Mechanical, Electronic, and Magnetic Properties of NiX2 (X = Cl, Br, I) Layers

Since the recent experimental discovery of the CrI3 and CrGeTe3 monolayers, van der Waals (vdW) layered transition metal compounds have been recognized as promising candidates to realize 2D ferromagnetic (FM) semiconductors. However, until now, only limited compounds have been proposed to be ferromagnetic semiconductors. Here, on the basis of first-principles calculations, we report that the monolayer, Janus monolayer, and bilayer of NiX2 (X = Cl, Br, I) are intrinsic 2D FM semiconductors. Our results show that exfoliation energy of the NiX2 monolayer is smaller than that of graphene, and all studied NiX2 layers show semiconducting band gaps. The predicted Curie temperature values for NiX2 (X = Cl, Br, I) monolayers ranged from 120 to 170 K with Monte Carlo simulations. For the Janus monolayer, we found that the spin interaction shows a very strong magnetoelectric coupling under an external electric field. Furthermore, for the bilayer of NiX2, our results show that the interlayer coupling is quite weak, indicating the possibility of tuning the magnetic coupling through external manipulations.


■ INTRODUCTION
During the past decade, two-dimensional (2D) materials, such as graphene, 1 h-BN, 2 MoS 2 , 3 and black phosphorus, 4 have received much attention because of their unique properties and potential applications in future nanodevices. 5 Magnetism, especially in 2D systems, is one of the most fascinating properties of materials, not only because of the complex magnetic behavior itself but also due to its interplay with the other important properties of materials such as superconductivity, ferroelectricity, and quantum Hall effects. However, in early studies, long-range magnetic order is predicted to be prohibited in a 2D system according to the Mermin−Wagner theorem. 6 Recently, Gong et al. 7 and Huang et al. 8 reported their discovery of long-range ferromagnetic (FM) order in 2D intrinsic semiconductors, Cr 2 Ge 2 Te 6 and CrI 3 , which are exfoliated down to atomically thin layers from their van der Waals (vdW) layered bulk materials. These exciting findings promote the studies of 2D magnetic semiconductors to a new stage. But the measured Curie temperature (T C ) values of these materials are very low (<45 K), which badly hinders their practical applications in spintronic devices.
Tremendous efforts have been devoted to realize ferromagnetic order in 2D semiconductors with higher Curie temperature. For example, embedding transition metal atoms and applying strain, defects, or boundaries 9−13 have been adopted as a strategy to induce magnetism in a nonmagnetic system. But these methods are usually difficult to control in an experiment, and the induced ferromagnetism is very weak. Recently, the vdW layered transition metal compounds have been recognized as promising candidates to realize 2D FM semiconductors because these materials usually possess intrinsic magnetism and can be easily exfoliated down to monolayers. Several 2D systems have been confirmed to be intrinsic FM materials such as VSe 2 , 14 MnSe 2 , 15 FePS 3 , 16 and MnO 2 , 17 but all these materials are metallic. 2D FM semiconductors are still very rare, and their T C values are much below room temperature because the virtual exchange interactions in a semiconductor are usually much weaker than the carrier-driven exchange interactions in a metallic system. Thus, to better understand the intrinsic mechanism of ferromagnetic semiconductors, it becomes more and more important to explore new families of semiconducting monolayers with ferromagnetism.
Here, on the basis of first-principles calculations, we predict that the family of NiX 2 (X = Cl, Br, I) layers can be promising 2D intrinsic FM semiconductors. Our results show that exfoliation energy of the NiX 2 monolayer is smaller than that of graphene, and all the NiX 2 monolayers show semiconducting band gaps, which vary from 1.24 to 2.60 eV.
Their Curie temperature (T C ) values were predicted from 120 to 170 K with Monte Carlo simulations. For the Janus monolayer, we found that the spin interaction shows a very strong magnetoelectric coupling. Furthermore, for the bilayer of NiX 2 , our results show that the interlayer coupling is quite weak, indicating the possibility of tuning the magnetic coupling through external manipulations.

■ RESULTS AND DISCUSSION
The large interlayer distance between vdW layers (3.08, 3.28, and 3.35 Å for NiCl 2 , NiBr 2 , and NiI 2 , respectively) of NiX 2 bulks implies a weak interlayer interaction between layers and the possibility to obtain 2D monolayers with a suitable exfoliation method. To demonstrate this, two fundamental questions should be taken into account. One is the cleavage energy, which gives a quantitative description of the strength of interlayer binding, and the other one is the in-plane stiffness of the corresponding 2D monolayer, which determines whether the exfoliated monolayer can have a large area and be freestanding. Generally, a small cleavage energy and strong inplane stiffness are what we need.
To achieve the 2D monolayer from the vdW bulk crystals, the most commonly used approaches are mechanical cleavage and liquid exfoliation. 1,18,19 To confirm that it is possible to exfoliate NiX 2 monolayers from their bulk phases in the experiment, we first calculated the cleavage energy. A gradually expanded fracture is introduced in the bulk to simulate the exfoliation procedure (Figure 1a,b). 20 As expected, the total energy increases as the separation between two fractured parts increases ( Figure 1c). It can be found that the total energy increases rapidly when the separation is less than 3.0 Å. As the separation becomes larger, the total energy barely changes. Then, we find that the cleavage energies for NiCl 2 , NiBr 2 , and NiI 2 are 0.223, 0.242, and 0.26 J/m 2 , which are smaller than that for graphite (∼0.36 J/m 2 ), 21,22 implying that NiX 2 can be easily exfoliated down to the monolayer in the experiment. We also calculate the cleavage energy of bilayer nanosheets and obtain very similar values, which are 0.224, 0.243, and 0.26 J/ m 2 .
To obtain a free-standing membrane during the exfoliation process in the experiment, it is important to avoid curling or buckling. To investigate the in-plane stiffness of NiX 2 monolayers, the 2D Young's modulus is evaluated according to the following equation where E is the total energy per unit cell, a 0 is the 2D lattice constant, and A is the corresponding surface area. Figure 1d shows the profile of total energy versus lattice constant a 0 for NiX 2 monolayers. The 2D Young's moduli for NiCl 2 , NiBr 2 , and NiI 2 are calculated to be 54, 50, and 45 N m −1 , respectively, which are close to that of the MnPSe 3 23 monolayer and comparable to that of the ultrastrong material graphene (∼340 N m −1 ). 24,25 Further, according to the elastic

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Article theory, the typical out-of-plane deformation h induced by gravity can be estimated by the formula 25 where p = 2.00 × 10 −6 , 3.06 × 10 −6 , and 3.78 × 10 −6 kg/m 2 are the densities of 2D NiCl 2 , NiBr 2 , and NiI 2 , respectively, and L is the edge length. Assuming that L ≈ 100 μm, we obtain h/L values of 3.31 × 10 −4 , 3.91 × 10 −4 , and 4.35 × 10 −4 for 2D NiCl 2 , NiBr 2 , and NiI 2 , respectively. These values are of the same order of magnitude as that of graphene. 25 These suggest that the NiX 2 monolayers are stiff enough to withstand its own weight and keep a free-standing planar structure during exfoliation.
The NiX 2 monolayers possess a structure similar to that of the T-MoS 2 monolayer, which belong to the P-3m1 layer group. No Jahn−Teller distortion is observed, and each Ni atom is coordinated to six ligands. The structural parameters are listed in Table 1. As the atomic radius increases from NiCl 2 to NiI 2 , the lattice constant a 0 , bond length of Ni−X atom marked as d Ni−X , and vertical distance between two halide planes d 0 also increase. The calculated lattice constants a 0 of the monolayers are very close to those of their bulks (3.483, 3.699, and 3.983 Å for NiCl 2 , NiBr 2 , and NiI 2, respectively), suggesting very weak interlayer interactions between vdW layers.
In NiX 2 monolayers, because of the octahedral crystal field caused by the ligands, the Ni d orbitals split into two parts, namely, the lower t 2g and the higher e g manifolds. Each Ni gives two electrons to form ionic bonding with the ligands and leaves eight electrons, which fully occupy the t 2g orbitals and the spin-up e g orbitals, and the spin-down e g orbitals are empty. Thus, the Ni 2+ ion shows an occupation state of d 8 with a magnetic moment of ∼2 μ B . A sizable electronic band gap is expected to be opened by the crystal field. To determine the magnetic ground state of monolayer NiX 2 , we carry out spinpolarized calculations. Two different magnetic configurations are considered in a 2 × 1 super cell, that is, the FM state and antiferromagnetic (AFM) state. The spin densities for these two states are shown in Figure 2a Table 2. It is found that the FM states for NiCl 2 , NiBr 2 , and NiI 2 monolayers are lower in energy than the corresponding AFM states by 11.2, 12.6, and 15.4 meV per unit cell, respectively, suggesting that the ground states are FM; this is consistent with recent studies reported by Mounet et al. 26 It is known that the results from GGA + U calculations for a magnetic system sometimes may depend on the adopted value of effective Hubbard U (U eff ). Thus, to verify our result, we repeat the GGA + U calculations with U eff = 2−5 eV for Ni d orbitals for the NiI 2 monolayer. The ground state remains FM with exchange energy ranging from 12 to 23 meV. Thus, the predicted ferromagnetism in NiI 2 is robust against the value of Hubbard U. Figure 3 shows the electronic structures of FM ground states for NiX 2 monolayers. Here, we focus on the results calculated by the PBE + U method. The effect of spin-orbit coupling (SOC) can be further elucidated by comparing the electronic band structure, as shown in Figure 3. It is found that all the three systems are semiconductors with indirect band gaps of 2.60, 1.97, and 1.24 eV for NiCl 2 , NiBr 2 , and NiI 2 , respectively, consistent with our above analysis. The decrease in band gap from NiCl 2 to NiI 2 is due to the decrease in strength of the crystal field, which is inversely proportional to the Ni−X bond length. Interestingly, the value of exchange energy increases from NiCl 2 to NiI 2 . This may be caused by two factors: (i) The reduction of energy gap between t 2g and e g orbitals will strengthen the t 2g −e g virtual exchange interaction, benefiting the FM coupling, which has been carefully discussed in our previous work. 27 (ii) I has a smaller electronegativity than Cl, namely, the on-site energy of I p orbitals is closer to that of Ni d orbitals than the Cl p orbitals do. Thus, the d−p−d superexchange in NiI 2 is stronger than that in NiCl 2 , which can be confirmed by comparing the PDOS between NiI 2 and NiCl 2 , where the broadening of Ni d orbitals in energy for NiI 2 is larger than that for NiCl 2 . Similar phenomena are also found in previous studies on transition metal halides monolayers such as CrX 3 28 and VX 3 . 29 After considering SOC, there is no noticeable change in the electronic band structure, as shown in Figure 3, with the indirect band gaps of 2.57, 1.82, and 1.03 eV for NiCl 2 , NiBr 2 , and NiI 2 , respectively.
For practical spintronic applications of 2D FM semiconductors, it is necessary to investigate the magnetic behavior under finite temperature. The Ising model has been widely used to describe the magnetic coupling in 2D magnetic systems. 17,30−32 Here, we also use the Ising model including nearest-neighboring exchange interactions to study the magnetic behavior of NiX 2 (X = Cl, Br, I) monolayers. The spin Hamiltonian where the summation i runs over all Ni sites, j runs over the six nearest neighbors of site i. J ij is the nearest-neighboring exchange parameter. Then, we performed Metropolis Monte Carlo simulations 33,34 to estimate the T C . A 30 × 30 hexagonal superlattice containing 900 magnetic sites and the periodic boundary condition is used. The average magnetization per formula unit and the specific heat [C v = (⟨E 2 ⟩ − ⟨E⟩ 2 )/k B T 2 ] are taken after the system reaches the equilibrium state at a given temperature (Figure 4). The T C can also be extracted from the peak of the specific heat profile. 35 The estimated T C values for NiX 2 (X = Cl, Br, I) monolayers are 120, 140, and 170 K, respectively, which are much larger than the reported values for 2D CrI 3 (∼45 K) 8 and Cr 2 Ge 2 Te 6 (∼30 K) 7 and are higher than the liquid nitrogen temperature (77 K).
For practical uses, a 2D material usually needs to be supported by a suitable substrate, which may apply an in-plane strain and affect the performance of the pristine material. Thus, it is necessary to investigate the magnetic and electronic properties of NiX 2 monolayers under a proper in-plane strain.

Article
Here, we focus on NiI 2 because it has the highest T C among the three systems. The applied biaxial in-plane strain is defined as ε = (a − a 0 )/a 0 × 100%, where a 0 and a are the lattice constants of 2D NiI 2 in its equilibrium and strained states, respectively. Positive and negative values of ε represent tensile and compressive strain, respectively. As shown in Figure 5a, a tensile strain will reduce the exchange energy. This is opposite to previous studies on CrI 3 28 and CrGeTe 3 36 where the exchange energy is increased by a tensile strain. It can be understood that the Cr 3+ ion in CrI 3 and CrGeTe 3 has a d 3 occupation state, which is less than half-filled. In this case, the AFM direct exchange mainly contributed by the t 2g −t 2g hybridizations, which can be fairly reduced by a tensile inplane strain that increases the distance between neighboring Cr 3+ ions. Thus, a proper tensile in-plane strain can enhance the FM couplings in CrI 3 and CrGeTe 3 . But in NiI 2 , the occupation state of Ni 2+ (d 8 ) is more than half-filled. In this case, the AFM direct exchange originates from e g −e g hybridizations, which is usually much weaker than the t 2g −t 2g hybridizations and is not sensitive to a tensile strain, while the tensile strain reduces the d−p exchange interactions, weakening the FM couplings between adjacent Ni 2+ ions. When applying a compressive strain, the exchange energy decreases. This is mainly because a compressive strain will strengthen the crystal field, increasing the energy gap between t 2g and e g orbitals ( Figure 5c) and weakening the FM t 2g −e g virtual exchange interactions. The indirect band gaps of NiI 2 under −4 and 4% strain are 0.90 and 1.43 eV, respectively. Overall, the electronic and magnetic properties of NiI 2 only slightly change under a moderate in-plane strain, which makes it a robust 2D FM semiconductor. PBE functional is known to usually underestimate the band gap; thus, we also repeat our calculation using HSE06 hybrid functional for NiI 2 monolayer. The band structure of NiI 2 calculated by HSE06 functional is also shown in Figure 5b, with an indirect band gap of 2.31 eV nearly twice as large as the one (1.24 eV) calculated by the PBE + U method. Also, the ground state is also FM with an exchange energy of 19.5 meV, which is a bit larger than the PBE + U result (15.4 meV).
Janus 2D materials with breaking mirror or inversion symmetry along out-of-plane orientations provide abundant new properties such as electric polarization and Rashba effect, which expands the promising applications of 2D materials. 37−39 Thus, it will be interesting to explore the properties of the Janus NiXY monolayer. Here, we calculated the NiICl monolayer. The calculated lattice constant is 3.74 Å, which is

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Article between those of NiI 2 and NiCl 2 . The breaking of inversion symmetry can be seen from the optimized structure ( Figure  6a). This induces a vertical electric polarization of 0.19 C/m 2 , making NiICl a 2D multiferroic material. The magnetic ground state is also FM for NiICl, with an exchange energy of 9.3 meV.
Interestingly, because of out-of-plane electric polarization, the response of magnetic coupling to a vertical external electric field also exhibits a "polar" behavior; that is, an electric field along the +z direction will enhance the FM coupling, while an electric field along the −z direction will weaken it (Figure 6b). On the other hand, the electronic structure of NiICl does not change much compared to that of the NiI 2 monolayer ( Figure  6c). On the other hand, 2D magnetic materials are usually used as a building block of vdW heterostructures and junctions to realize a highly functional nanodevice. For instance, the recently discovered 2D FM semiconductor CrI 3 shows AFM interlayer magnetic couplings in bilayer CrI 3 . This unusual property has attracted much attention. Thus, it is also important to explore the interlayer interactions between NiX 2 layers. Here, AA-and AB-stacked bilayers of NiCl 2 , NiBr 2 , and NiI 2 are considered in our calculations. The ABstacked bilayers are more favored than the AA-stacked ones with total energy values lower by 0.001, 0.001, and 0.002 eV for NiCl 2 , NiBr 2 , and NiI 2 , respectively. The lattice constants a 0 of AB-stacked bilayers NiCl 2 , NiBr 2 , and NiI 2 are 3.49, 3.70, and 4.01 Å, which are very close to their bulks, suggesting weak vdW interlayer interactions between layers.
For the magnetic coupling in such bilayer systems, two different magnetic configurations are considered, that is, the FM state and the interlayer-antiferromagnetic (I-AFM, where Ni ions in one layer possess spin-up magnetic moment and, in the other layer, possess spin-down magnetic moment) state. We find that the FM states are lower in energy than the I-AFM states. But the energy differences between FM and I-AFM states are very small, which are 0.27, 0.65, and 2.20 meV for AB stacking (0.17, 0.47, and 1.42 meV for AA stacking), suggesting that the magnetic coupling between NiX 2 layers can be easily tuned by external perturbations.

■ CONCLUSIONS
In summary, on the basis of first-principles calculations, we propose a class of 2D FM semiconductors, the NiX 2 (X = Cl, Br, I) monolayers by exfoliation from corresponding vdW bulk materials. The calculated cleavage energy of bulk NiX 2 (X = Cl, Br, I) is slightly smaller than that of graphite, indicating that exfoliation is possible in the experiment by mechanical cleavage or liquid exfoliation. Meanwhile, the calculated inplane stiffness implies that the 2D NiX 2 (X = Cl, Br, I) monolayers can keep their free-standing structures without curling or buckling. All three monolayers are semiconducting with band gaps of 2.60, 1.97, and 1.24 eV for NiCl 2 , NiBr 2 , and  . Specific heat C v with respect to temperature for the NiCl 2 , NiBr 2 , and NiI 2 monolayers; the inset shows the corresponding magnetization.

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Article NiI 2 , respectively. The ground states of the monolayers are FM with T C values ranging from 120 to 170 K. The T C increases from NiCl 2 to NiBr 2 and NiI 2 due to enhanced p−d exchange interactions and t 2g −e g virtual exchange interactions. Different from the usual case of tensile strain-induced enhancement of ferromagnetism in systems such as CrI 3 and CrGeTe 3 , FM couplings in NiI 2 are slightly decreased by both tensile and compressive strain due to its more than half-filled occupation state. Besides, for the Janus monolayer, we found that the spin interaction shows a very strong magnetoelectric coupling. Furthermore, for the bilayer of NiX 2, our results show that the interlayer coupling is quite weak, indicating the possibility of tuning the magnetic coupling through external manipulations. Our findings suggest that NiX 2 (X = Cl, Br, I) layers can have promising applications in spintronic devices such as spin valves, information transport, and storage between electric signals and spin signals. We hope that the present study will stimulate further experimental effort on this subject.

■ COMPUTATIONAL METHODS
The first-principles calculations based on spin-polarized density functional theory (DFT) are performed using the projected augmented plane-wave (PAW) 40 method as implemented in the Vienna ab initio simulation package (VASP). 41 Generalized gradient approximation (GGA) given by Perdew−Burke−Ernzerhof (PBE) was adopted for exchange−correlation functionals. 42 Considering the strongly correlated electrons in the partially filled d subshells, we use the GGA + U method introduced by Dudarev et al. 43 with U eff = 4 eV for Ni d orbitals. The electronic wave functions are expanded using a plane-wave basis set with a cutoff energy of 500 eV. For 2D monolayer system, a vacuum space of 20 Å along the z direction is adopted to avoid interactions between two neighboring images. The Γ-centered Monkhorst− Pack 44−46 point scheme with 8 × 8 × 1 and 4 × 8 × 1 grid meshes is used to sample the reciprocal space of the primitive cell and the 2 × 1 super cell, respectively. For the bulk crystals, the DFT-D3 method by Grimme 47 is used to account for the weak vdW interactions between different layers. During the optimization, both the lattice constants and positions of all

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Article atoms are relaxed without any symmetry restriction until the Hellmann−Feynman force on each atom is less than 0.01 eV Å −1 . The convergent criterion for the total energy is set as 1 × 10 −5 eV.