Electronic structure and magnetism of transition metal dihalides: bulk to monolayer

TL;DR

This study uses first-principles calculations to explore how the electronic and magnetic properties of transition metal dihalides change from bulk to monolayer, revealing potential ferromagnetic insulators suitable for 2D spintronics.

Contribution

It provides a comprehensive analysis of the evolution of electronic and magnetic states in MX$_2$ monolayers, identifying new ferromagnetic insulators with significant anisotropy.

Findings

FeX$_2$, NiX$_2$, CoCl$_2$, CoBr$_2$ monolayers are ferromagnetic insulators

Magnetocrystalline anisotropies are sizable in these monolayers

Spin-orbit coupling is crucial for accurate ground state prediction

Abstract

Based on first-principles calculations, the evolution of the electronic and magnetic properties of transition metal dihalides MX$_2$ (M= V, Mn, Fe, Co, Ni; X = Cl, Br, I) is analyzed from the bulk to the monolayer limit. A variety of magnetic ground states is obtained as a result of the competition between direct exchange and superexchange. The results predict that FeX$_2$, NiX$_2$, CoCl$_2$ and CoBr$_2$ monolayers are ferromagnetic insulators with sizable magnetocrystalline anisotropies. This makes them ideal candidates for robust ferromagnetism at the single layer level. Our results also highlight the importance of spin-orbit coupling to obtain the correct ground state.

Tables

Table 1. Table 1: Calculated in-plane lattice parameters for monolayer MX 2 within GGA.

$a$	Cl	Br	I
VX2	3.62	3.81	4.08
MnX2	3.64	3.84	4.12
FeX2	3.49	3.69	3.98
CoX2	3.49	3.73	3.92
NiX2	3.45	3.64	3.92

Table 2. Table 2: Magnetism of bulk MX 2 within GGA. The second column reflects the nature of the ground state (GS) where AFS stands for an antiferromagnetic striped configuration (in-plane stripes coupled AFM along c 𝑐 c ), AF for FM planes coupled AFM along c 𝑐 c , FM for ferromagnetic, I for insulator, and HM for half-metal. * indicates that FeX 2 become insulators after SOC and U 𝑈 U are included. The third column shows the energy difference (per unit cell) between ferromagnetic and corresponding antiferromagnetic ordering. The fourth column shows the magnetocrystalline anisotropy energy for the magnetic moments pointing in-plane versus out-of-plane (negative values reflect out-of-plane moments). The last column shows the calculated magnetic moment.

	GS	$\Delta$EFM-AFM (meV)	MAE (meV)	MM   (${\mu }_B$)
Bulk
VCl2	AFS-I	55.49	0.04	2.60
VBr2	AFS-I	25.57	0.03	2.61
VI2	AFS-I	10.13	0.62	2.62
MnCl2	AFS-I	18.45	0.09	4.50
MnBr2	AFS-I	12.27	0.03	4.52
MnI2	AFS-I	8.63	0.16	4.51
FeCl2	AF-HM*	14.74	-1.09	3.47
FeBr2	AF-HM*	5.44	-0.82	3.54
FeI2	AF-HM*	2.72	-1.05	3.40
CoCl2	AF-I	5.03	0.62	2.54
CoBr2	AF-I	1.49	0.76	2.49
CoI2	AF-I	13.60	0.17	2.24
NiCl2	AF-I	4.62	0.50	1.46
NiBr2	AF-I	2.99	0.26	1.39
NiI2	AF-I	27.33	0.30	1.25

Table 3. Table 3: Magnetism for monolayer MX 2 within GGA. The second column reflects the nature of the ground state where AFS stands for antiferromagnetic (stripe) in-plane order, FM for ferromagnetic in-plane order, I for insulator, and HM for half-metal. * indicates that FeX 2 become insulators after SOC and U 𝑈 U are included. The third column shows the energy difference between ferromagnetic and antiferromagnetic in-plane order. The fourth column shows the magnetocrystalline anisotropy energy for the magnetic moments pointing in-plane versus out-of-plane (negative values reflect out-of-plane moments). The last column shows the calculated magnetic moment.

	GS	$\Delta$EFM-AFS (meV)	MAE (meV)	MM   (${\mu }_B$)
Mono
VCl2	AFS-I	61.47	-0.53	2.67
VBr2	AFS-I	31.90	-0.30	2.67
VI2	AFS-I	10.70	-0.25	2.67
MnCl2	AFS-I	30.00	-0.20	4.54
MnBr2	AFS-I	15.36	-0.25	4.52
MnI2	AFS-I	12.92	-0.18	4.46
FeCl2	FM-HM*	-121.58	-0.89	3.57
FeBr2	FM-HM*	-67.66	-0.33	3.53
FeI2	FM-HM*	-35.90	-0.59	3.45
CoCl2	FM-I	-57.52	-0.69	2.54
CoBr2	FM-I	-10.33	-0.68	2.49
CoI2	AFS-I	14.15	-0.50	2.23
NiCl2	FM-I	-38.76	-0.12	1.68
NiBr2	FM-I	-32.70	-0.02	1.63
NiI2	FM-I	-33.73	-0.18	1.53

Table 4. Table 4: Magnetism for monolayer MX 2 (continued). The second column reflects the nature of the ground state where AFS stands for antiferromagnetic (stripe) order, FM for ferromagnetic order, I for insulator, and HM for half-metal. * indicates that FeX 2 become insulators after SOC and U 𝑈 U are included. The third column shows the value of the exchange interaction. The fourth column is an estimate of the Curie temperature.

	GS	$J$ (K)	Tc   (K)
Mono
FeCl2	FM-HM*	44.1	160
FeBr2	FM-HM*	24.5	89
FeI2	FM-HM*	13.0	47
CoCl2	FM-I	37.1	135
CoBr2	FM-I	6.7	24
CoI2	AFS-I	–	–
NiCl2	FM-I	56.2	205
NiBr2	FM-I	47.4	173
NiI2	FM-I	48.9	178

Equations

$$\begin{array}{r}\mid a_{1g}⟩=\frac{1}{\sqrt{3}}\left(\mid xy⟩+\mid xz⟩+\mid yz⟩\right),\\ \mid e_{g\pm }^{\ast }⟩=\pm \frac{1}{\sqrt{3}}\left(\mid xy⟩+e^{\pm 2\pi i/3}\mid xz⟩+e^{\mp 2\pi i/3}\mid yz⟩\right)\end{array}$$

$$J=\frac{\Delta E}{8S^2}$$

Full text

I Introduction

Long range magnetic order is a common phenomena in three-dimensional materials but not in lower dimensions. According to the Mermin-Wagner theorem, long-range magnetic order is not possible in 2D for spin-rotational-invariant systems.Mermin and Wagner (1966) However, magnetic anisotropies (i.e., those that break spin-rotational symmetry) remove this restriction. Magnetic van der Waals (vdW) materials are good candidates given their flexibility that can allow for a tuning of their magnetic anisotropy, as well as for their ease of exfoliation. They offer the possibility of obtaining a magnetic ground state even in the single-layer limit.

An example is CrI3, a layered Ising ferromagnet (FM) in which the Cr ions lie on a honeycomb lattice. In 2017, it was shown that ferromagnetism does survive at the single-layer level with a transition temperature near that of the bulk material (61 vs 45 K).Huang et al. (2017) Magnetism in 2D has the potential to open up a number of technological opportunities such as sensing, information, and data storage. 2D ferromagnets are particularly interesting for spintronic applications. Novel functionalities based on van der Waals heterostructures, in which magnetism adds a new ingredient, can also be anticipated. The potential use of CrI3 and other materials for building devices and tuning their properties through gating has already started to be explored, marking the birth of a new era of magnetism.Gong et al. (2017); Huang et al. (2018); Xing et al. (2017); Burch et al. (2018) However, pushing Tc to higher values will be necessary for real applications.

In this regard, there is a materials family related to CrI3 that holds equal promise: binary transition metal dihalides MX2 (M = transition metal, X = halogen: Cl, Br, I). They form low dimensional crystal structures composed of either one dimensional chains or two dimensional layers.McGuire (2017) The layered structure is shown in Fig. 1 where the vdW gap between layers is apparent. In contrast to trihalides, layered dihalides contain a triangular lattice of transition metal cations, and geometrical frustration in such a lattice is expected when the magnetic interactions are antiferromagnetic (AFM). This set of materials hence provides a rich playground for examining low dimensional magnetism. In the bulk, the magnetism of most of these materials was analyzed decades ago.McGuire (2017) At the monolayer level, some theoretical effort has been devoted to them,Ashton et al. (2017); Feng et al. (2018); Kulish and Huang (2017); Wasey et al. (2013); Torun et al. (2015) but the full trends in magnetism and electronic structure and in particular the magnetocrystalline anisotropy energy (crucial to establish 2D long range magnetic order) have not been completely studied for all possible 3$d$ transition metal and halide ions.

Here, we present a systematic study of the electronic structure and magnetism of MX2 compounds from the bulk to the monolayer limit by means of first principles calculations. After a general overview of the magnetic trends in bulk dihalides, we turn to results at the monolayer level, where a variety of magnetic states is found as a result of the competition between direct exchange and superexchange via the halogen $p$ states. We predict that FeX2, NiX2, CoCl2 and CoBr2 monolayers are ferromagnetic insulators with an easy axis normal to the planes and a sizable magnetocrystalline anisotropy making them ideal candidates for robust 2D ferromagnetism. Our results also highlight the importance of considering both the trigonal distortion of the MX6 octahedra as well as the spin-orbit coupling to obtain the correct ground state.

II Computational Methods

Our electronic structure calculations were performed using the all-electron, full potential code WIEN2k Blaha et al. (2001) based on the augmented plane wave plus local orbitals (APW + lo) basis set. The Perdew-Burke-Ernzerhof version of the generalized gradient approximation (GGA) Perdew et al. (1996) was used for structural relaxation and optimization. Then, the LDA+$U$ scheme within the fully localized limit was applied to treat the strong correlations.Lichtenstein et al. (1995) We have studied the evolution of the electronic structure with increasing $U$ ($U$= 2-6 eV, $J$= 0.8 eV). For all materials, we performed calculations in supercells of size 1\times$$\sqrt{3} to allow for the possibility of an in-plane antiferromagnetic ‘striped’ configuration.

For the calculations, we converged using RmtKmax = 7.0 with a fine $k$ mesh of 17$\times$17$\times$3 for the bulk materials and 19$\times$19$\times$1 for the monolayers. Muffin-tin radii of 2.48 a.u. for Fe, 2.45 a.u. for V, 2.45 a.u. for Co, 2.50 a.u. for Mn, 2.50 a.u. for Ni, 2.11 a.u. for Cl, 2.48 a.u. for Br, and 2.50 a.u. for I were employed.

To determine the magnetocrystalline anisotropy energy (MAE), we calculate how the direction of the spin of the metal atom affects the energy when spin-orbit coupling (SOC) is included with the moment orientation either in-plane or out-of-plane. SOC was introduced in a second variational procedure.Singh and Nordstrom (2006) The calculations of magnetic anisotropy require careful convergence of the total energy. We found that a converging criterion for the total energy to within 10*-6* eV yields stable results.

III Structural properties

As mentioned above, most of the bulk transition metal dihalides have a natural layered structure that contains triangular nets of cations in edge-sharing octahedral coordination forming MX2 layers separated by van der Waals gaps (Fig. 1). The octahedral crystal field will split the $3d$ orbitals of the metal atoms into higher lying eg ($d_{x^{2}-y^{2}}$, $d_{z^{2}}$) and lower lying t2g ($d_{xy}$, $d_{xz}$, $d_{yz}$) states. All of the nearest-neighbor distances within the MX6 octahedra are the same, but there is a trigonal distortion which further splits the t2g manifold. The consequences this has for the electronic structure are explained below.

MX2 compounds adopt either the trigonal CdI2 structure (so called 1-T with $P\bar{3}m1$ space group) or the rhombohedral CdCl2 ($R\bar{3}m$) one. These structures have different stackings along the $c$-axis. The Cr and Cu materials are slightly different in that their structure in the bulk is 1D-like and monoclinic, respectively, hence we do not analyze them here. The structure of monolayer dihalides is analogous to that of the intensively studied transition metal dichalcogenides in which ferromagnetism at the monolayer level has been anticipated by DFT calculations.Ma et al. (2012) Regardless, based on the known bulk values, larger magnetic moments in 2D dihalides can be expected.

It was previously found that all dihalides prefer the 1-T crystal structure at the monolayer level.Kulish and Huang (2017) The stability of single-layer dihalides was also evaluated from their formation energy confirming that not only are metal dihalide monolayers stable, but also that they could potentially be exfoliated.Mounet et al. (2018); Kulish and Huang (2017) Based on this, for the monolayer we will focus on 1-T structures only. The calculated structural parameters of MX2 monolayers are summarized in Table 1 and agree with Ref. Kulish and Huang (2017) and with the experimental bulk data.

IV Dihalides - Bulk magnetism

We analyze first the electronic structure and magnetism of the bulk materials, experimentally studied already, to test the validity of our predictions. All the materials are insulators as shown by experiment and confirmed by our calculations (to open up a gap, a $U$ is required in some cases). As mentioned above, there are two competing magnetic interactions in the planes: direct exchange between transition metal cations, and superexchange through the halogen anions. The magnetism of these materials can be understood from the Goodenough-Kanamori-Anderson rules.Goodenough (1963) In the case of a 90∘ M-X-M bond angle, the eg-eg exchange is always FM and weak, the direct t2g-t2g overlap can give rise to an AFM exchange, and depending on the particular orbital occupation, the t2g-t2g superexchange via halides can either be AFM or weakly FM. For the case of edge-sharing octahedra, the t2g orbitals on neighboring sites, pointing between the oxygens, are directed toward each other. The resulting $d-d$ hopping turns out to be very important for early $3d$ metals (i.e., V) and it can give rise to AFM exchange. As a note, the magnetic order found in most of these compounds either consists on ferromagnetic planes, stripes, or is helimagnetic.McGuire (2017) The results of the density functional theory (DFT) calculations are shown in Table 2. We discuss them below in order of increasing $d$ count. Three magnetic structures were explored: AFS (stripe) order, which was taken to be alternating rows of ferromagnetic spins stacked AFM along $c$, AF order taken to be ferromagnetic planes stacked AFM along $c$, and FM order. These allow us to capture the physics of most of the materials. However, the non-colinear nature of the order observed in some of the materials is beyond the scope of the present work.

VX2. In these materials, V is in a high spin $d^{3}$ configuration (S = 3/2). Neutron powder diffraction studies found that all three vanadium dihalides order antiferromagnetically with Néel temperatures of 36.0, 29.5, and 16.3 K for Cl, Br and I, respectively,Hirakawa et al. (1983) with the low values relative to the Weiss temperatures (437 K, 335 K, 143 K) Niel et al. (1977) being an indication of geometric frustration. Our AFS (stripe) calculations indeed confirm that the magnetic order involves antiferromagnetic in-plane coupling, with the moments predicted to lie in-plane. The actual moment orientation for VCl2 and VBr2 is non-colinear (120 degree orientation of the moments) as typical for triangular antiferromagnets, and moreover the moments appear to be tilted out of the plane.Hirakawa et al. (1983); Kadowaki et al. (1985, 1987). VI2 is more complicated in that the 120 degree order occurs first, and then slightly below this at 14.4 K a stripe phase (the same as simulated here) sets in, though the moments are probably also tilted out of the plane.Kuindersma et al. (1979); Hirakawa et al. (1983) The closeness of the two phase transitions indicates that the free energy difference between the 120 degree order and the stripe phase is small.

MnX2. In these materials, Mn is in a high spin $d^{5}$ configuration (S = 5/2). For this filling, superexchange is AFM. The magnetic structure for MnCl2 has ferromagnetic stripes of width two rows within the layers with antiferromagnetic coupling between neighboring stripes and between the layers.Wilkinson et al. (1958) There is evidence that the moments lie in-plane for the Cl and Br materials,Wollan et al. (1958) that our DFT calculations confirm. For simplicity, we approximate this state by the one width row AFS (stripe) order for the purposes of Table II. MnI2 adopts a complicated helical magnetic structure at low T stemming from competition with longer range exchange.Cable et al. (1962) The development of a ferroelectric polarization was recently reported, spurring interest in this compound as a multiferroic material.Kurumaji et al. (2011)

FeX2. Divalent iron, $d^{6}$, is expected to be in a high spin state, S = 2. Given the partially filled $t_{2g}$ levels, an orbital moment may be expected as well. The magnetic structure of FeCl2 and FeBr2 shows ferromagnetic order within the layers and antiferromagnetic stacking along $c$.Wilkinson et al. (1959) Our calculations do reproduce the observed magnetic order. The iodide counterpart, though, adopts the two-row width stripes as found in MnBr2.Gelard et al. (1974) The moments are along the $c$ axis in all cases. Our obtained magnetocrystalline anisotropy energies confirm that the moments lie out-of-plane.

CoX2. Co2+, $d^{7}$, can be in a high spin (S = 3/2) or low spin (S = 1/2) state. Neutron diffraction results show that the high spin state is preferred, at least for CoCl2 and CoBr2, and our DFT calculations confirm this picture.Wilkinson et al. (1959) Below their ordering temperatures, both of these compounds adopt a magnetic structure with ferromagnetic alignment within each layer and antiferromagnetic stacking. This is correctly reproduced by our calculations, along with the fact that the magnetic moments lie in-plane. The magnetic behavior in CoI2 is more complex - it is a helimagnet with a spiral spin structure, indicating the presence of longer range exchange.Mekata et al. (1992)

NiX2. Divalent nickel in these compounds has a $d^{8}$ configuration, with low spin being S=0 and high spin S=1, the later being favored. Both NiCl2 and NiBr2 show a magnetic configuration with moments lying within the plane that are ferromagnetically aligned within each layer, with antiferromagnetic stacking.Adam et al. (1980) Our DFT calculations are able to reproduce this. NiBr2 develops helimagnetic order below a lower second transition, that is seen as well below TN in the iodide material. Like MnI2 and CoI2 described above, NiBr2 and NiI2 also develop a ferroelectric polarization in their helimagnetic states. Kurumaji et al. (2013); Tokunaga et al. (2011)

The trend in the magnetic moments across the MX2 series is the expected one according to the spin states described above: starting with $\sim$3 $\mu_{B}$ per metal atom for the V-halides, increasing to $\sim$5 $\mu_{B}$ for the Mn-halides, and then gradually decreasing to $\sim$2 $\mu_{B}$ for the Ni-halides (Table 2). The halogens develop a magnetic moment of 0.16-0.20 $\mu_{B}$ and hence show a significant spin-polarization.The magnetic moment on the metal atoms decreases with the Z of the halide atom (Cl - Br - I). In the Fe and Co compounds, an orbital moment $\sim$ 0.1 and 0.2 $\mu_{B}$, respectively, is found when SOC is included.

V dihalides-monolayer magnetism

After confirming the correct magnetic ordering trends can be reproduced in the bulk dihalides, we turn our attention to the monolayers. The obtained ground states are very similar to those obtained in the bulk, a reasonable outcome given the weak interlayer coupling in vdW materials. In a similar fashion to the above bulk description, comparison has been established between ferromagnetic, and antiferromagnetic (AFS stripe) ordering within the layers. FeX2, CoX2 (X= Cl, Br) and NiX2 prefer a FM ground state, while VX2, MnX2 and CoI2 are (striped) AFS (Table 3). These preferences can be justified based on the above described competition between direct exchange and superexchange for different fillings. CoI2 breaks the trend among the Co compounds due to the sensitivity of superexchange for M-X-M bond angles near 90∘ to the relative position of the anion $p$ states (Ref. Goodenough (1963)).

We will focus on the description of materials with a FM ground state at the monolayer level starting with the Fe compounds. FeX2 monolayers were the focus of previous studies in which they were predicted to be FM-half metals.Ashton et al. (2017) We have analyzed the electronic structure of these materials in further detail, in particular, the evolution of the electronic structure with $U$ within the LDA+$U$ method and upon inclusion of SOC. Fig. 2 shows the band structure and density of states of the half-metallic FeCl2 monolayer obtained within GGA. The electronic structure is consistent with the description in Ref. Ashton et al., 2017: Fe2+ being HS, the majority spin channel $d$ states are completely occupied; in the minority spin channel, there are two partially occupied t2g bands and three unoccupied $d$ bands (a flat t2g one, and above this, two $e_{g}$ ones). The electronic structure for the minority spin channel is identical to the bulk electronic structure in which FM layers are stacked in an AFM fashion (not shown). FeBr2 and FeI2 display the same basic electronic structure around EF. The spin moment per unit cell is 4$\mu_{B}$ with most of it residing on the Fe site ($\sim$3.5$\mu_{B}$).

One interesting question is the origin of the t2g splitting into an e${}_{g}^{*}$ doublet and a higher-lying a1g singlet (Fig. 3).Kuneš et al. (2003) This is due to the above mentioned trigonal distortion (compression along the [111] axis of the octahedra, i.e., the c-axis of the crystal). This distortion is quantified by the angle $\theta$ (Fig. 3) that is larger than the value for an undistorted octahedron $\theta_{0}$= 54.74∘ = arccos(1/$\sqrt{3}$). Additionally, one has to take into account the contribution to the crystal field of neighboring transition metal atoms to the a1g-eg splitting. The corresponding wavefunctions for these states can be written in the form:

[TABLE]

The a1g orbital has a very simple shape in local coordinates. It is analogous to a $z^{2}$- eg orbital with its $z$ axis directed along the [111] axis (in this case, the c-axis of the crystal). The other two $t_{2g}$ orbitals, denoted as e${}_{g}^{*}$, have a more complicated shape. These are states with $\mid{l^{z}}=\pm 1\rangle$ (with the quantization axis along c), the a1g state being the $\mid{l^{z}}=0\rangle$ one.

This clearly explains why, once SOC is included, the e${}_{g}^{*}$ orbitals are split. The overall band structure is similar to that of Fig. 2 with an orbital moment of 0.10 $\mu_{B}$ being induced on Fe. This splitting increases once an on-site $U$ is included, with an insulating gap opening up once $U$ exceeds 3 eV (Fig. 3). As a consequence, a larger orbital moment ($\sim$0.6 $\mu_{B}$) develops along the the trigonal axis, parallel to the spin moment. It should be noted that regardless of the $U$ value, a gap is not opened up unless SOC is included and the e${}_{g}^{*}$ states are split. The degeneracy lifting due to spin orbit coupling is similarly required to open a gap in FeX2 bulk materials, which are known to be Mott insulators.

The electronic structure of monolayer CoCl2 within GGA (d7 filling, HS) is shown in Fig. 2. The electronic structure for the bulk (with a magnetic order consisting of FM layers stacked AFM) is equivalent to that of the monolayer. At the GGA level, a gap already opens up in spite of the partial ($d^{7}$) filling. The reason is once again the trigonal distortion of the octahedra that gives rise to the t2g splitting into an a1g singlet and e${}_{g}^{*}$ doublet (the flat band right above the Fermi level in the minority spin channel has a1g character, the two e${}_{g}^{*}$ are occupied). In a similar fashion to the Fe case, the total moment per unit cell is 3$\mu_{B}$ with most of it residing on the Co site site ($\sim$2.5$\mu_{B}$). Once a $U$ is included, the gap between the occupied e${}_{g}^{*}$ and unoccupied a1g states increases (Fig. 3). The orbital moment derived for LDA+SOC+U (U= 4 eV) is 0.2 $\mu_{B}$. This, unlike for the Fe case, is the same as for $U$=0, an expected result given the complete filling of the e${}_{g}^{*}$ doublet.

NiCl2 also is a FM insulator at the monolayer level. The insulating character with Ni being d8 (HS) is simpler to understand. The band gap opens up between t2g occupied and eg unoccupied minority spin bands. The derived magnetic moment agrees with this picture (Table 3). In this case, a $U$ simply increases the gap as expected (Fig. 3).

As mentioned above, the character of the ferromagnetic ordering in these 2D materials is determined by the magnetocrystalline anisotropy energy. It is magnetic 2D materials with an easy axis that can have a ferromagnetically ordered phase at finite temperature. All Fe and Ni dihalides, as well as CoCl2 and CoBr2, are predicted to have their easy axis along $c$. This makes them a very promising platform for stable and robust ferromagnetism.

As in Ref. Kulish and Huang, 2017, we estimate the strength of the magnetic interactions from the energy difference between a FM and an AFM configuration (denoted as $\Delta$E). In our case, for the AFS state, a given magnetic ion has four AFM and two FM near neighbors, and for the FM state, six FM near neighbors. The same counting applies for next near neighbors. This leads to an effective $J$ of:

[TABLE]

where $S$ is the spin per metal atom (2, 3/2 and 1 for Fe, Co and Ni). The calculated values for materials with a FM ground state are shown in Table 4. The derived values are similar to those in previous works in which pseudopotentials were used,Kulish and Huang (2017) with predicted values for FeCl2 and NiX2 being particularly high, noting that our estimates are higher than theirs since they assumed a factor of 12 rather than 8 in the denominator of Eq. (2). This $J$, which can be considered as a sum of the near-neighbor $J_{1}$ and next near-neighbor $J_{2}$, is relevant if longer range $J$s are not important (the next next near-neighbor $J_{3}$ drops out of this energy difference). As a cautionary note, the values we estimate for $J$ typically exceed those extracted from inelastic neutron scattering.Birgeneau et al. (1972) But, they should give some idea of the trends in $J$ as M and X are varied.

For uniaxial anisotropy, the appropriate model is the Ising one. For a triangular 2D lattice, the critical temperature is 3.641$J$/$k_{B}$.Wannier (1950); Fisher (1967) The Tcs based on this, from our estimate of $J$, are shown in Table 4 for each of the monolayers. Given the above caveats, these should be considered as overestimates. Still, we would like to point out that in most cases, MX2 materials have Néel temperatures that exceed those of their MX3 counterparts.McGuire (2017) Moreover, it is entirely conceivable that the monolayer Tc could exceed the bulk TN. This is particularly relevant for NiX2, which have the highest TN of the MX2 materials besides TiCl2.McGuire (2017) If the spins do convert from xy-like (in-plane) in the bulk to Ising-like (along $c$) in the monolayer, as we predict, then the NiX2 monolayer Curie temperatures could indeed be high. As suggested in Ref. Kulish and Huang, 2017, Tc could also be increased by strain from a suitable substrate.

VI summary

In summary, motivated by recent experiments for CrI3, we have explored the evolution of the magnetism in transition metal dihalides from the bulk to the monolayer limit. FeX2, NiX2, CoCl2 and CoBr2 monolayers are predicted to be ferromagnetic insulators with out-of-plane moments and sizable magnetocrystalline anisotropies. Our results highlight the importance of considering the symmetry lowering at the transition metal site due to the trigonal distortion of the MX6 octahedron, as well as including the effect of spin-orbit coupling, to obtain the correct ground states. This work confirms the potential for stable and robust 2D ferromagnetism in transition metal dihalide-based monolayers, and we hope this will stimulate experimental efforts to realize them in the laboratory.

VII acknowledgments

This work was supported by the U.S. Dept. of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. We acknowledge the computing resources provided on Blues, a high-performance computing cluster operated by Argonne’s Laboratory Computing Resource Center.