Since synthesis of CrI311. C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546, 265 (2017). https://doi.org/10.1038/nature22060 and CrGeTe3,22. B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Nature 546, 270 (2017). https://doi.org/10.1038/nature22391 two-dimensional (2D) magnetic materials have drawn huge interest. Numerous ferromagnetic (FM) materials are predicted to exhibit excellent properties such as half-metal,3–53. M. Ashton, D. Gluhovic, S. B. Sinnott, J. Guo, D. A. Stewart, and R. G. Hennig, Nano Lett. 17, 5251 (2017). https://doi.org/10.1021/acs.nanolett.7b013674. S.-S. Wang, Z.-M. Yu, Y. Liu, Y. Jiao, S. Guan, X.-L. Sheng, and S. A. Yang, Phys. Rev. Mater. 3, 084201 (2019). https://doi.org/10.1103/PhysRevMaterials.3.0842015. X. Xin, W. Li, R. Pang, H. Wang, C. Guo, X. Shi, and Y. Zhao, Appl. Phys. Lett. 111, 253102 (2017). https://doi.org/10.1063/1.5007166 bipolar magnetic semiconductor,66. X. Li, X. Wu, Z. Li, J. Yang, and J. G. Hou, Nanoscale 4, 5680 (2012). https://doi.org/10.1039/c2nr31743e quantum anomalous Hall insulators,7–97. S.-J. Zhang, C.-W. Zhang, S.-F. Zhang, W.-X. Ji, P. Li, P.-J. Wang, S.-S. Li, and S.-S. Yan, Phys. Rev. B 96, 205433 (2017). https://doi.org/10.1103/PhysRevB.96.2054338. Z. Liu, G. Zhao, B. Liu, Z. F. Wang, J. Yang, and F. Liu, Phys. Rev. Lett. 121, 246401 (2018). https://doi.org/10.1103/PhysRevLett.121.2464019. M. Mogi, R. Yoshimi, A. Tsukazaki, K. Yasuda, Y. Kozuka, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Appl. Phys. Lett. 107, 182401 (2015). https://doi.org/10.1063/1.4935075 magnetic Dirac fermions, etc. In addition to 2D FM materials, 2D antiferromagnetic (AFM) materials also draw attention for their promising properties in spintronics,1010. S. Rahman, J. F. Torres, A. R. Khan, and Y. Lu, ACS Nano 15, 17175 (2021). https://doi.org/10.1021/acsnano.1c06864 e.g., ultra-fast dynamics, strong spin-transfer torque,1111. H. Yan, Z. Feng, P. Qin, X. Zhou, H. Guo, X. Wang, H. Chen, X. Zhang, H. Wu, C. Jiang, and Z. Liu, Adv. Mater. 32, 1905603 (2020). https://doi.org/10.1002/adma.201905603 and topological property.1212. P. Zhou, C. Q. Sun, and L. Z. Sun, Nano Lett. 16, 6325 (2016). https://doi.org/10.1021/acs.nanolett.6b02701As a type of high-performance spintronic device, 2D magnetic topological materials are especially desirable for their low power consumption, chiral edge state, and various emergent magnetotransport property such as intrinsic anomalous Hall effect. It is intriguing that the topological phase transition could be realized by the spin direction in some topological trivial 2D materials such as FM semiconductor CrI3,1313. P. Jiang, L. Li, Z. Liao, Y. X. Zhao, and Z. Zhong, Nano Lett. 18, 3844 (2018). https://doi.org/10.1021/acs.nanolett.8b01125 half-metallic CrO2,1414. B. Zhang, J. Sun, J. Leng, C. Zhang, and J. Wang, Appl. Phys. Lett. 117, 222407 (2020). https://doi.org/10.1063/5.0031443 T-LuN2,1515. Y. Yu, X. Chen, X. Liu, J. Li, B. Sanyal, X. Kong, F. M. Peeters, and L. Li, Phys. Rev. B 105, 024407 (2022). https://doi.org/10.1103/PhysRevB.105.024407 NiAsO3, and PdSbO3.1616. Z. Li, Y. Han, and Z. Qiao, Phys. Rev. Lett. 129, 036801 (2022). https://doi.org/10.1103/PhysRevLett.129.036801 As to the FM state, tuning the spin direction does not change the magnetic state, because all the spin will tilt along the same direction. While for the AFM state, especially for a noncollinear AFM state, the magnetic state will transform to a weak ferromagnetic state along the tuned direction, and this may give rise to unexpected change to the electronic structure.Most of the proposed 2D magnetic materials are collinear because of their high crystal symmetry. By breaking inversion symmetry, the skyrmion could be induced by the Dzyaloshinskii–Moriya interaction.17–1917. J. Yuan, Y. Yang, Y. Cai, Y. Wu, Y. Chen, X. Yan, and L. Shen, Phys. Rev. B 101, 094420 (2020). https://doi.org/10.1103/PhysRevB.101.09442018. Q. Cui, J. Liang, Z. Shao, P. Cui, and H. Yang, Phys. Rev. B 102, 094425 (2020). https://doi.org/10.1103/PhysRevB.102.09442519. C. Xu, J. Feng, S. Prokhorenko, Y. Nahas, H. Xiang, and L. Bellaiche, Phys. Rev. B 101, 060404(R) (2020). https://doi.org/10.1103/PhysRevB.101.060404 However, noncollinear AFM materials are rarely investigated. Although triangular lattice MnBr2 and MnI2 are predicted to be 120° AFM state,2020. J. Luo, G. Xiang, Y. Tang, K. Ou, and X. Chen, J. Appl. Phys. 128, 113901 (2020). https://doi.org/10.1063/5.0015936 the investigation of magnetic phase transition and the noncollinear electronic band structure is not considered. Therefore, it is still challenging and interesting to explore material candidates that can realize noncollinear AFM, especially for ones with a tunable topological electronic band structure.

In this work, we predict that triangular lattice T-FeO2 is a stable 2D noncollinear AFM material, whose ground state is metallic planar 120° AFM, and the magnetic easy axis is in plane. The exchange coupling of nearest neighbor (NN) is antiferromagnetic, and the next nearest neighbor (NNN) is ferromagnetic. The NNN interaction is stronger than that of the NN interaction because of the RKKY interaction. In addition, by tuning the spin direction toward out of plane, the induced spin component perpendicular to the basal plane transforms the magnetic state to weak ferromagnetic, and the electronic band structure is distinct around the Fermi energy level. It is interesting that tuning the spin direction also gives rise to nontrivial topological phase transition.

We adopt density-functional theory (DFT) methods to carry out systematic calculations implemented in the Vienna Ab initio Simulation Package (VASP).2121. G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). https://doi.org/10.1103/PhysRevB.47.558 The generalized gradient approximation (GGA) in the form of the Perdew–Burke–Ernzerhof (PBE) functional is used as the exchange-correlation interactions.2222. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). https://doi.org/10.1103/PhysRevLett.77.3865 The projector augmented wave potentials explicitly include six valence electrons for O (2s22p4) and eight for Fe (4s23d6). The cutoff energy of 600  eV is taken, and the convergence criterion is set at 10−6  eV. The Γ-centered 21×21×1 Monkhorst-pack k-points mesh is used for the Brillouin-zone integrations in the nanosheets. The total energies are calculated by using the linear tetrahedron method with Bloch corrections. During structural relaxation, atomic positions are relaxed until the Hellman–Feynman forces are less than 10–3 eV/Å. The phonon dispersion of the corresponding crystal structure is computed by PHONOPY package2323. A. Togo and I. Tanaka, Scr. Mater. 108, 1–5 (2015). https://doi.org/10.1016/j.scriptamat.2015.07.021 integrated with the density functional perturbation theory (DFPT).2424. S. Baroni, S. D. Gironcoli, A. D. Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001). https://doi.org/10.1103/RevModPhys.73.515 To verify the thermal stability, ab initio molecular dynamics (AIMD) calculation with canonical (NVT) ensemble is performed. The AIMD calculation lasts for 10 ps with a time step of 1.0 fs, and the temperature is controlled by the Nose´-Hoover method.2525. G. J. Martyna, M. L. Klein, and M. Tuckerman, J. Chem. Phys. 97, 2635 (1992). https://doi.org/10.1063/1.463940 To consider the interlayer correction, the van der Waals (vdW) interaction is included with optB86b-wdW correction.26,2726. M. Dion, H. Rydberg, E. Schroder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004). https://doi.org/10.1103/PhysRevLett.92.24640127. J. Klimes, D. R. Bowler, and A. Michaelides, J. Phys.: Condens. Matter 22, 022201 (2010). https://doi.org/10.1088/0953-8984/22/2/022201There are generally two types of structures for the 2D MX2 family,2828. F. Aguilera-Granja and A. Ayuela, J. Phys. Chem. C 124, 2634 (2020). https://doi.org/10.1021/acs.jpcc.9b06496 which is named with T and H phases, respectively. H-structures possess mirror symmetry without inversion symmetry, whereas T-structures possess inversion symmetry without mirror symmetry. The geometric structure of 2D T-FeO2 is shown in Fig. 1(a).The lattice of T-FeO2 is a=b=2.80 Å, which is close to the result of the previous work.2828. F. Aguilera-Granja and A. Ayuela, J. Phys. Chem. C 124, 2634 (2020). https://doi.org/10.1021/acs.jpcc.9b06496 To verify its mechanical stability, we calculated the phonon band structure [see Fig. 1(b)]. The phonon dispersion exhibits one tiny imaginary frequency around the Γ point without multiple imaginary frequencies.2929. A. Lopez-Bezanilla, Phys. Rev. Mater. 2, 011002(R) (2018). https://doi.org/10.1103/PhysRevMaterials.2.011002 The result demonstrates the mechanical stability of the free-standing T-FeO2 monolayer in the absence of imaginary frequencies.To verify its thermal stability, we performed AIMD calculation at 300 K (see Fig. 2) with a 6×6×1 supercell of 108 atoms. The geometric structure of the supercell at the end of the AIMD calculation is shown Fig. 2(a), which indicates that the bonds maintained well. We also captured the variation in the bond lengths from one Fe atom to its six neighbor O atoms [see Fig. 2(b)]. One sees that, the bonds maintained well in the whole range of AIMD time steps, and the results confirm the thermal stability of T-FeO2 at 300 K.We further calculated its elastic constants Cij, which can be written as Cij=1A0∂2E∂εi∂εj to investigate the mechanical characteristic. Here, A0, E, and ε are the equilibrium area, total energy, and strain of the unit cell, respectively. The calculated elastic constants are C11=C22=116.4, C12=29.0, and C66=44.4  N/m. The results indicate that the elastic constants fulfill the criteria of the mechanical stability (C11>|C12| and C66>0).3030. Z. J. Wu, E. J. Zhao, H. P. Xiang, X. F. Hao, X. J. Liu, and J. Meng, Phys. Rev. B 76, 054115 (2007). https://doi.org/10.1103/PhysRevB.76.054115The planar Young's moduli Y2D could be evaluated by the formula: Y2D=(C11C22−C122)/C11, and the resultant Y2D=109.2  N/m, which is about 30% of ultrastrong material graphene.3131. T. J. Booth, P. Blake, R. R. Nair, D. Jiang, E. W. Hill, U. Bangert, A. Bleloch, M. Gass, K. S. Novoselov, M. I. Katsnelson, and A. K. Geim, Nano Lett. 8, 2442 (2008). https://doi.org/10.1021/nl801412yConsidering the possibility of synthesis for T-FeO2, relevant experiments have been reported. α-Fe2O3 was synthesized by means of liquid exfoliation. A chemical exfoliation technique has also been developed to extract a single layer such as MXene.3232. M. Naguib, M. Kurtoglu, V. Presser, J. Lu, J. Niu, M. Heon, L. Hultman, Y. Gogotsi, and M. W. Barsoum, Adv. Mater. 23, 4248 (2011). https://doi.org/10.1002/adma.201102306 Hence, the T-FeO2 layers could be chemically exfoliated from the bulk phase of CuFeO2,3333. T.-R. Zhao, M. Hasegawa, and H. Takei, J. Cryst. Growth 154, 322 (1995). https://doi.org/10.1016/0022-0248(95)00172-7 similar to exfoliating a MXene single layer from the MAX phase.To confirm the ground state of T-FeO2 monolayer, we constructed a 4×4×1 supercell and compared three different magnetic states: the ferromagnetic state (FM), chain-antiferromagnetic state (C-AFM), and zigzag-antiferromagnetic state (Z-AFM) (see Fig. 3). The magnetic moment is 2μB for each Fe atom, and the energy of the Z-AFM state is lower than the others (see Table I). We further calculated the magnetic anisotropic energy (MAE) of T-FeO2 monolayer with spin–orbit coupling (SOC). MAE is defined as EMAE=Ex−Ez=−0.8 meV, which suggests that the spin direction is parallel to the basal plane.

TABLE I. Energies relative to the Z-AFM state of different magnetic states, MAE, and corresponding exchange coupling parameters (EP) for T-FeO2 monolayer.

States and EPFMC-AFMZ-AFMMAEJJ′Energy (meV)169.12454.860−0.8−2.443.55Based on the energies mentioned above, we could derive the magnetic exchange coupling strength with consideration of the NN Fe atoms and the NNN Fe atoms based on the Heisenberg model H=−J∑⟨ij⟩S→i·S→j−J′∑⟨⟨ij⟩⟩S→i·S→j−D∑kSz,k2,(1)where J and J′ are the NN and NNN exchange coupling parameters, respectively. D is the magnetic anisotropic energy, S→i/S→j are spin of the i/jth site, and Sz,k is the spin moment along the z axis of the kth site. The results are listed in Table I, and the exchange coupling parameters reveal that the NN couple spins prefer the antiparallel state while the NNN ones prefer the parallel state. Another interesting result is that exchange coupling of the NNN is even larger than that of NNs because of the RKKY interaction of metallic itinerant electrons.3434. D. Zhang, A. Rahman, W. Qin, X. Li, P. Cui, Z. Zhang, and Z. Zhang, Phys. Rev. B 101, 205119 (2020). https://doi.org/10.1103/PhysRevB.101.205119 As to the triangular lattice, the N e´ el antiferromagnetic state is forbidden because the three NN magnetic atoms are all antiparallel coupling interaction, while a frustrated AFM state (F-AFM) with a magnetic propagation vector q→ = (π3, π3,0) (120° AFM state) will be the ground state (see Fig. 3).Different from the planar FM state, in which long range spin order is absent and the Kosterlitz–Thouless (KT) phase transition emerges, the 120° AFM state in the triangular lattice could maintain below KT transition temperatures.3535. L. Seabra, T. Momoi, P. Sindzingre, and N. Shannon, Phys. Rev. B 84, 214418 (2011). https://doi.org/10.1103/PhysRevB.84.214418 However, most of the theoretical works do not include the NNN exchange coupling interaction and MAE. The situation of the RKKY induced strong NNN interaction and planar easy axis spin direction is crucial for the magnetic phase transition. To unambiguously reveal the phase transition mechanism, we performed Monte Carlo simulations according to the results of J, J′, and D based on the Heisenberg model at different temperatures. In our Monte Carlo simulations, we adopted a 60 × 60 supercell, and 2×108 steps are performed to capture the magnetic characteristics of each temperature. After 2×108 Monte Carlo steps, we further performed 107 steps Monte Carlo and captured the average spin direction ⟨mi⟩, and the average absolute value of the spin direction ⟨|mi|⟩ (see Figs. S1–S4). The resultant magnetic states with different temperatures are shown in Fig. 4. It reveals that the spin direction of the triangular lattice mainly stays in the basal plane with spin frustration states of a 120° spin angle between the neighbor Fe atoms at 5 K. At 10 K, the stripy state, which forms a couple of antiparallel spin and a perpendicular spin for the three neighbor Fe atoms, will emerge slowly. When the temperature is larger than 20 K [see Fig. 4(c)], the stripy area will dominate the lattice. Around 90 K, the lattice will be paramagnetic (see Fig. S4). The result reveals that the magnetic phase transition is complicated than that of only NN exchange coupling Ising, XY, and Heisenberg model,35–3735. L. Seabra, T. Momoi, P. Sindzingre, and N. Shannon, Phys. Rev. B 84, 214418 (2011). https://doi.org/10.1103/PhysRevB.84.21441836. D. H. Lee, J. D. Joannopoulos, J. W. Negele, and D. P. Landau, Phys. Rev. B 33, 450 (1986). https://doi.org/10.1103/PhysRevB.33.45037. M. Mekata, J. Soc. Jpn. 42, 76 (1977). https://doi.org/10.1143/JPSJ.42.76 and they also exhibit a similar mechanism. We also performed Monte Carlo of the XY model, whose transition temperature is about 320 K. The result suggests that the XY model will overestimate the phase transition temperature (see Fig. S5).We further investigated the electronic band structure of T-FeO2 monolayer with a 3×3×1 supercell to construct the F-AFM state [see Fig. 5(a)] with SOC. The band structure exhibits metallic property, and the band characteristic along K−Γ and K′−Γ is degenerate.Although the spin direction of ground state T-FeO2 monolayer is parallel to the basal plane rather than the out of plane, the MAE is vulnerable. The easy axis could be tuned to be tilted under correction of the substrate or proximity effect. We further considered the band characteristic when the spin canted along the direction perpendicular to the plane to investigate the band evolution. As shown in Figs. 5(b) and 5(c), we plotted the band structures of two canted F-AFM states. The magnetic moment perpendicular to the plane are around 0.4 μB and 0.7 μB, respectively. The results suggest that, when the spin direction is canted perpendicular to the plane, multiple gapped Dirac states emerge around the Fermi energy level and degeneration of band characteristic along K−Γ and K′−Γ is broken.For C3v point group symmetry of T-FeO2 monolayer, five d orbitals split to three groups: (dxy+dx2−y2), (dxz+dyz), and dz2. To confirm the crystal field effect and orbital contribution of Fe atoms, we further plotted the projection band structures of five d orbitals (see Fig. 5). One sees that, the dz2 is the lowest one in the three group, and the multiple Dirac states are mainly from the dxy+dx2−y2 and dxz+dyz orbitals. As to Fe4+ in T-FeO2 monolayer, the crystal field energy split between dz2 and the other d orbitals is larger than Hund's coupling, and it gives rise to a middle spin configuration of 2μB per Fe atoms (Fig. 6).We further calculated the Berry curvature [Ωn(k)]3838. H.-J. Kim, C. Li, J. Feng, J.-H. Cho, and Z. Zhang, Phys. Rev. B 93, 041404(R) (2016). https://doi.org/10.1103/PhysRevB.93.041404 of the lower band in the gapped Dirac states of F-AFM (mz=0.0μB) and canted F-AFM states (mz=0.7μB) by the following equation: {A→nx=i⟨n(k)|∂∂kx|n(k)⟩,A→ny=i⟨n(k)|∂∂ky|n(k)⟩Ωn(k)=∇×A→n,,(2)where A→nx and A→ny are the Berry connection of the nth band. The results are shown in Fig. 7, from which one sees that, when the spin canted to out of plane, the Berry curvature is obviously increased around the Dirac states.

We further calculated the Chern number of the lower band of the Dirac state by the formula C=12π∫BZΩd2k. It is especially interesting that for the canted F-AFM state, it exhibits nontrivial topological property with a high Chern number C = 6. While for the F-AFM state, the Chern number is 0. It indicates that F-AFM to weak ferromagnetism topological phase transition could be induced by tuning the spin direction.

We also calculated the FM state to compare the topological property. The electronic band structure of FM T-FeO2 monolayer is shown in Fig. S6, which exhibits half-metallic property. Its insulator for a spin majority channel and the indirect bandgap is about 1.59  eV, whereas the spin minority channel exhibits metallic property. In addition, for a spin minority channel, there are two Dirac states around 0.5  eV above the Fermi energy (set to 0  eV). One is in the high symmetric K(13,13) point and the other is in the off symmetric point between Γ and K. Our previous works14,39,4014. B. Zhang, J. Sun, J. Leng, C. Zhang, and J. Wang, Appl. Phys. Lett. 117, 222407 (2020). https://doi.org/10.1063/5.003144339. B. Zhang, J. Sun, J. Leng, C. Zhang, H. Chen, and J. Wang, Phys. Rev. B 102, 165404 (2020). https://doi.org/10.1103/PhysRevB.102.16540440. B. Zhang, X. Chen, and J. Wang, Appl. Phys. Lett. 119, 162401 (2021). https://doi.org/10.1063/5.0069680 reveal that multiple Dirac states could be induced by an electron counting rule associated with bilayer stacking crystal field effect. The geometric symmetry of T-FeO2 monolayer is the same with that of T-CrO2, and the symmetric property gives rise to multiple Dirac states around the Fermi energy for the FM state. In addition, we calculated the Chern number of the bottom band of the Dirac states and the resultant Chern number C = 2, which is topological nontrivial. The corresponding one-dimensional topological edge state is shown in Fig. S6, from which one sees that there are two edge states cross from the top band to the bottom band of the gapped Dirac point.

In summary, we predicted a stable 2D triangular lattice T-FeO2 monolayer. The stability is verified by phonon and AIMD calculations, which confirms that the monolayer is stable above room temperature. The planar Young's modulus of T-FeO2 monolayer is 109.2  N/m, which is about 30% of that of graphene. The exchange coupling and MAE parameters indicate that the ground state is planar 120° AFM. Our Monte Carlo results reveal that, the magnetic state will evolve from the planar 120° AFM state to the stripy AFM state and then to disorder by increasing temperature up to 100 K. As a 120° AFM triangular lattice, tuning the spin direction is an effective method to manipulate the electronic band structure from the planar AFM state to the weak ferromagnetic state, which produces a spin exchange field. It is interesting that, tuning the spin direction could give rise to the nontrivial topological phase transition in T-FeO2 monolayer. Our work provides a 2D metal oxide, which exhibits the magnetic phase transition and spin direction induced nontrivial topological phase transition. The magnetic and electronic properties are very promising for spintronics.

See the supplementary material for Monte Carlo results of T-FeO2 monolayer, electronic band structures, and edge state of FM state FeO2 monolayer.

This work was supported by the NSFC (Nos. 52101225, 12174172, and 12004153), the Natural Science Foundation of Fujian (No. 2021J011041), the Fuzhou Institute of Oceanography Project (No. 2021F06), and the Science and Technology Project of Fujian Provincial Department of Education (No. JAT200439).

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Bingwen Zhang: Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Xuejiao Chen: Formal analysis (equal); Investigation (equal); Software (equal). Fenglin Deng: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal). Xiaodong Lv: Investigation (equal); Visualization (equal); Writing – review & editing (equal). Cheng Zhang: Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Biao Zheng: Data curation (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal). Huining Wang: Investigation (equal); Writing – review & editing (equal). Jun Wang: Funding acquisition (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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