3+ ion to its surrounding Ge/Te ions. However, there is no evident charge transfer in the case of Fe 2+ ions. Following this picture, we find that the Fe 2+ ions are localized relative to the Fe 3+ ions in 2D Fe 3 GeTe 2 , which is consistent with a previous report by ref. 27 . The coexistence of localized and itinerant magnetism has also been found in ironbased superconductors and double perovskite materials, such as LaOFeAs, Sr 2 FeMoO 6 , and La 1−x Sr x MO 3 (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 Fe 3+ and Fe 2+ ions of 2D Fe 3 GeTe 2 , 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 a 1 (d z 2 ), two twofold-degenerate states e 1 (d x 2 Ày 2 /d xy ) and e 2 (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 Ày 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 orbitals 28 . Moreover, the 3d bands of the majority spin for both Fe 3+ and Fe 2+ 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 Ày 2 , d xy , d xz , and d yz orbitals of Fe n GeTe ultrathin films are listed in Supplementary Note 4, which further confirms the valance states of Fe ions. The main difference between Fe 3+ and Fe 2+ 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 Fe 3+ and Fe 2+ ions, respectively. That is to say, the Fe 2+ ion has~0.3 more electrons than Fe 3+ ion to occupy the minority d z 2 orbital. The occupied minority d z 2 orbital results in the Fe 2+ ion being more localized than the Fe 3+ ion. In addition, compared with the Fe 2+ ion, the energy level of the minority d xz / d yz state of the Fe 3+ 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 Fe 3+ and Fe 2+ ions, respectively. Similar to Fe ion in Li 3 FeN 3 , (PMe 3 ) 2 FeCl 3 and FePc/Ti 3 C 2 T x compounds [32][33][34] , the magnetic moments values may suggest that both Fe 3+ and Fe 2+ 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 Fe 3+ and Fe 2+ ions, respectively, in bulk Fe 3 GeTe 2 . The localized a 1 and delocalized e 1 /e 2 states result in the unique magnetic properties of 2D Fe 3 GeTe 2 , with coexistence of local and itinerant magnetism, consistent with a previous report by ref. 25 Magnetic coupling mechanism of 2D Fe 3 GeTe 2 . Because of the coexistence of localized and itinerant magnetism, the magnetic behavior of metallic ferromagnetic Fe 3 GeTe 2 will deviate from the itinerant Stoner model. The recent experiment by Yang et al. 25 indeed confirmed that metallic Fe 3 GeTe 2 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 temperaturedependent exchange splitting that disappears above T C 25 . Moreover, Tovar et al. 29 used a corrected Stoner parameter to describe the magnetic behaver in polyvalent Sr 2 FeMoO 6 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 measurements 29 . Therefore, we need a new model to describe the complicated ferromagnetism in Fe 3 GeTe 2 systems.
In this paper, two main magnetic exchange mechanisms have been introduced in 2D Fe 3 GeTe 2 , namely, the itinerant magnetism between e 1 and e 2 electrons and the localized magnetism in a 1 spins. Therefore, we propose a multipath magnetic interaction mechanism to understand the localized magnetic exchange in 2D Fe 3 GeTe 2 . According to the splitting of Fe 2+ and Fe 3+ orbitals in the crystal field and the multilayer structure of 2D Fe 3 GeTe 2 , three possible exchange paths are considered. Figure 2 shows the exchange paths between unoccupied d z 2 orbitals (Fe 3+ -Fe 3+ ), between occupied d z 2 orbitals (Fe 2+ -Fe 2+ ), and from an unoccupied to an occupied d z 2 orbital (Fe 2+ -Fe 3+ ), 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 P 1 ), which occurs between magnetic ions in different oxidation states, i.e., double exchange 37 . Double exchange plays an essential role in polyvalent ferromagnetic materials such as La 1−x Sr x MnO 3 that also exhibit both localized and itinerant magnetism 38 . However, spin crossover between both unpaired Fe-d z 2 orbitals (path P 2 ) and paired Fe-d z 2 orbitals (path P 3 ) 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 Ày 2 /d xy are mainly contributed at the Fermi level, and their lower kinetic energy makes them contribute to the itinerant ferromagnetism in 2D Fe 3 GeTe 2 . Therefore, the interaction between itinerant electrons in e 1 states favors intralayer FM (I 1 ), while the coupling between electrons in e 2 states favors interlayer FM (I 2 ). Of these, the intralayer FM is contributed only by itinerant electrons (I 1 ), but there is competition between interlayer FM (P 1 and I 2 ) and AFM (P 2 and P 3 ) coupling. This explains why there is some debate regarding Fe atoms behaving ferromagnetically or antiferromagnetically in Fe 3 GeTe 2 39 . Moreover, Fu et al. 40 have also found that the coexistence of localized and itinerant 3d electrons in BiFeO 3 / SrTiO 3 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 Fe 3 GeTe 2 system 14 . In 2D Fe 3 GeTe 2 , there are three types of exchange interaction between Fe ions, corresponding to the first, second, and third nearest neighbor magnetic exchange constants J 1 , J 2 , and J 3 , as shown in Fig. 2. The values of J 1 , J 2 , and J 3 for 2D Fe 3 GeTe 2 can be extracted from the total energy difference between different spin orderings. As summarized in Table 1, the derived exchange interaction parameters are J 1 = −0.44 meV, J 2 = 3.27 meV, and J 3 = 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 J 1 of −0.44 meV yields weak AFM coupling, which occurs mainly through the path P 2 . The path P 1 corresponds to strong FM coupling, with J 2 having a value of 3.27 meV. Moreover, itinerant magnetism (I 1 and I 2 ) gives a value of 0.47 meV for J 3 , 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 Fe 3 GeTe 2 . 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 Fe 3 GeTe 2 monolayer. According to our picture, tensile strain will shorten the Fe 3+ -Fe 2+ distance (P 1 path) but lengthen the other interatomic distances, which in turn will enhance FM double exchange between Fe 3+ and Fe 2+ ions.
MAE as an important parameter of ferromagnets counteracts thermal fluctuations and preserves long-range FM ordering 42 . From noncollinear calculations with inclusion of the spin-orbit coupling (SOC) effect, the MAE of 2D Fe 3 GeTe 2 has been determined as 0.94 meV/Fe, favoring perpendicular anisotropy, whereas the previously reported value was 0.67 meV/Fe 14 . For comparison, the MAE of 2.5 meV/Fe in bulk Fe 3 GeTe 2 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 atom 43 . 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 spindown 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 xz; j; L z ; j; yz and xz; yz; j; L x ; j; z 2 . 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 Fe 3+ ions, while the coupling of occupied d z 2 and unoccupied d xz /d yz orbitals of Fe 2+ ions make a negative contribution to the MAE. Such a mechanism also accounts for the variation in MAE for Fe 3 GeTe 2 monolayer when the Fe 3+ 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 Fe 3 GeTe 2 layer and oxidized Fe 3 GeTe 2 layers. Their DFT calculations further revealed that such AFM coupling mainly originates from the oxygen atoms located at the bilayer interface, while bilayer Fe 3 GeTe 2 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 P 2 path between two Fe 3 GeTe 2 layers, thereby inducing AFM coupling. Dai et al. 24 reported a pressure-dependent phase diagram of Fe 3 GeTe 2 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 (P 1 path). On the other hand, by analyzing structural characteristics, we found that the Fe 3+ -Te distance clearly decreases at pressures below 7 GPa. The corresponding Fe 3+ -Fe 3+ exchange though the Te-mediated P 2 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 subband origin from the Fe-d z 2 , d xz , and d yz orbitals, inducing roomtemperature ferromagnetism in Fe 3 GeTe 2 14 . 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 Fe 3 GeTe 2 comes mainly from localized double exchange (P 1 path).
Structure and magnetic behavior of Fe n GeTe 2 ultrathin films. The above discussions on the one hand again demonstrate the coexistence of itinerant and localized magnetism in the Fe 3 GeTe 2 system. On the other hand, the interlayer competition between localized exchange coupling (paths P 1 , P 2 , and P 3 ) and itinerant electrons (I 1 and I 2 ) is also crucial in determining the nature of the magnetic ground states and the values of the Curie temperature and MAE of 2D Fe 3 GeTe 2 . 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−x GeTe 2 46 , 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 Fe 2+ /Fe 3+ 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 n GeTe 2 system. To satisfy these requirements, we have designed a series of Fe-rich Fe n GeTe 2 (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 Fe 3 GeTe 2 , these Fe n GeTe 2 ultrathin films also belong to the P-3m1 space group. The effective thicknesses (Table 1) of Fe 4 GeTe 2 , Fe 5 GeTe 2 , Fe 6 GeTe 2 , and Fe 7 GeTe 2 ultrathin films are 5.63 Å, 6.79 Å, 7.56 Å, and 8.73 Å, respectively, which are moderately larger than that of Fe 3 GeTe 2 (5.14 Å). The atomic arrangements of Fe n GeTe 2 ultrathin films can be regarded as six, seven, eight, and nine atomic layered thickness (001) surfaces of a Te-substituted Fe 7 Ge 4 crystal 47 . Fortunately, the atomic arrangement of a five atomic layered thickness Te-substituted Fe 7 Ge 4 crystal is the same as that of the experimentally reported Fe 3 GeTe 2 phase. To further check the experimentally feasibility of Fe n GeTe 2 , we have calculated their formation energies, defined as E f ¼ ½EðFe n GeTe 2 Þ À EðFe 2 GeÞ À EðTe 2 Þ À ðn À 2ÞEðFeÞ=n; where E(Fe n GeTe 2 ) is the total energy of the 2D Fe n GeTe 2 compound, and E(Fe 2 Ge), E(Te 2 ), and E(Fe) are the total energies of Fe 2 Ge, Te, and Fe in their most stable bulk phases 48 . The formation energies of four Fe n GeTe 2 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 Fe 3 GeTe 2 . All these negative values indicate that the formation processes are exothermic. More importantly, we find that the total energy of our proposed Fe 5 GeTe 2 ultrathin film is 0.24 eV per atom lower than that of the experimentally reported layered phase with the same stoichiometry 16 . It should be noted, however, that our DFT simulation results only mean that our proposed Fe 5 GeTe 2 ultrathin film is energetically favorable than the experimentally reported one at 0 K. Anyway, the satisfactory stability of these Fe n GeTe 2 ultrathin film implies that they are feasible from a theoretical point of view. It is noteworthy that ultrathin films of Cr 2 S 3 , 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 n GeTe 2 films could be grown on the surface of hexagonal Si phase. The calculated lattice mismatches between Si(001) and (5 × 5) Fe n GeTe 2 superlattices are 0.5%, 3.7%, 0.7%, and 1.1% for Fe 4 GeTe 2 , Fe 5 GeTe 2 , Fe 6 GeTe 2 , and Fe 7 GeTe 2 , respectively. The optimized structures of Fe n GeTe 2 /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 n GeTe 2 superlattices.
We further discuss the electronic and magnetic properties of the proposed Fe n GeTe 2 ultrathin films. Similar to 2D Fe 3 GeTe 2 , all the Fe n GeTe 2 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 n GeTe 2 (3 ≤ n ≤ 7) can be revealed by the Bader charge (see Supplementary Note 4) and the PDOS. The distributions of Fe 2+ and Fe 3+ ions vary with the thickness and composition of the 2D Fe n GeTe 2 ultrathin films. With increasing Fe content, the Fe 2+ /Fe 3+ (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 n GeTe 2 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 n GeTe 2 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 Fe 2+ /Fe 3+ 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 Fe 3+ ions contribute a larger magnetic moment than Fe 2+ ions.
Because the T C in Fe n GeTe 2 systems is determined mainly by localized double exchange, we consider the exchange parameters J of Fe n GeTe 2 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 n GeTe 2 (n = 5-7) with increasing Fe content  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.
(see Supplementary Note 5). For Fe 2+ /Fe 3+ 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 Fe 2+ / Fe 3+ ratio is increased further, localized magnetic exchange becomes dominant, and the magnitude decreases with the distance between Fe ions. Meanwhile, the variations in J 1 , J 2 , and J 3 can also be interpreted in terms of the magnetic interaction mechanism, as has earlier been established for 2D Fe 3 GeTe 2 .
To further clarify the magnetic ground states of 2D Fe n GeTe 2 ultrathin films, the relationship between the exchange-pathdependent parameters J 1 , J 2 , and J 3 and the Fe 2+ /Fe 3+ 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 (P 1 , P 2 , and P 3 paths) and the itinerant electron coupling of d xz /d yz orbitals (I 2 ) are stronger than the intralayer interactions (I 1 path) in Fe n GeTe 2 systems. Second, for all the Fe n GeTe 2 systems considered here, the dominant J parameter for FM coupling comes mainly from double exchange of localized d z 2 orbitals (P 1 path) and coupling of itinerant electrons in d xz /d yz orbitals (I 2 ). However, the major J parameter for AFM ordering comes mainly from coupling between localized d z 2 orbitals in Fe 3+ -Fe 3+ and Fe 2+ -Fe 2+ exchange (P 2 and P 3 paths). Therefore, the competition between interlayer AFM and FM coupling results from that between itinerant and localized magnetism in Fe 2+ -Fe 2+ or Fe 3+ -Fe 3+ coupling. In an Fe 5 GeTe 2 ultrathin film, when the distance between interlayer Fe layers is shorter, the localized magnetic exchange through the P 2 path can compete with itinerant e 2 electrons. However, the itinerant e 2 electrons become dominant as the Fe-Fe distance increases, such that the value of J 4 becomes 5.9 meV. Subsequently, the itinerant magnetism weakens as the Fe-Fe distance continues to increase, with the value of J 5 becoming 0.1 meV. Two competing ferromagnetisms of localized and itinerant are responsible for these complicated behaviors of the magnetic exchange parameters. As the Fe 2+ /Fe 3+ ratio increases, the itinerant behavior of d orbitals is weakened. That is to say, more localized P 1 /P 3 paths (FM/AFM) and fewer I 2 (FM) appear. Therefore, there is no simple trend of variation of the J parameters.
The MAE values for all the Fe n GeTe 2 ultrathin films with different Fe 2+ /Fe 3+ ratios are also shown in Fig. 3a. One can see that MAE first increases from 0.91 meV/Fe atom for x = 0 (Fe 5 GeTe 2 ) to 1.05 meV/Fe atom for x = 0.2 (Fe 6 GeTe 2 ). Then, it decreases almost monotonically with increasing Fe 2+ /Fe 3+ ratio in the mixed-valence Fe n GeTe 2 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 Fe 2+ /Fe 3+ ratio. As we have discussed with regard to 2D Fe 3 GeTe 2 , the Fe 3+ and Fe 2+ 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 Fe 5 GeTe 2 to Fe 6 GeTe 2 , 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 Fe 2+ /Fe 3+ ratio is 0 (Fe 5 GeTe 2 ), 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 Fe 2+ /Fe 3+ ratio is increased from 0 (Fe 5 GeTe 2 ) to 0.2 (Fe 6 GeTe 2 ), 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 n GeTe 2 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 Fe 3 GeTe 2 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 (Fe 3+ and Fe 2+ ) in Fe 3 GeTe 2 compounds are also presented in Supplementary  Fig. 11a. One can see that the magnetizations of both Fe 3+ and Fe 2+ sublattices indeed behave like ferromagnets. Additionally, the estimated T C for bulk Fe 3 GeTe 2 crystal is 280 K (see Supplementary Note 5), which is also comparable to the experimental value of 230 K 27 .
For various stoichiometries, three kinds of M-T curves are observed. Similar to Fe 3 GeTe 2 , Fe 4 GeTe 2 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 symmetry 15 . For Fe 5 GeTe 2 , 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 Fe 1 and Fe 5 layers are thermally disturbed more easily, and their magnetic moments decrease almost linearly with increasing temperature, while the other three layers (Fe 2 , Fe 3 , and Fe 4 ) drastically decrease around 320 K. The correlation here between magnetization and temperature is characteristic of a Néel's P-type ferrimagnet 50 . The ferrimagnetic (FiM)-to-paramagnetic transition occurs at a critical temperature T C = 320 K. Such complicated magnetic behavior of Fe 5 GeTe 2 has also been discussed in previous papers. For example, Ramesh et al. 51 found that the Fe 5−x GeTe 2 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 Fe 5 GeTe 2 , the results of which support the existence of the magnetic transition but not that of a spin glass state. Compared with 2D Fe 5−x GeTe 2 , Fe 6 GeTe 2 and Fe 7 GeTe 2 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 Fe 5 GeTe 2 from Fe 6 GeTe 2 and Fe 7 GeTe 2 is the existence of a frustration effect. In this situation, the magnetic moments of Fe 5 GeTe 2 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 Fe 6 GeTe 2 , the spin moments of the Fe 1 , Fe 3 , Fe 4 , Fe 5 , and Fe 6 layers and those of the Fe 2 layers show opposite directions, although all the moments decrease with temperature. For the M-T curves of 2D Fe 7 GeTe 2 , the spin directions of the Fe 2 , Fe 4 , and Fe 5 layers and those of the Fe 1 , Fe 3 , Fe 6 , and Fe 7 layers are opposite. The FiM-toparamagnetic transition occurs at a T C of 570 K. The Néel's Rand P-type magnetization profiles seen here have also been reported in mixed-valence complex alloys (Mn 1.5 FeV 0.5 Al 52 and Mn 2 V 0.5 C 0.5 Z 53 ), complex oxides (NiCo 2 O 4 54 ), layered materials (AFe II Fe III (C 2 O 4 ) 3 55 ) and core-shell nanoparticles 56 . The competition between interlayer AFM and FM coupling resulting in the transition from FM to FiM states in Fe n GeTe 2 is derived from the coexistence of different electronic states.
The simulated values of T C are 138 K for Fe 3 GeTe 2 , 68 K for Fe 4 GeTe 2 , 320 K for Fe 5 GeTe 2 , 450 K for Fe 6 GeTe 2 , and 570 K for Fe 7 GeTe 2 . For truly FM systems, T C drops from 138 K for Fe 3 GeTe 2 to 68 K for Fe 4 GeTe 2 because of the flipping of out-ofplane MAE brought about by the increased ratio of Fe 2+ ions. Further increases in Fe content lead to a transition of magnetic ordering from FM to FiM at Fe 5 GeTe 2 . For n ≥ 5, the T C of the FiM Fe n GeTe 2 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−x Cr x Ge and Fe 3−x V x Ge alloys 57,58 . Extrapolating to even thicker films, Fe n GeTe 2 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 phase 15 (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 FeGe 59 to 485 K for Fe 2 Ge 60 , and then to 1043 K for pure Fe 15 , 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 0.18 Ge 0.82 Te films is 100 K 61 . The T C of our Fe 3 GeTe 2 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 Fe 6 GeTe 2 (450 K) is slightly lower than that of the bulk Fe 2 Ge 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 n GeTe 2 ultrathin films with different Fe contents and thicknesses, which are experimentally more feasible than the reported 2D layered phase. By firstprinciples 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 n GeTe 2 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 Ày 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 variablevalence 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 n GeTe 2 ultrathin films with variable-valence elements. They also indicate that 2D magnetic Fe n GeTe 2 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 code 63 . The projected augmented wave (PAW) potential was used to describe ion-electron interaction 64 . 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 Fe 2+ / Fe 3+ ratios x were also tested, as shown in Supplementary Note 5. Our results shown that the best choice for all Fe n GeTe 2 compounds is U = 4.3, although a different value would not affect our conclusions. Correction of van der Waals interactions using the DFT-D3 scheme 67 was included in the bulk Fe 3 GeTe 2 calculations.
Magnetic parameters. To describe the magnetic properties of Fe n GeTe 2 crystals, the magnetic anisotropy energy (MAE) is defined as 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, respectively 68 . 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 as 69 MAE % 4E dd ¼ ξ 2 ∑ o À u À o À ; j; L z ; j; u À 2 À o À ; j; L x ; j; u À 2 ε u À Àε o À : 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 4E 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 constant 70 . The MAE can then be expressed approximately as MAE dd % xz; j; L z ; j; yz 2 ε xz À ε yz À xz; yz; j; L x ; j; z 2 2 ε xz;yz À ε z 2 : ð4Þ Monte Carlo simulations. Monte Carlo simulations were carried out to determine the magnetic transition temperatures. The Hamiltonian of the system was expressed as follows: where S i is the unit vector of the magnetic moment at site i, J ij is the exchangecoupling 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 and 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 10 5 MC steps were discarded for thermal equilibration, and the successive 10 5 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.

Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.

Code availability
The electronic structure and magnetic properties calculations were performed using the proprietary code VASP 63 . The codes of Monte Carlo simulations in this paper are available from the authors upon request.