High throughput study on magnetic ground states with Hubbard U corrections in transition metal dihalide monolayers

Abstract

We present a high throughput study of the magnetic ground states for 90 transition metal dihalide monolayers TMX₂ using density functional theory based on a collection of Hubbard U values. Stable geometrical phases between 2H and 1T are first determined. Spin-polarized calculations show that 50 out of 55 magnetic TMX₂ monolayers are energetically prone to the 1T phase. Further, the magnetic ground states are determined by considering four local spin models with respect to different U values. Interestingly, 23 out of 55 TMX₂ monolayers exhibit robust magnetic ground orderings which will not be changed by the U values. Among them, NiCl₂ with a magnetic moment of 2 μ_B is a ferromagnetic (FM) insulator, while the VX₂, MnX₂ (X = Cl, Br and I), PtCl₂ and CoI₂ monolayers have noncollinear antiferromagnetic (120°-AFM) ground states with a tiny in-plane magnetic anisotropic energy, indicating flexible magnetic orientation rotation. The exchange parameters for both robust FM and 120°-AFM systems are analyzed in detail with the Heisenberg model. Our high-throughput calculations give a systematic study of the electronic and magnetic properties of TMX₂ monolayers, and these two-dimensional materials with versatile magnetic behavior may have great potential for spintronic applications.

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I. Introduction

The demand of miniaturization and feasibility of integration of functioning devices makes two-dimensional (2D) materials remarkably promising in a wide spectrum of technological applications. Due to the breaking of translational symmetry, the 2D materials display distinct properties from the bulk systems, as supremely spotlighted by graphene.^1,2 Recent highlights on 2D materials studies have moved from graphene to other layered van der Waals materials. Among them, transition metal based ultrathin materials^3,4 with hexagonal structures similar to that of graphene have captured intensive attention because of their abundant electronic properties, including semiconductors with sizable band gaps, 100% spin-polarized half metals,^5–7 and nontrivial topological states.^8–10

Such versatile electronic or magnetic properties have lead to comprehensive research in various fields like catalysis, opto-electronics, spintronics, electrodes and supercapacitors. In particular, transition metal dichalcogenide monolayers (TMDs),¹¹ transition metal trichalcogenides MAX₃,¹² and 2D transition metal carbides (nitrides) as so-called MXenes¹³ have been systematically studied. For instance, MoS₂, MoSe₂, WS₂ and WSe₂ monolayers have sizable direct band gaps allowing applications in transistors, photodetectors, and electroluminescent devices.^14,15 VS₂ and VSe₂ monolayers are typically metallic ferromagnets with strain-tunable magnetic ground states.⁵ MXenes with semiconducting band gaps around 0.25–2.0 eV yield very large Seebeck parameters around 100 K allowing high thermoelectric performance.⁴ van der Waals layered Cr₂Ge₂Te₆ has been experimentally proved to be a magnetic insulator, which are regarded as a key resource for next-generation spintronic devices.¹⁶

Beyond transition metal chalcogenides, carbide and nitride transition metal halides are also found to be interesting.¹⁷ Recently, layer-dependent ferromagnetism in van der Waals CrI₃ crystals has been systematically studied. It is demonstrated that the monolayer CrI₃ is an Ising ferromagnet with an out-of-plane spin orientation.¹⁸ Soon after, (quasi) 2D ferromagnetic insulators have revealed many important applications in phototronics, spintronics and magnetoelectronics. For instance, Seyler et al. unveiled spontaneous circularly polarized excitation in the CrI₃ monolayers.¹⁹ Jiang et al. revealed that a direct-to-indirect band gap transition can be caused by rotating the magnetic direction of CrI₃ from out-of-plane to in-plane.²⁰ Klein et al. detected the magnetic ground state and interlayer coupling in CrI₃ and uncovered a field-induced metamagnetic transition.²¹ In addition, transition metal dihalides (TMX₂) with intriguing simple lattice structures, relative large ionicity, and magnetic diversities have invigorated the enthusiasm of systematic researches. Early in 1971, layered NiCl₂ and CoCl₂ separated by one or two layers of graphite was proved to be a two-dimensional Heisenberg ferromagnet.²² It was shown that intercalation of graphite can weaken significantly the interaction between the TMX₂ layers, so that these compounds can be regarded as 2D paramagnets. In recent years, TMX₂ monolayers, especially TMX₂ magnets, have attracted more attention. For instance, the ferromagnetic or antiferromagnetic properties of TMX₂ (TM = 3d transition metals; X = Cl, Br, I) in both the 1T and 2H phase were investigated and the Curie temperature of magnetic transition were evaluated using Monte Carlo simulations.²³ Moreover, FeX₂ were proven to be half-metals²⁴ with sufficient stability to be exfoliated from bulk layered compounds.⁶ Despite the achievements made on 3d transition metal dihalide monolayers, more pending questions still need to be investigated in a deeper way: (1) since fullerene-like CdCl₂ closed cage structures have been successfully synthesized,²⁵ there is a high possibility to achieve 4d or 5d TMX₂ monolayers. In this case, the exotic electronic or magnetic properties of 4d or 5d TM dihalide monolayers are open to be explored. (2) Most current magnetic analyses of TMX₂ monolayers are focusing on collinear magnetic ordering. However, as the TM atomic arrangements in TMX₂ monolayers of both the 1T and 2H phase are triangular lattices, geometric frustration on the spin configurations is inevitable. Thus, non-collinear antiferromagnetic ordering needs to be further identified.²⁶ (3) Previous density-functional-theory (DFT) calculations on TMX₂ either neglected the local Coulomb interactions (U values) or used specific U value to estimate the electronic or magnetic properties, which may not be in accordance with future experimental results. In this regard, investigating TMX₂ with a batch of U values to identify property–U relations is instructive to practical experiments.

In this work, we implement a high throughput method to identify the magnetic ground states of 90 TMX₂ (TM = transition metals from Sc to Hg; X = Cl, Br, I) with four magnetic ordering configurations based on first principles calculations. 55 out of 90 TMX₂ monolayers are magnetic cases and 50 magnetic TMX₂ are stable in the 1T phase instead of the 2H phase. Due to the lack of experimental data when describing the correlations on d electrons of TM atoms in TMX₂ monolayers, the magnetic ground states for magnetic TMX₂ systems are systematically investigated with respect to Hubbard U values from 0 to 5 eV. We predict that the NiCl₂ is a robust ferromagnetic insulator with magnetic moments of 2 μ_B and Curie temperature of 57 K. Besides, 8 TMX₂ monolayers are stable in noncollinear antiferromagnetic ground states with different Hubbard U values, and the in-plane magnetic anisotropy energy is estimated to be smaller than 10 μeV indicating an easily overcome thermal fluctuation and a high-flexibility of magnetic orientation.

II. Computational method

Geometrical optimization and electronic structure calculations of the TMX₂ monolayers with different magnetic orderings were carried out using the projected augmented wave method as implemented in the Vienna ab initio Simulation Package (VASP),^27,28 where the exchange-correlation functional was approximated by the generalized gradient approximation (GGA)²⁹ with Perdew–Burke–Ernzerhof (PBE)³⁰ parametrization. The energy cutoff of 500 eV is applied to ensure convergence. The 2D Brillouin zone integration was done with a k-mesh density of 60/a, where a denotes the length of the lattice constant in the unit of Å. In order to obtain accurate magnetic anisotropic energy, a k-mesh of 120/a is applied. A vacuum slab of 25 Å was induced to prevent the image interaction from periodic supercells. A Gaussian smearing factor of 0.01 V is used. The atomic coordinates were fully optimized with a maximum residual force of 0.01 eV Å⁻¹. Spin polarized calculations with large initial magnetic moments were carried out in order to identify magnetic and nonmagnetic systems. For the magnetic systems, four magnetic states with ferromagnetic (FM), 2 × 1 × 1 antiferromagnetic (AFM), 4 × 1 × 1 AFM and noncollinear 120° AFM orderings are generated as the prototypes, implemented in our HTP calculations. The relaxation of magnetic ground state and net magnetic moment are performed manually to avoid local minimum optimization of TMX₂ systems. Regarding the effect of the correlations on the properties of the TM atoms, the Hubbard U correction from U = 1 to U = 5 was applied to account for the electronic correlation of d electrons.

III. Results and discussions

A. Geometrical stability: 2H and 1T phases

Similar to the TM dichalcogenides like MoS₂,³¹ there are two common crystal structures for the TM dihalide monolayers, i.e., the 2H phase with trigonal prismatic D_3h symmetry and the 1T phase with octahedral O_h symmetry. TMX₂ with both geometries are fully optimized and the energy difference between the 2H and 1T phases with U = 0 and U = 2 are shown in Fig. 1. Obviously, most TMX₂ monolayers are stable in the 1T structure, while only 21 (when U = 2) out of 90 TMX₂ monolayers prefer the 2H structural phase. A stable 2H phase occurs with large possibilities in group-3 (d¹) and group-4 (d²) TMX₂ monolayers. For 2H systems, the energy difference between the 2H and 1T phases ranges from 10 to 139 meV per atom. Meanwhile, the energy difference between the 1T and 2H phases of MoS₂ and WS₂ calculated by DFT are around 283 and 300 meV per atom, respectively, which are much larger than those of TM dihalides.³² Thus, we suspect that structural phase transition can be induced in TM dihalides by plasmonic hot electron injection,³³ infrared laser induction,³⁴ and employing lithium molten salts.³⁵ We note that the stability of the 2H and 1T structures is independent of the U value in most cases. However, as shown in Fig. 1b, in some cases such as TiBr₂, TcBr₂, ScI₂ and TiI₂, the ground structural phase can be changed by employing different U values, which are consistent with a recent theoretical study on NbSe₂ which shows that DFT with the U method can tune the energy difference and improve the stability of metastable phase, explaining its detection in experiments.³⁶

Fig. 1 Energy difference between 2H and 1T phase of 90 TMX₂ monolayers with U = 0 (blue ball) and U = 2 (gray ball), respectively. Solid perpendicular lines indicate that geometrical phase transition happens in these four TMX₂ monolayers.

On the other hand, the dynamic stability of 1T monolayers with a space group symmetry of p3̄m1 has been studied thoroughly by previous high throughput studies. Haastrup et al. introduced the Computational 2D Materials Database (C2DB) containing 1500 2D materials distributed over 30 different crystal structures, including the 1T-CdI₂ prototype. To determine the local minimum of the potential energy surface, the dynamical stabilities are analyzed by calculating the Γ-point phonons.³⁷ It shows that 90 out of 315 1T-CdI₂ prototype monolayers are of high stability. Besides, phonon dispersions without imaginary frequencies of first row 1T-TMX₂ (TM = V, Mn, Fe, Co, Ni; X = Cl, Br, I) monolayers have been investigated by Kulish et al., indicating high dynamic stabilities of the 1T-TMX₂ monolayers.²³ Furthermore, since the most stable structure generally locates on the global free energy minimum of the potential energy surface, the corresponding potential energy surface for each TMX₂ will be explored in detail in our further studies by CALYPSO software using the particle swarm optimization (PSO) algorithm.³⁸

B. Magnetic ground states

The spin structure and magnetic properties of some bulk TMX₂ compounds like NiBr₂ have been studied in a previous work, indicating a transition from a collinear easy-plane metamagnetic structure to an in-ab-plane spin structure.³⁹ Since the broken translation symmetry along the z direction in a 2D magnetic monolayer can remove the magnetic ordering perpendicular to the layer plane, the magnetic properties of a 2D structure may differ from its bulk phase.¹⁷ Here, one important question is to determine the magnetic ground state of 2D TMX₂ monolayers. Clearly, the TM sublattice in both the 2H and 1T phases are of the same type, i.e., a triangular lattice of TM atoms. We consider the following four possible magnetic configurations with high symmetries in 1T-TMX₂ as shown in Fig. 2: (a) Ferromagnetic (FM) ordering with all initial magnetic moments pointing to the z direction perpendicular to the monolayer plane. (b) Antiferromagnetic (AFM) ordering in a 2 × 1 × 1 supercell (211-AFM). The magnetic arrangements have FM ordering within the stripes, and have AFM ordering between neighboring stripes. (c) AFM ordering in 4 × 1 × 1 (411-AFM). The magnetic arrangements have FM ordering with adjacent stripes, while they have AFM ordering between the two next neighboring stripes. (d) Due to the triangular networks of TM atoms, frustrated magnetic orderings are considered to have a supercell with the angle of spin directions between adjacent TM atoms being 120°.

Fig. 2 Four non-equivalent magnetic orderings in TMX₂ monolayers: (a) FM ordering. (b) AFM ordering in a 2 × 1 × 1 supercell. (c) AFM ordering in a 4 × 1 × 1 supercell. (d) Noncollinear AFM with 120° clockwise ordering in a supercell.

Based on the magnetic structure prototypes, DFT calculations are carried out for four such spin configurations of all 90 TMX₂ monolayers. According to the total energies of different magnetic orderings obtained by the GGA+U (U = 2 eV) method, the most favorable configurations of each TMX₂ in the corresponding stable geometrical phase are listed in Fig. 3. There are 55 out of 90 systems with a total magnetic moment larger than 0.2 μ_B in the FM configuration, which will be considered as magnetic. In accordance with Fig. 1, the stable 2H phases are mainly concentrated in the nonmagnetic (NM) TMX₂ monolayers with TM atoms with d¹–d² configuration, while magnetic FeX₂ (X = Cl, Br, I) and TcX₂ (X = Br, I) monolayers with a stable 2H phase are exceptions to this rule. Besides, the Zn-group monolayers with a stable 1T phase are also nonmagnetic.

Fig. 3 Overview of ground geometrical structure and magnetic states for all 90 transition metal dihalides with U = 2.

Since the majority of magnetic TMX₂ monolayers are more stable in the 1T phase, in the following discussions the magnetic and electronic structures are mainly discussed in 1T-TMX₂ monolayers. As for FeX₂, the energy differences between the 1T and 2H phase range from 10 to 63 meV per atom under different U, which are quite small, and 1T FeX₂ monolayers have been proved to be stable in 2D materials database.^6,40 Thus, our analysis of the electronic and magnetic properties of FeX₂ are discussed in the 1T phases. Tc is a highly radioactive element which needs to be avoided, thus no further discussions will continue. From the magnetic point of view, it can be clearly noticed by comparing TMX₂ monolayers in the sequence from 3d to 4d to 5d elements that the tendency of magnetic systems has a remarkable right-shift. It can be well explained by d band broadening which will prevent higher d transition metal based compounds from being spin polarized, i.e. magnetic. The argument is in good accordance with the magnetic behavior of d electrons in transition metal monolayers on noble metal (001) substrates in ref. 41.

At current stage, we have obtained an overall landscape of 90 TMX₂ monolayers with both geometrical and magnetic ground states clarified under the condition of U = 2. However, one threatening problem is that there is no experimental result for such 2D systems. In this case, the effect of the correlations on d electrons of TM atoms may not be described precisely without supportive experimental data. To solve these problems and to make the theoretical study more instructive, we implement HTP calculations on 55 magnetic 1T-TMX₂ systems with sequential U values from 0 to 5 in a batch mode to determine the relations between U values and magnetic ground states. Magnetic ground states are evaluated by the energy difference between FM and the most stable AFM states (ΔE). The concrete classification is shown in Fig. 4a. We can clearly see from the pie-chart diagram that 55 out of 90 are magnetic systems. Taking magnetic TMCl₂ monolayers for example, the energy difference as shown in Fig. 4b can intuitively reflect the changing tendency of magnetic ground state with respect to different U values from 0 to 5. Among all magnetic systems, the spin configurations are classified into two cases with respect to U values: first, 23 U-independent magnetic systems which indicate that the magnetic ground states will not be changed by different U values exhibiting robustness. The 23 TMX₂ with robust magnetic ground states include TMCl₂ (TM = V, Cr, Mn, Ni, Ag, and Pt), TMBr₂ (TM = Ti, V, Cr, Co, Ni, Ag, and Pt), and TMI₂ (TM = Sc, Ti, V, Mn, Fe, Co, Ni, Pd, and Au). Results indicate that U-independent magnetic TMX₂ monolayers are mainly dominated by 3d TM dihalides, which can be explained by weaker exchange correlation potentials for lighter TM atoms. Besides, TM dihalides with IB and IIB TM atoms can also be robust magnetic systems against Hubbard U values. Second, 32 U-dependent magnetic systems which show that the magnetic properties will change among different U. Among 32 U-dependent TMX₂ monolayers, there are 11 systems (U-regular cases) with magnetic ground states which can be tuned regularly by enlarging U values by the linear change of the energy difference. Meanwhile the magnetic ground states of the other 21 systems (U-irregular cases) cannot be linearly tuned by enlarging the U values.

Fig. 4 (a) Specific classification of 90 TMX₂ systems with relation to different U values (from 0 to 5). (b) Energy difference between FM and the most stable AFM states of magnetic TMCl₂ monolayers with different U values.

It is worth mentioning that among the 23 magnetism-robust systems, 4 of them are robust ferromagnetic monolayers including 1 ferromagnetic insulator (NiCl₂) and 3 half metals including AgCl₂, AgBr₂ and AuI₂, while 8 out of 19 U-independent cases are noncollinear AFM monolayers. Our results for the ground states of all magnetic TMX₂ monolayers are summarized in the Table in the ESI,† corresponding to the classification shown in Fig. 4a. We will make explicit analysis for these 12 1T-TMX₂ monolayers with robust magnetic orderings with respect to U values in the following discussions.

Band structures with both spin-up and spin-down channels for four robust FM 1T-TMX₂ monolayers with U = 2 are as shown in Fig. 5. Band curve dispersions in the whole Brillouin zone indicate similar state distributions due to the influence of the same crystalline field. The spin splitting near the Fermi level of NiCl₂ is about 2.3 eV, that is, larger than the spin splittings of AgCl₂, AgBr₂ and AuI₂, which are in the range of 0.2 to 0.5 eV. The magnetization for NiCl₂, dominated by the Ni atom, is close to 2 Bohr magneton (μ_B) per atom indicating 8 electrons in d orbital filling with high-spin states. Meanwhile for the other Au or Ag based dihalides, the magnetization is distributed and delocalized in both Au (Ag) and halide atoms smaller than 0.5 (μ_B) per atom, which can also be explained from particle density of states as shown in Fig. 5. For AgCl₂, AgBr₂ and AuI₂ monolayers, the d orbitals of Au or Ag are coupled with halide p orbitals which result in strong interaction between adjacent atoms. Besides, AgCl₂, AgBr₂ and AuI₂ monolayers are half metals with a Fermi level only crossing one spin channel with 100% spin polarization. To ensure the reliability of the band structures of TMX₂, the screened hybrid functional Heyd–Scuseria–Ernzerhof (HSE) is further implemented.⁴² Results indicate that both GGA+U and HSE functional do not change the energy dispersions, and orbital occupations remain similar with the two methods. An exception is represented by the band gap which in the HSE functional is larger with respect to GGA+U results. As for NiCl₂, the band gap in the HSE functional is 4.48 eV. Our results are consistent with previous studies.²³ However, this difference does not change other electronic and geometric properties, including exchange interactions and magnetic orderings.

Fig. 5 Spin polarized band structures with U = 2 for 4 U-independent FM cases: (a) NiCl₂, (b) AgCl₂, (c) AgBr₂ and (d) AuI₂, respectively.

It is known that ferromagnetic insulators are in high demand in the development of next-generation quantum-spintronic devices, in particular those high symmetric systems without chemical doping or tensile strain that can only work at very low temperatures (below 16 K).⁴³ Thus, studies of the magnetic mechanism of ferromagnetic insulator NiCl₂ monolayers are further required. Since the magnetic moments are mostly located on Ni atoms, the total energies can be mapped onto a Heisenberg model:

where J_ij are exchange coupling parameters between spins at site i and site j, is the spin operator at site i. By considering four unequal magnetic configurations and calculating three independent total energy differences between two random configurations, we obtain the exchange parameters for the nearest neighbor J₁, the second nearest neighbor exchange parameter J₂ and the third nearest neighbor J₃ as illustrated in Fig. 6:

E_FM − E_211AFM = −8(J₁ + J₂)S² (2)

E_FM − E_411AFM = (−4J₁ − 8J₂ − 8J₃)S² (3)

E_FM − E_120°AFM = −6(J₁ + J₃)S² (4)

Fig. 6 Schematic diagrams of four representative magnetic ordering arrangements corresponding to Fig. 2. Blue balls indicate TM atoms which form six trigonal lattices with nearest neighbors. The black circles identify the exchange coupling J₁ between nearest neighbors (NN) of the central TM atom, the green circles identify the exchange coupling J₂ between second nearest neighbors (SNN) and the orange circles identify the exchange coupling J₃ between the third nearest neighbors (TNN).

By combining these three equations, the three exchange parameters of ferromagnetic NiCl₂ are calculated as shown in Table 1. The exchange parameters: J₁ = 5.46 meV, J₂ = −0.23J₁, and J₃ = 0.14J₁ which notably indicate strong ferromagnetic ground state orderings. The Curie temperature of NiCl₂ is further evaluated by mean field theory using , where .^44,45 Results show that NiCl₂ is a U-independent ferromagnetic insulator with a Curie temperature of 57 K.

Robust AFM states with respect to U parameters in TMX₂ are also worth mentioning. Previous studies have shown that antiferromagnets have typically much faster dynamics than ferromagnets, as the reorientation of antiferromagnetic moments will cost exchange energy. In particular, the insulating antiferromagnets have been proved to have significant application in optical devices. Early in 2004, Kimel et al. uncovered that the spins of the antiferromagnet TmFeO₃ can be tuned on a timescale of a few picoseconds, in contrast to hundreds of picoseconds in a ferromagnet.⁴⁶ Besides, Takei et al. proved theoretically that spin–superfluid transport can be realized in AFM insulator.⁴⁷ In TMX₂ systems, we filter 8 noncollinear AFM insulators whose magnetic ground state will not change with respect to various U, and the corresponding band structures are listed in Fig. 7. The results show that the band gaps of these noncollinear AFM insulators are in the range between 0.5 and 2.5 eV with U = 2. The NN, SNN and TNN exchange parameters listed in Table 1 are analyzed by the Heisenberg model. The positive J₁ is 3.8 to 5.6 times larger than the absolute value of J₂, which is consistent with the favored noncollinear AFM spin configurations. Besides, the in-plane magnetic anisotropy energy (MAE) for 8 noncollinear AFM are estimated by a rotating magnetic orientation with an interval of 10°. These results indicate that the in-plane MAE is smaller than 10 μeV per unit-cell for each noncollinear AFM TMX₂, which shows that the in-plane magnetic orientation can overcome small thermal fluctuation and is of high flexibility. There is one thing worth mentioning for low-dimensional magnetic systems, especially for noncollinear cases, which is that a spin–spiral structure may exist. In previous studies, the exchange interaction parameters of the 3d transition metal chains have been evaluated by using energy dispersion relations of the spin–spiral waves. Results show that the magnetic coupling in V, Mn, Cr chains are frustrated.⁴⁸ The results are in consistent with the frustrated magnetic behaviors in Mn or V based TMX₂ systems. For a better understanding of the spin–spiral waves and the exchange interaction of transition metal dihalides, further studies need to be explored.

Fig. 7 Band structures for noncollinear TMX₂ systems with U = 2.

Table 1 Exchange parameters J₁, J₂ and J₃ (in the unit of meV) and Curie temperatures (in the unit of K) estimated by mean field theory of ferromagnetic NiCl₂, and 8 noncollinear antiferromagnetic TMX₂ monolayers

TMX2	J 1	J 2	J 3	T C	μ B	GS
NiCl2	5.46	−1.28	0.75	57	2	FM
VCl2	−28.54	6.26	−3.97	—	3	120°-AFM
MnCl2	−3.02	0.68	−0.65	—	4	120°-AFM
PtCl2	−18.06	4.03	−7.63	—	1	120°-AFM
VBr2	−17.07	3.91	−2.32	—	3	120°-AFM
VI2	−7.68	1.84	−1.21	—	3	120°-AFM
MnI2	−2.93	0.52	−0.87	—	4	120°-AFM
CoI2	−2.80	0.69	−2.3	—	3	120°-AFM

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IV. Conclusions

In summary, by employing a high throughput study combined with first principles calculations, we systematically study the magnetic ground states of transition metal dihalide monolayers with respect to different Hubbard U values. The geometrical phase stability is first investigated by considering both the 2H and 1T phase in a hexagonal TMX₂ lattice. The results show that 50 out of 55 magnetic TMX₂ are more stable in the 1T phase except for FeX₂ (X = Cl, Br and I) and TcX₂ (X = Br and I), while the stable 2H phase is mainly concentrated in nonmagnetic TMX₂ monolayers with a TM belonging to d¹–d² elements. The magnetic ground states for 1T-TMX₂ monolayers are considered thoroughly under a batch of Hubbard U values from 0 to 5 eV by considering four typical magnetic orderings. Interestingly, we have found 23 U-independent magnetic TMX₂ systems with robust magnetic ground states not changed by the U values. Among them, NiCl₂ is a ferromagnetic insulator with a magnetic moment of 2 μ_B on Ni, and the Curie temperature estimated by Heisenberg model is 57 K. Besides, 8 TMX₂ monolayers including VX₂ (X = Cl, Br and I), MnX₂ (X = Cl, Br and I), PtCl₂ and CoI₂ are proved to be noncollinear antiferromagnetic insulators. The in-plane MAE is estimated to be smaller than 10 μeV per unit-cell and such tiny MAE will not stabilize the magnetic order against thermal fluctuation, which indicates the in-plane magnetic orientation is of high flexibility. Generally, we have presented a thorough understanding of the magnetic behavior of transition metal dihalides TMX₂ with respect to Hubbard U. Our studies demonstrate that TMX₂ monolayers are promising for spintronic applications, and especially TMX₂ monolayers with robust noncollinear antiferromagnetic orderings are possible candidates for skyrmions and quantum spin liquids.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the Priority Program 1666 of the Deutsche Forschungsgemeinschaft (DFG). Xinru Li gratefully acknowledges the financial support by the Sino-German (CSC-DAAD) postdoc scholarship program and the special funds for theoretical physics of National Natural Science Foundation of China (No. 11947121). We acknowledge computing time on the Lichtenberg supercomputer at TU Darmstadt.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9na00588a

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