Abstract
Magnetic anisotropy is crucially important for the stabilization of twodimensional (2D) magnetism, which is rare in nature but highly desirable in spintronics and for advancing fundamental knowledge. Recent works on CrI_{3} and CrGeTe_{3} monolayers not only led to observations of the longtimesought 2D ferromagnetism, but also revealed distinct magnetic anisotropy in the two systems, namely Ising behavior for CrI_{3} versus Heisenberg behavior for CrGeTe_{3}. Such magnetic difference strongly contrasts with structural and electronic similarities of these two materials, and understanding it at a microscopic scale should be of large benefits. Here, firstprinciples calculations are performed and analyzed to develop a simple Hamiltonian, to investigate magnetic anisotropy of CrI_{3} and CrGeTe_{3} monolayers. The anisotropic exchange coupling in both systems is surprisingly determined to be of Kitaevtype. Moreover, the interplay between this Kitaev interaction and single ion anisotropy (SIA) is found to naturally explain the different magnetic behaviors of CrI_{3} and CrGeTe_{3}. Finally, both the Kitaev interaction and SIA are further found to be induced by spin–orbit coupling of the heavy ligands (I of CrI_{3} or Te of CrGeTe_{3}) rather than the commonly believed 3d magnetic Cr ions.
Introduction
Twodimensional (2D) magnetic materials are receiving a lot of attention, due, e.g., to the search for longrange ferromagnetism (FM),^{1,2} which can facilitate various applications from sensing to data storage.^{3,4} According to Mermin and Wagners theorem,^{2} however strong the shortrange isotropic couplings are, the realization of 2D magnetism relies on magnetic anisotropy, as a result of spin–orbit coupling (SOC). The requirement of strong magnetic anisotropy in lowdimensional systems therefore explains the rareness of 2D FM materials.
The recent observation of FM in monolayers made of CrI_{3} and CrGeTe_{3},^{5,6,7} therefore opens a new chapter in the field of 2D materials. The chromium in both compounds share the same valence state of Cr^{3+}, with the 3d^{3} configuration and S = \(\frac{3}{2}\).^{7,8,9,10} FM arises there from the super exchange between nearestneighbor Cr ions, that are linked by I or Te ligands through nearly 90° angles.^{10,11} CrI_{3} has been demonstrated to be well described by the Ising behavior,^{1,5,12} for which the spins can point up and down along the outofplane zdirection. In contrast, the magnetic anisotropy of CrGeTe_{3} was determined to be consistent with the Heisenberg behavior,^{2,7,12} for which the spins can freely rotate and adopt any direction in the threedimensional space. Interestingly, structural and electronic similarities between these two compounds strongly contrast with their difference in magnetic behaviors, which implies subtle origins for their magnetic anisotropy. A recent theoretical work adopted the XXZ model, for which the exchange coupling is identical between the inplane xdirection and ydirection but different along the outofplane zdirection, to explain the outofplane magnetization of CrI_{3}.^{8} However, there is no definite proof that the XXZ model is accurate enough to describe the magnetic anisotropy of CrI_{3}, and there is a current paucity of knowledge for the mechanism responsible for the magnetic anisotropy of CrGeTe_{3}. Hence, a thorough microscopic understanding of the difference between the Ising behavior of CrI_{3} and the Heisenberg behavior of CrGeTe_{3} is highly desired.
In particular, it is tempting to investigate if the Kitaev interaction,^{13} which is a specific anisotropic exchange coupling, can also be significant in CrI_{3} and CrGeTe_{3}. This temptation is mainly based on the fact that these two materials adopt a honeycomb lattice and edgesharing octahedra, exactly as the layered Na_{2}IrO_{3} and aRuCl_{3} compounds which exhibit magnetic behaviors that are close to spin liquids^{14}—as a result of significant Kitaev interactions. Interestingly, finding finite Kitaev interaction in Cr3dbased CrI_{3} and CrGeTe_{3} compounds would enlarge the types of systems possessing such interaction, not only from 4d or 5d to 3d transitionmetalbased insulators, but also from S = 1/2 to S = 3/2 systems. Such broadening is inline with recent theoretical predictions of Kitaev interaction in d^{7} or 3d systems.^{15,16}
The main goal of this manuscript is to report results of firstprinciples calculations, along with the concomitant development of a simple but predictive Hamiltonian, to demonstrate that: (i) significant Kitaev interaction does exist in CrI_{3} and CrGeTe_{3} (which also invalidates the XXZ model in these two compounds); and (ii) the different interplay between this Kitaev interaction and the single ion anisotropy (SIA) naturally explains the observed magnetic anisotropy in these 2D ferromagnetic materials. Another surprising result is that the Kitaev and SIA anisotropies are both dominantly induced by the SOC of the heavy ligand elements rather than the 3d element Cr.
Results
Magnetic Hamiltonian and coupling coefficients
To precisely describe the magnetic anisotropy and explore differences between CrI_{3} and CrGeTe_{3}, we consider a Hamiltonian containing both exchange coupling, \({\cal H}_{{\mathrm {ex}}}\), and SIA, \({\cal H}_{{\mathrm {si}}}\), terms:
where \({\cal J}_{ij}\) and \({\cal A}_{ii}\) are 3 × 3 matrices gathering exchange and SIA parameters, respectively. The sum over i in Eq. (1) runs over all Cr sites, while the sums over i, j run over all nearestneighbor Cr pairs (note that the anisotropy in exchange coupling between more distant Cr neighbors is at least an order smaller and is thus negligible, see details in Supplementary Table 1). Density functional theory (DFT) calculations are performed on CrI_{3} and CrGeTe_{3} monolayers to extract the components of \({\cal J}\) and \({\cal A}\) using a precise fourstates method^{17,18} (see Supplementary Discussion for details). Note that (i) all the results shown below are based on the use of an effective Hubbard U = 0.5 eV, unless stated (see the effects of the choice of other U's in Method section); and (ii) the Dzyaloshinski–Moriya (DM) interaction is absent in our studied systems because of the existence of an inversion center between nearestneighbor Cr ions.^{19}
Let us first focus on the exchange coupling \({\cal J}\) matrix for CrI_{3} and CrGeTe_{3}. The \({\cal J}\) matrix is expressed in the {xyz} basis, for which the x–y plane is the film plane while the zaxis is the outofplane direction of the film. We choose the Cr0–Cr1 pair (see Fig. 1) to calculate the exchange coupling parameters, from which the parameters for Cr0–Cr2 and Cr0–Cr3 pairs can be deduced via threefold rotational symmetry. It is numerically found that this matrix is symmetric, i.e., J_{xy} = J_{yx}, J_{yz} = J_{zy}, and J_{xz} = J_{zx}, which is consistent with the fact that there is no DM interaction in our investigated compounds. As shown in Table 1, J_{xx}, J_{yy}, and J_{zz} of CrI_{3} possess quite different values of −2.29, −1.93, and −2.23 meV, respectively, while the offdiagonal elements of \({\cal J}\) in the {xyz} basis are smaller but nonnegligible. Such results contrast with the XXZ model adopted in ref. ^{8}, which assumes that (i) \(J_{xx} = J_{yy} \ne J_{zz}\) and (ii) J_{xy}, J_{yz} and J_{xz} can all be neglected. Different schemes and strategies, such as changing the value of U, using experimental structures rather than the computationally optimized ones and even replacing other Cr ions by nonmagnetic Al, are used to check their influence on J_{xx}, J_{yy}, and J_{zz} of CrI_{3}. It is numerically found that they all qualitatively give the same results (as detailed in Supplementary Discussion) in the sense that the aforementioned assumption (i) of the XXZ model providing equality between J_{xx} and J_{yy} is not satisfied, which automatically implies that such latter model is not accurate enough to precisely describe magnetic anisotropy in CrI_{3} and CrGeTe_{3} systems.
The symmetric \({\cal J}\) matrix is then diagonalized to obtain its eigenvalues (to be denoted as J_{a}, J_{β}, and J_{γ}) and corresponding eigenvectors (to be coined α, β, and γ) for the Cr0–Cr1 pair. As seen in Table 2, J_{α} and J_{β} are the strongest eigenvalues in magnitude and are close to each other in CrI_{3} (−2.46 and −2.41 meV, respectively), while J_{γ} is smaller in magnitude by about 1 meV. The same hierarchy exists between J_{α}, J_{β}, and J_{γ} in CrGeTe_{3}, but with J_{α} and J_{β} being now stronger in strength (about −6.65 meV), while J_{γ} = −6.28 meV is about 0.4 meV smaller in magnitude than the other two exchange coefficients. As shown in Fig. 1, the αaxis points from Cr1 to Cr0 in both systems, and therefore belongs to the x–y plane. On the other hand, the βaxis is roughly along the direction joining the two ligands bridging Cr0 and Cr1, and thus does not belong to the x–y plane. Similarly, the γaxis, which is perpendicular to both the αaxis and βaxis, does not lie in the x–y plane. It is important to realize that the {αβγ} basis diagonalizing the \({\cal J}\) matrix is specific to each considered Cr pairs, unlike the “global” {xyz} basis. In other words, the {αβγ} basis differs for the Cr0–Cr1, Cr0–Cr2, and Cr0–Cr3 pairs, as shown in Fig. 1 by means of red, green, and blue arrows.
The \({\cal H}_{{\mathrm {ex}}}\) exchange coupling Hamiltonian can now be rewritten, gathering these three local orthogonal {αβγ} coordinate bases (one for each Cr pair) and assuming that J_{α} = J_{β} (note that it is different with J_{xx} = J_{yy} in XXZ model^{8}), as
where J = J_{α} = J_{β} is the isotropic exchange coupling and K = \(J_\gamma  J_\alpha > 0\) is the socalled Kitaev interaction that characterizes the anisotropic contribution. Table 1 provides the values of both J and K and, in particular, indicates that the Kitaev interaction cannot be neglected in CrI_{3} and CrGeTe_{3}.
Let us now investigate the other energy of Eq. (1), that is the SIA, which involves the \({\cal A}\) matrix. For that, one needs to go back to the global {xyz} basis, since only the A_{zz} term can be finite by symmetry. It is numerically found that A_{zz} = −0.26 meV in CrI_{3}, while it adopts a similar magnitude but with a change of sign in CrGeTe_{3} (since A_{zz} = 0.25 meV there). Such significant values of A_{zz} (which is of the same order of magnitude than the K Kitaev parameter) implies that SIA is not negligible, which contrasts with the results in ref. ^{8}. \({\cal H}_{{\mathrm {si}}}\) can thus be simplified as
The total Hamiltonian of Eq. (1) can then be rewritten by combining Eqs. (2) and (3) as
This simplified Hamiltonian gathers (i) isotropic exchange coupling from J; (ii) anisotropic Kitaev interaction from K in the different local {αβγ} bases; and (iii) SIA in the global {xyz} basis. Let us now try to express the total energy associated with magnetism in an unified coordinate system. Equation (4) shows that the anisotropic part of the exchange energy (arising from K) is only related to the projections of spins on the three different γ axes (one for each Cr–Cr pair). Due to the fact that these three γ axes (to be denoted as γ_{1}, γ_{2}, and γ_{3}, respectively) are normally not perpendicular to each other, we now orthogonalize them using the Löwdin’s symmetric orthogonalization scheme.^{20} The resulting orthogonal axes form the {XYZ} coordinate system that is shown in Fig. 1. In this global {XYZ} basis, the outofplane zaxis of the film is along the [111] direction, and γ_{1}, γ_{2}, and γ_{3} can be expressed as (1, a, a), (a, 1, a), and (a, a, 1), where a\(\in\)[−\(\frac{1}{2}\), 1]. As consistent with the relatively largeinmagnitude and negative value of the isotropic exchange coupling J (see Table 1), ferromagnetic states are considered here. When expressing their spin in the {XYZ} basis, i.e., S(S_{X}, S_{Y}, S_{Z}), and considering a magnitude \(S = \frac{3}{2}\), it is straightforward to prove that the energy per Cr ion associated with Eq. (4) can then be rewritten as
where \(b = \frac{{a^2 + 2a}}{{2a^2 + 1}}\) and \(C = \frac{9}{8}(3J + K) + \frac{3}{4}A_{zz}\) are independent of the spin direction (see Supplementary Discussion for details). One can also easily demonstrate that the symmetric form of \((S_XS_Y + S_YS_Z + S_ZS_X)\) implies that the magnetization within the x–y plane of the film (for which \(S_X + S_Y + S_Z = 0\)) is fully isotropic. In other words, any direction of the spin within this x–y plane generates the same energy. Note that we further conducted DFT calculations (not shown here) that indeed numerically confirm that such inplane isotropy is mainly obeyed in CrI_{3} and CrGeTe_{3} (the maximal energetic difference we found between inplane directions of spins is 0.006 and 0.004 meV/f.u. in CrI_{3} and CrGeTe_{3}, respectively), which attests of the relevance and accuracy of the simple Hamiltonian of Eq. (4) and the resulting energy of Eq. (5). Note also that, although such isotropy in the x–y plane is in line with the results of ref. ^{8}, its origin is totally different: here it arises from the Kitaev interaction and its subsequent frustration, while in ref. ^{8} the isotropy in the x–y plane lies in the assumption of the XXZ model (see a detailed comparison between the two models in Supplementary Discussion).
It is also worthwhile to emphasize that ref. ^{8} assumed that SIA is negligible small, while according to Eq. (5) and as we will show below, both the Kitaev interaction (K) and SIA (A_{zz}) play an important role on the overall magnetic anisotropy of CrI_{3} and CrGeTe_{3}. To demonstrate such fact, one can realize that Eq. (5) involves \((S_XS_Y + S_YS_Z + S_ZS_X)\), which adopts (i) its maximum when \(S_X = S_Y = S_Z = \frac{{\sqrt 3 }}{2}\), which corresponds to spins being aligned along the outofplane zdirection; versus (ii) a minimum when \(S_X + S_Y + S_Z = 0\), that is when spins are lying within the x–y plane. The sign and value of the \(bK + \frac{2}{3}A_{zz}\) coefficient appearing in front of \((S_XS_Y + S_YS_Z + S_ZS_X)\) in Eq. (5) should therefore determine the magnetic anisotropy: a negative \(bK + \frac{2}{3}A_{zz}\) favors an easy axis along the outofplane direction while a positive \(bK + \frac{2}{3}A_{zz}\) will encourage spins to lie within the x–y plane. To characterize the strength of such anisotropy between the outofplane direction and the x–y plane, we also computed the energy difference, \(\Delta \varepsilon\), between the energy of the state having a fully outofplane magnetization and the averaged energy of states having inplane magnetization.
In the case of CrI_{3}, the b parameter is numerically found to be negative. Together with the positive K and negative A_{zz} from Table 1, both bK and \(\frac{2}{3}A_{zz}\) are thus negative. They are determined to be −0.16 and −0.17 meV, respectively, as shown in Table 2. Such negative values indicate that both Kitaev interaction and SIA lead to an outofplane easy axis, which is further confirmed by the negative value of −1.11 meV/f.u for \(\Delta \varepsilon\), as calculated from Eq. (5). Such value is not only consistent with the result of −0.82 meV/f.u. obtained from DFT calculations (confirming once again the validity and accuracy of our rather simple Eqs. (4) and (5) but also explains the previously determined Ising behavior of CrI_{3} favoring the outofplane direction for the magnetization.^{1,5,12}
In the case of CrGeTe_{3}, the b parameter is also found to be negative and leads to bK adopting a negative −0.17 meV value that is similar to the one of CrI_{3}. On the other hand, \(\frac{2}{3}A_{zz}\) is positive and yields \(\simeq\) 0.17 meV, which therefore results in a nearly vanishing bK + \(\frac{2}{3}A_{zz}\) and thus to a \(\Delta \varepsilon\) being nearly zero—that is, −0.01 meV/f.u. according to Eq. (5), which also compares well with the result of 0.02 meV/f.u. directly obtained from DFT (the difference in sign between the energies from Eq. (5) and DFT can be overlooked since both calculations provide vanishing \(\Delta \varepsilon\)). Such nearly zero value for bK + \(\frac{2}{3}A_{zz}\) therefore implies that Eq. (5) predicts that CrGeTe_{3} is basically isotropic in the whole threedimensional space, that is any spatial direction of the magnetization (inplane, outofplane or even combination between inplane and outofplane components) should provide similar magnetic energy. Such finding is fully consistent with the observed Heisenberg behavior of CrGeTe_{3}.^{2,7,12} Note that, although the isotropic Heisenberg behavior of CrGeTe_{3} is confirmed by both previous experiments and the present computational work, the anisotropy (Kitaev interaction and SIA) still play a crucial important role in stabilization of the longrange magnetic ground state, as indicated by Mermin and Wagners theorem.^{2}
Microscopic origin of Kitaev interaction and SIA
It is also worthwhile to realize that both the Kitaev parameter K and the SIA coefficient A_{zz} originate from SOC. One may wonder what specific ions contribute to such coefficients via SOC. To address such issue, Fig. 2a and b display the atomically resolved contribution of the K parameter as a function of the SOC strength in CrI_{3} and CrGeTe_{3}, respectively. Such parameter mainly arises from the SOC of the heavy ligands (namely, I of CrI_{3} or Te of CrGeTe_{3}) in a quadratic way, while the SOC of Cr has almost no effect on K (the effect from SOC of Ge on K of CrGeTe_{3} is also negligible). Such predictions for CrI_{3} and CrGeTe_{3} are consistent with the results of ref. ^{8} that anisotropy of CrI_{3} mainly arises from the SOC of I ligand. This can be understood by the fact that the super exchange between nearestneighbor Cr sites is mostly mediated by these ligands. To further confirm such results, we developed a tightbinding model (see details in Supplementary Discussion) that contains only two Cr ions and two bridging ligands, which form the x′y′ plane (see Fig. 3a for details). Such model confirms that K > 0 and \(K \propto \lambda ^2\), where λ is the SOC strength of ligands.
We now work on understanding the analytical results that K > 0 and \(K \propto \lambda ^2\). If SOC is not considered, the magnetic coupling is isotropic with the strength of J. When the SOC of ligands is included, an extra hopping path emerges for spins that are along the z′ direction, as shown in Fig. 3a and b, since \(\langle p_{x\prime \uparrow }{\mathbf{L}} \cdot {\mathbf{S}}p_{y\prime \uparrow }\rangle \ne 0\). Such an extra path provides an additional energy term K to the magnetic coupling, as \(J_{z\prime z\prime } = J + K = \frac{1}{{2S^2}}(E_{{\mathrm {FM}},z\prime }  E_{{\mathrm {AFM}},z\prime })\).
In the antiferromagnetic case, the two Cr have opposite spins. The electrons can hop from the occupied t_{2g} orbitals of one Cr to the unoccupied t_{2g} orbitals of another Cr, which can lower the \(E_{{\mathrm {AFM}},z\prime }\) by the amount of \(K \propto \lambda ^2\). The form of λ^{2} can be understood as that the whole hopping procedure includes two times of ligands’ SOC effects, as shown in Fig. 3b. On the other hand, in the ferromagnetic case, t_{2g} orbitals of both Cr are occupied with electrons having the same spin direction. In such case, although the aforementioned extra hopping path still exists, it cannot lead to the energy lowering of \(E_{{\mathrm {AFM}},z\prime }\). In contrast, there is no such extra hopping path when spins lie in the Cr_{2}L_{2} (x′y′) plane (denoted with the “–“ mark), since \(\langle p_{x\prime  }{\mathbf{L}} \cdot {\mathbf{S}}p_{y\prime  }\rangle = 0\). Such effects are further illustrated in Fig. 3c. As a result, the total effective \(J_{z\prime z\prime } = J + K\) is larger than that \(J_{x\prime x\prime } = J_{y\prime y\prime} = J\), i.e., K > 0. The findings reported here thus demonstrate, for the first time, that (i) the Kitaev interaction not only exists in 4d or 5d transition metal insulators, but also can occur in 3d systems; and (ii) the SOC of the ligands can play a crucial role on that interaction. Such findings can facilitate the ongoing efforts to realize Kitaevtype interactions in 3d systems.^{15,16}
Regarding SIA, Fig. 2c shows that the SOC of I is basically responsible for the negative A_{zz} of CrI_{3}, and that Cr does not significantly contribute to such parameter. Such finding of a ligandinduced SIA is also novel, since SIA is typically believed to arise from the transition metal ion.^{21,22} Similarly and as indicated by Fig. 2d, the Te ligand in CrGeTe_{3} produces a rather strong (and positive, in that case) A_{zz}, which is even twice as large as the one resulting from the SOC of all considered ions. It is in fact the combination of SOC from both Te and Cr that provides a value of A_{zz} that is close to the total one in CrGeTe_{3} (note that Ge does not significantely contribute to this total SIA).
Discussion
We now explain on magnetic behaviors of related 2D systems. Another 2D ferromagnetic system, CrBr_{3}, was also studied. It is numerically found (not shown here) that, as similar to CrI_{3}, both the Kitaev interaction and SIA favor an outofplane easy axis in this material. Such finding is consistent with measurements determining that the net anisotropy of CrBr_{3} is outofplane.^{23} Interestingly, Eq. (5) of the manuscript can also be useful to shed some light into controversial issues, such as which magnetic model is more pertinent to CrSiTe_{3}. As a matter of fact, early neutron work suggested an Isinglike model^{24} for this system, while recent measurements argue between an Heisenberglike model^{25} and an Ising model coupled with longrange interaction.^{26} CrSiTe_{3} shares similarities with CrGeTe_{3} in the sense that it has a negative bK associated with Kitaev interaction, as well as a positive \(\frac{2}{3}A_{zz}\) induced by SIA. However, the former is equal to −0.21 meV and is larger in magnitude than the latter (that is equal to +0.11 meV) in CrSiTe_{3}, therefore leading to a less negative bK + \(\frac{2}{3}A_{zz}\) of −0.07 meV and thus slightly tipping the balance towards outofplane magnetization (as confirmed by our DFT results providing \(\Delta \varepsilon\) = −0.10 meV for the difference between the energy for an outofplane magnetization and the averaged energy of ferromagnetic states having inplane magnetization). One can thus propose that the correct magnetic model for CrSiTe_{3} should be related to a slight perturbation of the Ising model, in order to account for possible (weaker) inplane components of the magnetization in addition to (stronger) outofplane ones.
The present general model and the XXZ model are compared here. The present general model (Eqs. (1), (4) and (5)) adopts the most generalized form of the \({\cal J}\) and \({\cal A}\) matrices, which can capture the microscopic details of different anisotropy. Such model is powerful, as it explains the origin of Kitaev interaction, as well as the competition and collaboration between Kitaev interaction and SIA. It also allows for antisymmetric exchange coupling, i.e. the socalled DM interaction, when the inversion centers between Cr–Cr pair are somehow removed. In contrast, the XXZ model is more macroscopic in nature, since it starts from the overall effects of the frustration among Cr–Cr pairs. Nevertheless, this XXZ model can still somehow describe the competition and collaboration between Kitaev interaction (since \(J_x\) and \(J_z\) are different from each other) and SIA. As a result, the XXZ model can be technically applied to both CrI_{3} and CrGeTe_{3}, as well as to the aforementioned related systems, but, as documented in Section 4 of the Supplementary Discussion, one really has to include SIA there to be more accurate.
Moreover, our work implies potential Kitaevtype quantum spin liquids in related systems. The predicted presence of Kitaev interaction in CrI_{3} and CrGeTe_{3} systems hints towards the possibility of realizing quantum spin liquid state in 3d systems. As a matter of fact, further studies we performed indicate that varying inplane strain can make the isotropic exchange coupling vanishing while the Kitaev interaction remains finite, which is promising to realize quantum spin liquid state in CrI_{3} and CrGeTe_{3} systems. Furthermore, our predictions also provide another way to enhance Kitaev interaction in the “traditional” 4d and 5d systems, that is, for example, to substitute the light Cl ligand with the heavier I ion in RuCl_{3} compound. The hybrid source of SOC to produce strong Kitaev interaction should result in interesting physics and phenomena in related systems.
We hope that our firstprinciples calculations and concomitant development of a simple insightful Hamiltonian (see Eqs. (4) and (5)), along with a tightbinding model, deepens the understanding of magnetic anisotropy in lowdimensional systems. The decomposition of the total magnetic anisotropy into Kitaev and SIA effects further sheds light into the behaviors of other related systems, such as CrBr_{3} and CrSiTe_{3}, therefore further demonstrating its relevance and importance.
Methods
DFT parameters
Simulations are performed using the Vienna abinitio simulation package (VASP).^{27} A 350 eV plane wave cutoff energy is used, which is 130% larger than the highest ENMAX of all involved elements. A 12 Å vacuum layer is adopted for all calculations, which is thick enough to get converged results according to former studies.^{9,28} The projector augmented wave (PAW) method^{29} is employed with the following electrons being treated as valence states: Cr 3p, 4s and 3d, I and Te 5s and 5p, Ge 4s and 4p, and Al 3s and 3p. The local density approximation (LDA)^{30} is used, with an effective Hubbard U parameter chosen to be 0.5 eV for the localized 3d electrons of Cr ions. Such relatively small U value is based on the facts that (i) the shallow p orbitals of heavy ligands (I or Te) strongly hybridize with 3d orbitals of Cr; and (ii) such strong hybridization leads to less localized 3d electrons. Nevertheless, other U choices of 0, 1, 2, 4 eV and U = 4 eV together with J = 0.7 eV are also presently tested and qualitatively render the same results, in the sense that K and A_{zz} have different (the same, respectively) signs for CrI_{3} (CrGeTe_{3}, respectively), as shown in Supplementary Table 1. Gaussian smearing with a smearing width of 0.05 meV is used in the Brillouin zone integration. Tetrahedron method with Bl\(\ddot o\)chl corrections is also performed, which yields the same K and A_{zz} (to the precision of 0.01 meV). Note that, exchange couplings between more distant neighbors other than the first nearest neighbors are not considered in the magnetic Hamiltonian, due to the reasons that (i) we focus on the magnetic anisotropy (more distant neighbors may have strong isotropic couplings, but they do not contribute to the stabilization of the longrange order); and (ii) the \({\cal J}\) matrix of second nearest neighbors is also calculated (see Supplementary Table 1) and its anisotropy is found to be an order smaller than that of the first nearest neighbors. kpoint meshes are chosen, such as they are commensurate with the choice of 4 × 4 × 1 for the unit cell (that contains eight atoms for CrI_{3} and 10 atoms for CrGeTe_{3}). For instance, the kpoint mesh of a 2 × 2 × 1 supercell is 2 × 2 × 1. Denser kpoint meshes up to 4 × 4 × 1 for the 2 × 2 × 1 supercell are also tested and qualitatively yield the same results, as shown in Supplementary Table 2. All magnetic parameters are calculated using the fourstate energy mapping method (see Section 3 of Supplementary Discussion for details).^{17,18} The Hellman–Feynman forces are taken to be converged when they become smaller than 0.001 eV/Å on each ion. Schematization of crystal structures are prepared using the VESTA software.^{31}
Data availability
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Acknowledgements
We thank Zhenglu Li and Jianfeng Wang for useful discussion. C.X. thanks the support of the Office of Basic Energy Sciences under contract ER46612. L.B. acknowledges the support of the ARO Grant No. W911NF1610227. H.X. is supported by NSFC (11374056), the Special Funds for Major State Basic Research (2015CB921700), Program for Professor of Special Appointment (Eastern Scholar), Qing Nian Ba Jian Program, and Fok Ying Tung Education Foundation. J.F. aknowledges Open Foundation of Key Laboratory of Computational Physical Sciences (Ministry of Education). The Arkansas High Performance Computing Center (AHPCC) is also acknowledged.
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C.X. and J.F. contributed equally to this work. H.X and C.X. conceived the idea. H.X. and L.B. supervised this work. C.X. performed the DFT calculations and related analysis, J.F. worked on the tightbinding model. All authors contributed to the discussion of the results and the writing of the manuscript.
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The authors declare no competing interests.
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Correspondence to Hongjun Xiang or Laurent Bellaiche.
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