Abstract
We study the microscopic origin of magnetism in suspended and dielectrically embedded CrI_{3} monolayer by downfolding minimal generalized Hubbard models from ab initio calculations using the constrained random phase approximation. These models are capable of describing the formation of localized magnetic moments in CrI_{3} and of reproducing electronic properties of direct ab initio calculations. Utilizing the magnet force theorem, we find a multiorbital superexchange mechanism as the origin of magnetism in CrI_{3} resulting from an interplay between ferro and antiferromagnetic CrCr d coupling channels, which is decisively affected by the ligand p orbitals. We show how environmental screening, such as resulting from encapsulation with hexagonal boron nitride, affects the Coulomb interaction in the film and how this controls its magnetic properties. Driven by a nonmonotonic interplay between nearest and nextnearest neighbor exchange interactions we find the magnon dispersion and the Curie temperature to be nontrivially affected by the environmental screening.
Introduction
Ferromagnetic (FM) layered materials hold high promises for becoming one of the key ingredients in future spintronic nanodevices^{1,2,3,4} based on vanderWaals heterostructures. Recent observations of magnons in thin layers of chromium trihalides via inelastic electron tunneling spectroscopy (IETS)^{1,5} and magnetoRaman spectroscopy^{6} have also opened new ways to explore magnonbased lowconsumption spintronics at the twodimensional (2D) limit^{7}. Whether or how the magnetic properties are modified or stabilized in these heterostructures is, however, still under debate. Recently reported theoretical predictions to suggest, for instance, a strong dependence of the magnetooptical response on the film thickness and the spin–orbit coupling of CrI_{3}^{8}, while recent experimental studies point towards the possibility to tune the magnetic properties of monolayer and bilayer CrI_{3} using electrostatic gating^{9,10,11,12}. These observations highlight the importance of addressing how the environment of layered magnetic systems modifies their electronic and magnetic properties via proximity, gating, or screening effects.
For the case of chromium trihalides, CrX_{3} (X = I, Br, Cl), the nearest neighbor magnetic exchange couplings have been theoretically reported to be between 1 and 3.2 meV^{13,14,15,16,17,18} depending on the material and calculation scheme, which renders rigorous theoretical descriptions absolutely necessary. This holds especially with respect to the Coulomb interactions in a layered material, which are enhanced due to reduced polarization in the surrounding. At the same time changes to the environmental polarization and thus to the environmental screening can significantly modify the Coulomb interaction within the layered material. Thus, properties of layered materials affected by the Coulomb interaction, such as the formation of excitons and plasmons or the stabilization of charge and superconducting order, are in principle strongly dependent on the material’s environment allowing for Coulomb engineering of these manybody properties^{19,20,21,22,23,24}. So far it is not known how efficient Coulomb engineering can be for tailoring the magnetic ground state or magnonic excitations of layered magnetic systems. For the case of CrI_{3}, both can be described by effective spin Hamiltonians describing exchange interactions between the Cr atoms as the spin carriers. Thus any changes from environmental screening to the exchange couplings of this effective Hamiltonian can give rise to changes in the magnetic transition temperature or the magnon spectrum opening new routes towards the control and potential tailoring of magnetic properties of chromium trihalides.
Computationally it is, however, a nontrivial task to reliably study how screening from the environment modifies the exchange couplings, as this is beyond the capabilities of conventional density functional theory (DFT) based approaches, which describe environmentally induced modifications to the Coulomb interactions and thus to the magnetic properties only poorly. To overcome these DFTbased shortcomings, higherlevel theories, such as the GW approximation, are needed which take screening explicitly on the level of the random phase approximation (RPA) into account. Full ab initio GW calculations for layered heterostructures are, however, not feasible due to the large supercells needed to account for lattice mismatches between the involved materials. We utilize a multistep approach to overcome these computational problems of the conventional approaches to study the impact of Coulomb interactions on the magnetic properties of suspended and dielectrically embedded monolayer CrI_{3}. We start from spinunpolarized ab initio band structure calculations for monolayer CrI_{3} using DFT, which we use to downfold two minimal models defined by localized Wannier orbitals describing the Cr d states only and a combined (d + p)basis also taking the I p states into account. After constructing the corresponding Coulomb tensors from constrained RPA (cRPA) calculations we solve the resulting multiorbital Hubbard models within meanfield Hartree–Fock (HF) theory. The resulting interacting and spinresolved quasiparticle band structures are used afterward to evaluate the orbitally resolved magnetic exchange interactions using the magnetic force theorem (MFT). Using our Wannier function continuum electrostatics (WFCE) approach we additionally include the environmental screening effects to the Coulomb interactions, which finally allows us to study how the microscopic exchange coupling channels, the magnetic transition temperature, and the magnon dispersions are affected by different parts of the Coulomb tensor and/or by the dielectric encapsulation of the CrI_{3} monolayer. This combination of carefully validated stateoftheart approaches allows us to microscopically study magnetism in CrI_{3} heterostructures.
Results
Ab initio calculations
In the crystal structure of monolayer CrI_{3}, the Cr atoms are arranged on a honeycomb lattice surrounded by the I ligands forming an edgesharing octahedral environment around each metal ion, as depicted in Fig. 1a. The ligand field splits the Cr dorbitals into two sets, namely, the t_{2g} (d_{xy}, d_{xz}, d_{yz}) and the e_{g} (\({d}_{{x}^{2}{y}^{2}},{d}_{{z}^{2}}\))^{13}. Spin unpolarized firstprinciples calculations in the generalized gradient approximation (GGA) level predict a metallic ground state with halffilled bands of predominantly Cr t_{2g} character separated by sizable gaps from fully occupied bands of mostly I p character and unoccupied bands of Cr e_{g} type, as shown in the atomic resolved density of states (DOS) in Fig. 1b. Spinpolarized calculations in the GGAs + U approximation^{25} shift down the majority spin t_{2g} bands which hybridize with the ligand pbands. The unoccupied majority spin e_{g} bands appear at approximately 1.5 eV above the valence states followed by the minority spin t_{2g} and e_{g} bands, respectively, resulting in an FM insulating state, as depicted in Fig. 1 (c). Although LSDA+U approaches seem to reasonably describe the magnetic properties of CrI_{3}, its validity still needs to be benchmarked with higherlevel theories such as dynamical meanfield theory, as the local Coulomb repulsion, U is rather large compared to the t_{2g} bandwidth as discussed in detail later. Thus, to fully microscopically understand each part of the problem, ranging from hybridization effects to the Coulomb tensor to the choice of basis set, we proceed in the following with a downfolding procedure to generate reliable materialrealistic minimal models.
Model Hamiltonian
In the following, we will construct model Hamiltonians of the form
with \({{{{\mathcal{H}}}}}^{W}\) describing the noninteracting kinetic terms and \({{{{\mathcal{H}}}}}^{U}\) describing the local Coulomb interactions of the Cr d orbitals. Depending on the chosen basis we additionally use a doublecounting correction \({{{{\mathcal{H}}}}}^{DC}\). The noninteracting part of the Hamiltonian is defined by longrange hopping matrix elements t_{mn} and reads
while the Coulomb term takes local densitydensity \({U}_{mm^{\prime} }\) and Hund’s exchange \({J}_{mm^{\prime} }^{{{{\rm{H}}}}}\) interactions on the Cr atoms into account and is defined by
where \({\hat{n}}_{\sigma }={d}_{m\sigma }^{{\dagger} }{d}_{m\sigma }\) is the number operator and σ is the spin index. These Hamiltonians are solved within a meanfield HF ansatz and analyzed in terms of its microscopic magnetic properties as described in the “Methods” section.
Basis sets and noninteracting Hamiltonians
Motivated by the nonmagnetic ab initio DOS depicted in Fig. 1b, we utilize two minimal basis sets: one including only effective Cr d Wannier orbitals and a second one also including I p states, which are, respectively, referred to as donly and (d + p) models in the following. To this end, we start with conventional DFT calculations utilizing the Perdew–Burke–Ernzerhof GGA exchangecorrelation functional^{26} within a PAW basis^{27,28} as implemented in the Vienna Ab initio Simulation Package (vasp)^{29,30} for CrI_{3} monolayers with a lattice constant of a_{0} ≈ 6.97 Å embedded in a supercell with a height of 35 Å. We use (16 × 16 × 1) kmeshes together with an energy cutoff of 230 eV and apply a Methfessel–Paxton smearing of σ = 0.02 eV. For the donly model, we project the 10 Kohn–Sham states per spin between −0.8 eV and +2.2 eV (inner and outer Wannierization windows) onto 10 Crlocalized d orbitals (with a rotating axis parallel to the tetragonal main axis) and maximally localize them afterward using the WANNIER90 package^{31}. For the (d + p)model, we similarly project all Kohn–Sham states between −6.0 and +2.2 eV to Crcentered d orbitals as well as to Icentered p orbitals. In this case, we do not perform a total maximal localization of the total spread of all Wannier orbitals since it would increase the localization of the I p orbitals at the cost of delocalization of Cr d orbitals. This way we get in both models maximally localized Cr d orbitals (at the cost of delocalized p orbitals in the (d + p)model). These Wannier orbitals are used in a subsequent step to calculate the needed hopping matrix elements for the definition of \({{{{\mathcal{H}}}}}^{W}\). As shown in Supplementary Methods 1, the resulting Wannier models perfectly interpolate all Kohn–Sham states between −0.8 eV and +2.2 eV in the donly model as well as all Kohn–Sham states between −6.0 eV and +2.2 eV in the (d + p)model.
Constrained random phase approximation
The Coulomb interaction matrix elements of \({{{{\mathcal{H}}}}}^{U}\) are evaluated using the Cr d Wannier orbitals from these two models within the cRPA scheme^{32} according to
Here, \({{{\mathcal{U}}}}\) represents the partially screened Coulomb interaction defined by
with \({{{\mathcal{V}}}}\) being the bare Coulomb interaction and Π_{rest} the partial or restpolarization from all states except those of the correlated subspace defined by the Cr d states. Π_{rest} thus describes screening from all other Cr as well as all I states including a significant amount of empty states from the full Kohn–Sham basis. In detail, we use in total 128 bands and define Π_{rest} by explicitly excluding all Kohn–Sham states between −0.5 and 3 eV from the full polarization. To this end, we use a recent cRPA implementation by Kaltak available in vasp^{33} (see Supplementary Methods 2 for further details). This way we remove the unrealistically strong metallic screening of the halfoccupied t_{2g} band of the spinunpolarized GGA starting point. From this we can extract the full rank4 Coulomb tensor in the basis of the correlated Cr d states. Within \({{{{\mathcal{H}}}}}^{U}\) we restrict ourselves however to static local densitydensity elements \({U}_{mm^{\prime} }={U}_{mm^{\prime} m^{\prime} m}\) and local Hund’s exchange elements \({J}_{mm^{\prime} }^{{{{\rm{H}}}}}={U}_{mm^{\prime} mm^{\prime} }\) with m and \(m^{\prime}\) labeling d orbitals on the same Cr atom. As shown in ref. ^{34}, nonlocal interactions do not significantly modify the magnon dispersions of bulk CrI_{3}, while the local interactions have a strong impact on the latter. We thus neglect both nonlocal density–density and nonlocal Hund’s interactions here and use only the corresponding onsite interactions. Casula et al.^{35} furthermore showed that using \({{{\mathcal{U}}}}(\omega =0)\) instead of the fully retarded \({{{\mathcal{U}}}}(\omega )\) is justified when renormalized hopping parameters are utilized. The corresponding renormalization factor was shown to scale with the characteristic cRPA plasmon frequency ω_{p}. In the Supplementary Methods 3, we show that for CrI_{3} ω_{p} is rather large, which renders these renormalizations small here. We therefore neglect them in the following.
Minimal Cr donly basis
We start with analyzing the local bare and cRPA screened densitydensity Coulomb matrix elements as obtained from the donly basis. The full matrices are shown in Table 1 together with the corresponding Hund’s exchange elements. In all cases the resulting density–density matrices are approximately of the form
where v_{t} (\({j}_{\,{{\mbox{t}}}\,}^{{{{\rm{H}}}}}\)) and v_{e} (\({j}_{\,{{\mbox{e}}}\,}^{{{{\rm{H}}}}}\)) are intraorbital density–density (interorbital Hund’s exchange) matrix elements within the t_{2g} and e_{g} manifolds, respectively, v_{et} = 1/2(v_{e} + v_{t}), and \({j}_{1\ldots 4}^{{{{\rm{H}}}}}\) are interorbital Hund’s exchange elements between the two manifolds. This form of the density–density Coulomb matrix is similar to the one obtained for a fully rotationalinvariant d shell which is here, however, perturbed due to the ligandinduced crystalfiled splitting. Therefore, instead of five U_{0}, \({J}_{1\ldots 4}^{{{{\rm{H}}}}}\) or three U_{0}, F_{2}, F_{4}^{36} parameters we need here eight to represent the full density–density matrix. As a result, the t_{2g} and e_{g} channels themselves are easily parameterized using two Hubbard–Kanamori parameters (v and j^{H}). The interchannel elements show, however, a significant orbital dependence which can be represented by the four \({j}_{1\ldots 4}^{{{{\rm{H}}}}}\) Hund’s exchange elements.
Since the Wannier orbital spread \({{{\Omega }}}_{\alpha }=\left\langle {w}_{\alpha } {r}^{2} {w}_{\alpha }\right\rangle {\left\langle {w}_{\alpha } r {w}_{\alpha }\right\rangle }^{2}\)^{31} of the t_{2g} wave functions (\({{{\Omega }}}_{{t}_{2g}}\approx 3\) Å^{2}) is smaller then the corresponding e_{g} one (Ω_{eg} ≈ 5.3 Å^{2}) the bare intraorbital density–density elements v_{t} ≈ 15.5 eV are larger than the v_{e} ≈ 12.3 eV elements and also represent the largest elements in \({v}_{mm^{\prime} }\). This is also reflected in the intrachannel \({j}_{\,{{\mbox{t}}}\,}^{{{{\rm{H}}}}}\approx 0.54\) eV and \({j}_{\,{{\mbox{e}}}\,}^{{{{\rm{H}}}}}\approx 0.51\) eV elements which are, however, still rather similar. The interchannel \({j}_{1\ldots 4}^{{{{\rm{H}}}}}\) vary between about 0.25 and 0.47 eV. We note that these bare Hund’s exchange interactions are significantly smaller than the approximated values in bulk Cr of J^{H} ≈ 0.75 eV^{37}, which we attribute here to the enhanced Wannier function spread of the effective Cr donly basis.
The cRPA densitydensity matrix elements \({U}_{mm^{\prime} }\) are significantly reduced by factors between 1/4 to nearly 1/6 by the restspace screening from the other Cr and I states. Although this effective screening is strongly orbital dependent, it does not change the overall orbital structure of the Coulomb matrix depicted in Eq. (6). We find U_{t} ≈ 3.6 eV and U_{e} ≈ 3.1 eV, which are still rather large compared to the bandwidth of the halffilled Cr t_{2g} band of about 1 eV. Thus even taking cRPA screening into account correlation effects can be expected to play an important role. The screened Hund’s exchange interactions, \({J}_{\,{{\mbox{t}}}\,}^{{{{\rm{H}}}}}\approx 0.49\) eV and \({J}_{\,{{\mbox{e}}}\,}^{{{{\rm{H}}}}}\approx 0.44\) eV, are reduced by no more than 10% in comparison to the bare values, which also holds for the interchannel ones. The nearestneighbor cRPA screened density–density interactions are nearly orbital independent U_{01} ≈ 1.4 eV. The nearestneighbor Hund’s interactions are vanishingly small ranging between J_{01} ≈ 1 and 7 meV.
We proceed with analyzing how the local cRPAscreened Coulomb interactions affect the band structure of the Cr donly model within the meanfield theory. As depicted in Fig. 2 the Coulomb interactions drive the systems into an insulator with a sizeable bandgap and the same band ordering as known from LSDA(+U) calculations (i.e., fully occupied t_{2g} followed by completely empty e_{g,↑}, t_{2g,↓}, and e_{g,↓} manifolds). On a qualitative level, this minimal basis thus seems to be capable of reproducing the Cr d band structure of full ab initio calculations. To analyze to what extent this minimal model is also capable of reproducing the superexchange mechanism responsible for the FM ordering in CrI_{3} we have applied the MFT to these HF solutions in a subsequent step (for details see Methods). The resulting orbitally resolved exchange couplings are given in Table 2 and show a strong antiferromagnetic (AFM) total coupling of J = −1.964 meV for nearest neighbors and \(J^{\prime} =0.042\) meV for nextnearest neighbors, driven by a strong AFM J_{t2g–t2g} interaction. This is obviously in contradiction to LSDA(+U) calculations and to all available experimental data. This wrong prediction can be understood from the Kugel–Khomskii (KK) formalism^{15,38}, which describes the total magnetic exchange as the sum of the t_{2g}–t_{2g} and t_{2g}–e_{g} contributions as J(r) = J_{t2g–t2g}(r) + J_{eg–eg}(r) with
Here, \({\tilde{t}}_{{{{t2g}}}{{{t2g}}}}(r)\) and \({\tilde{t}}_{{{{t2g}}}{{{eg}}}}(r)\) are effective hopping matrix elements between the different orbital channels, Δ represents the crystal field splitting, and \(\tilde{U}\) and \({\tilde{J}}^{{{{\rm{H}}}}}\) are the averaged Coulomb and Hund’s exchange parameters. We immediately understand that J_{t2g–t2g} is by definition of AFM nature and controlled only by the effective t_{2g}–t_{2g} hopping and the density–density channel of the Coulomb interaction (\(\tilde{U}\)). In contrast, the FM channel, J_{eg–eg}, is controlled by the effective t_{2g}–e_{g} hopping and both \(\tilde{U}\) and J_{H}. By using the hopping values of our Wannier construction, given in the Supplementary Methods 5 and the averaged electron–electron interactions from the cRPA Coulomb tensor (\(\tilde{U}\approx 3.4\) and \({\tilde{J}}_{H}\approx 0.4\) eV), we obtain an AFM intralayer exchange coupling in line with the MFT results. We carefully checked that this wrong KK prediction also holds in the case of the extended (d + p) basis by using effective hoppings from this models with and without pmediated hopping, as summarized in the Supplementary Methods 5. In both cases, using only the direct hoppings as well as those after integrating our the I p contributions, we find that the effective nearestneighbor \({\tilde{t}}_{{{{t2g}}}{{{t2g}}}}\) is significantly larger than the \({\tilde{t}}_{{{{t2g}}}{{{eg}}}}\) one so that the AFM coupling channels always dominate. Thus the correct microscopic origin of the FM coupling in CrI_{3} cannot be described on a Cr donly basis and can neither be modeled within the KK approach.
Extended (d + p)basis
Motivated by the wrong predictions of the minimal Cr donly model, we expanded the basis to also include the ligand p contributions. This (d + p)basis has a significant impact to the local Cr Coulomb matrix elements as summarized in Table 3. Due to the presence of the I p orbitals the Cr d Wannier functions are more localized (Ω ≈ 1 Å^{2}) so that the resulting bare matrix elements are significantly enhanced as compared to the donly case. Also the density–density matrix elements in the e_{g} channel are now larger than in the t_{2g} channel. The cRPA screening due to all other Cr and all I states, again significantly reduce all matrix elements in an orbitaldependent manner. The final intraorbital densitydensity interactions are now on the order of 4 eV with interorbital elements of the order of 3 eV. Notably, the cRPA screened Hund’s exchange interactions are also enhanced by the increased localization of the Cr d states. They are, however, still on the order of 0.5–0.7 eV and thus still significantly smaller than the common approximation of 0.9 eV. The nearestneighbor cRPA screened density–density interactions are again nearly orbital independent U_{01} ≈ 1.4 eV, while the maximal nearestneighbor Hund’s interactions is J_{01} ≈ 2 meV.
As before in the donly basis, the interaction term \({{{{\mathcal{H}}}}}^{U}\) of our Hamiltonian from Eq. (1) acts on the Cr d states only. The kinetic part, \({{{{\mathcal{H}}}}}^{W}\), however, now also includes I p contributions. Thus, to counteract double counting effects we subtract here the doublecounting potential defined in Eq. (14) from the Cr d states using a nominal Cr^{3+}d occupation of \({N}_{imp}^{\sigma }=3\). This corrects for the relative positioning of the interacting Cr d bands with respect to the uncorrelated I p states.
The resulting HF DOS is shown in Fig. 3. In contrast to the donly model we now find a DOS which is vastly reminiscent of the full ab initio GGAs + U results shown in Fig. 1 (see Supplementary Methods 3 for corresponding band structures). Next to the spinsplitting and ordering also the full and subband gaps are in good agreement with GGAs + U predictions. Also, the atomic contributions to the unoccupied bands are very similar in our HF calculations as compared to the GGAs + U calculations. The most prominent difference is the orbital composition of the upmost valence states. While the full ab initio GGAs + U calculations finds mostly I contributions here, our HF calculations also show a significant Cr admixture. At this stage, we cannot finally judge whether this difference results from our minimal localized basis sets, the fully rotationally invariant GGAs + U implementation we have used, or from the charge selfconsistency within the full GGAs + U calculations, which we lack in our approach. Nevertheless, we see in the following that this model reliably predicts the magnetic properties of CrI_{3} monolayer.
The orbitalresolved intralayer exchange couplings calculated using the MFT are given in Table 2. In contrast to the donly model, the (d + p)model gives the correct FM exchange coupling J ≈ 1.76 meV (\(J^{\prime} \approx 0.35\) meV), similar to previously reported values from MFT calculations based on LSDA + U input^{15}. To microscopically explore how the different Coulomb interaction channels (U, \(U^{\prime}\), J^{H}) affect the electronic band structure and the magnetic properties, we present in Fig. 4 the bandwidths and positions together with the orbitallyresolved exchange interactions upon individual reductions of each Coulomb channel by 20%. Reducing the intraorbital Coulomb interactions (ΔU) results in a lower spin splitting of the t_{2g} and e_{g} bands, similar to the singleorbital Hubbard model. The other two cases, namely, \({{\Delta }}U^{\prime}\) and ΔJ^{H}, are more difficult to understand since these terms mix different types of orbitals. When reducing \(U^{\prime}\), we observe a strong reduction of the splitting between the t_{2g,↑}/e_{g,↑} manifolds, accompanied by a strong gap reduction of about 1 eV. In contrast, the splitting between the t_{2g,↓}/e_{g,↓} manifolds is weakly affected by \(U^{\prime}\) compared to the initial full cRPA case. Finally, when we reduce Hund’s exchange J^{H}, the \({t}_{2g,\sigma }/{t}_{2g,\bar{\sigma }}\) and the \({e}_{g,\sigma }/{e}_{g,\bar{\sigma }}\) splittings are both reduced, while the splitting between the majority spin occupied t_{2g,↑} bands and empty e_{g,↑} bands increases with respect to the unperturbed case, leading to a small increase in the bandgap.
These bandstructure renormalizations have nontrivial effects to the microscopic exchange interactions, which we depict in Fig. 4 (bottom panel) and which can be just partially understood within the KK formalism. In line with KK a reduction of U or \(U^{\prime}\) yields an enhancement of the AFM J_{t2g–t2g} channel, as shown in the ΔU and \({{\Delta }}U^{\prime}\) columns of Fig. 4 (bottom panel), while modifications to J^{H} do not affect J_{t2g–t2g} at all. Also in qualitative agreement with KK we find that a reduction of J^{H} reduces the FM J_{t2g–eg}. In contrast to the simple model predictions from KK we find from MFT that J_{t2g–t2g} is slightly reduced upon reducing U while modifications to \(U^{\prime}\) yields no modifications. This underlines the need for the full microscopic MFT in combination with an extended basis to quantitatively understand magnetism in CrI_{3}. We expect that just an extended model combining the superexchange mechanism from the GoodenoughKanamori description and the multiorbital KK mechanism might yield a qualitatively understanding in line with our MFT findings, which will be studied separately. Here, we conclude that the extended (d + p)basis together with the corresponding cRPA Coulomb matrix elements and using an HF solver results in a reliable bandstructure and microscopically correct magnetic properties.
Substrate tunability
In the following, we proceed with the analysis of dielectric screening effects to the magnetic properties of CrI_{3} monolayer. We investigate how a dielectric encapsulation, as depicted in the inset of Fig. 5a, modifies the Coulomb interactions in CrI_{3} and how this affects its bands structure and eventually its microscopic magnetic properties.
Figure 5a summarizes how the orbital averaged local intra and interorbital densitydensity as well as Hund’s exchange matrix elements scale upon increasing the environmental screening (ϵ). Since ϵ affects the macroscopic monopolelike interactions only, U and \(U^{\prime}\) are equally reduced by ϵ, while J^{H} is not modified at all. The resulting screeninginduced effects will thus be a combination of the ΔU and \({{\Delta }}U^{\prime}\) columns from Fig. 4.
In Fig. 5b, we show the HF DOS for three different values of ϵ. As discussed in the previous section, a decreasing U results in a reduction of the dbands splitting \({{{\Delta }}}_{{t}_{2g,\sigma }/{t}_{2g,\sigma }}\) and \({{{\Delta }}}_{{e}_{g,\sigma }/{e}_{g,\sigma }}\) and a decrease of the interorbital Coulomb terms \(U^{\prime}\) reduces the gap between t_{2g,↑} and e_{g,↑}. The overall effect of increasing ϵ is thus to decrease all gaps between all subbands.
In Fig. 5c, we summarize the resulting effects to the magnetic properties. Upon increasing ϵ we find the total nearestneighbor magnetic exchange decreasing from about J ≈ 1.1 meV at ϵ = 1 to J ≈ 0.2 meV at ϵ = 20, while the total nextnearestneighbor exchange interaction slightly increases from \(J^{\prime} \approx 0.3\)meV to \(J^{\prime} \approx 0.7\) meV. From Table 4, we understand that the decreasing trend in J is mostly driven by the enhancement of the AFM coupling within the J_{t2g–t2g} channel, while the FM J_{t2g–eg} channel is barely affected. The nextnearestneighbor \(J^{\prime}\) is affected simultaneously by both, increasing FM and AFM microscopic exchange interactions. The FM channel grows, however, slightly faster so that the overall FM \(J^{\prime}\) is enhanced. From this, we expect nontrivial effects of the environmental dielectric screening to the magnetic properties of CrI_{3} monolayer relevant for most experimental setups dealing with supported and/or encapsulated films.
We start with the analysis of the magnon dispersion which reacts to the environmental screening differently in different parts of the Brillouin zone, as shown in Fig. 5d. At the Γ point the optical (high energy) branch is continuously lowered in energy upon increasing the screening, while the Dirac point at K is nonmonotonously affected. Upon increasing ϵ from 1 to 8 the magnon energy at K first increases before it starts to decrease for larger ϵ. This behavior becomes clear from the spinwave Hamiltonian Eq. (18) evaluated at K yielding the degenerated magnonenergies \(E(K)=3S(J+3J^{\prime} +\lambda )\). As one can see from Fig. 5c, \(J+3J^{\prime}\) is indeed a nonmonotonic function of ϵ with a maximum around ϵ = 6. For the vanHove singularities of both magnon branches at the M point we find similar nontrivial and nonmonotonic behaviors with ϵ yielding a partially extended flat dispersion of the optical branch for intermediate ϵ. These nontrivial modifications to the magnon dispersion due to changes in ϵ are also reflected in the total magnon spectrum. Thereby we can relate each (partial) maximum in the magnon spectrum with either the Γ or the M point. As a result, these most prominent spectral features either monotonously decrease in energy or follow the nonmonotonous trends from the M point.
In Fig. 5e, we additionally show the Curie temperature (T_{c}) for the same ϵ. Again, we find a nonmonotonic behavior with an initially increasing T_{c} upon increasing the screening, a maximal T_{c} around ϵ = 6, and a subsequently strongly decreasing trend. This trend approximately follows the spectral peak arising from the optical magnon branch at the M point. Due to its similarity with the Kpoint behavior, we conclude that the initial increasing trend in T_{c} is driven by the increasing trend of \(J^{\prime}\) while the final decreasing trend is driven by J. The nonmonotonic behavior of T_{c} is thus due to the nonmonotonic interplay between nearest and nextnearest neighbor exchange interactions as a function of the environmental screening.
Discussion
By combining stateoftheart cRPAbased ab initio downfolding with our WFCE approach and the MFT method, we were able to study on a microscopic level how magnetism in CrI_{3} monolayer builds up and how it is controlled by environmental screening properties. We showed that a meanfield description within the HF approximation to treat local Cr Coulomb interactions is sufficient to reproduce all characteristics of full ab initio GGAs + U calculations. From a thorough investigation of different minimal models, we understood that only an I pbased superexchange mechanism together with the full multiorbital t_{2g}–e_{g} structure of the Cr atoms allows for a realistic description of CrI_{3} magnetism. A minimal model thus needs to involve all Cr d and I p states with sizable Coulomb interactions acting on the Cr d states. We also showed how environmental screening significantly reduces the local Coulomb interactions, which decisively modifies the electronic band structure and which finally yields nonmonotonic changes to the microscopic magnetic exchange interactions. In detail, we found that dielectric encapsulation of the CrI_{3} monolayer strongly reduces the nearestneighbor exchange interaction, while the nextnearestneighbor interaction is just slightly enhanced, which leaves remarkable footprints in the magnon spectral function and the Curie temperature.
These findings point to a variety of questions and problems to tackle in the future: On the modeling side, we found sizeable local Coulomb interactions as compared to the noninteracting bandwidth, possibly rendering dynamical meanfield theory or similar approaches necessary to capture all correlation effects^{39,40,41}. Our extended (d + p)model is an optimal starting ground for studies like these. Together with sizeable magnon–phonon couplings^{42}, light–matter interactions, and longrange Coulomb interaction effects^{8,34}, we can expect a plethora of correlation effects in this material to be found in the future.
In light of our findings on the environmental screeninginduced modifications to the magnetic properties of monolayer CrI_{3}, we expect that magnetism in multilayer CrI_{3} is more involving than expected. Next to electronic interlayer hybridization effects, layerdependent changes to Coulomb interactions seem to be important as well to gain a full understanding. Also, the role of anisotropic magnetic interactions, including both symmetric and asymmetric forms, needs to be studied in further detail. Substrate effects might be especially relevant in the context of Dzyaloshinskii–Moriya interactions (DMI), as they could be considerably enhanced by breaking/lowering inversion symmetries. At the same time, DMI appears to be promising for the stabilization of topological magnons^{43} and skyrmionic phases^{44,45} in chromium trihalides. Coulomb engineering of DMI could be considered using a computational scheme similar to the one proposed in ref. ^{46}.
Finally, our findings render CrI_{3} monolayerbased heterostructures with spatially structured environments a possibly fascinating playground to noninvasively structure the magnetic properties of layered magnetic materials similar to what has been discussed for correlation effects in layered semiconductors^{20,21,23,47,48}. Together with the recent discovery of other 2D ferromagnets and antiferromagnets we thus expect that magnetic van der Waals heterostructures are the most promising platforms to engineer and design nextgeneration magnetic and optomagnetic devices. The encapsulationmediated tunability of the magnetic exchange has important implications in the future application of 2D ferromagnets as spintronic devices. The possibility to combine strong and weak FM regions on 2D ferromagnets using different substrates may find application as memory storage devices. Also, realspace manipulation of the magnon dispersion can open possibilities for lowconsumption magnonbased devices.
Methods
HF solver and double counting corrections
We approximately solve the Hamiltonian from Eq. (1) utilizing a variational singleparticle wave function which allows for the breaking of the spinsymmetry. The variational energy is computed by decoupling the interaction terms in a conventional way. The resulting intraatomic HF Hamiltonian takes the form of an effective singleparticle one with local occupations n_{i} which need to be selfconsistently evaluated. It can be divided into spinconserving (h^{↑↑}, h^{↓↓}) and spinmixing terms (h^{↑↓}, h^{↓↑}). In a general form, the full effective singleparticle Hamiltonian reads
where the i, j are orbital indices. The explicit expressions for each of the terms in the Hamiltonian are
where ρ is the selfconsistent density matrix containing the local occupations: \({\rho }_{ii}^{\sigma \sigma }={\hat{n}}_{i}^{\sigma }\) and \({\rho }_{ij}^{\sigma \bar{\sigma }}=\langle {d}_{i,\sigma }^{{\dagger} }{d}_{j,\bar{\sigma }}\rangle\).
In order to minimize doublecounting errors in the (d + p)model due to interaction effects in the hopping matrix elements from the ab initio calculations, we subtract a doublecounting potential based on the fully localized limit^{25} approximation
where \(\bar{U}=1/(2l+1){\sum }_{i}{U}_{ii}\) and \(\bar{J}=1/(4{l}^{2}+2l){\sum }_{i\ne j}{J}_{ij}\) are the mean Coulomb and Hund exchange interactions obtained from the cRPA tensors, N_{imp} is the total occupancy of the Cr dorbitals and \({N}_{\rm{imp}}^{\sigma }\) is the occupancy per spin (\({N}_{\,{{\mbox{imp}}}}^{\sigma }={N}_{{{\mbox{imp}}}}/2\) in the paramagnetic ground state).
Magnetic force theorem
Magnetism in CrI_{3} and related compounds results from a detailed interplay between local and nonlocal kinetic and local Coulomb interactions terms yielding an effective magnetic exchange between neighboring Cr atoms. Generally speaking, it can be understood as a superexchange mechanism mediated by the ligand atoms which follows approximately the Goodenough–Kanamori mechanism^{13}. With an approximate 90° angle between neighboring Cr–I–Cr atoms we thus expect an FM coupling. On a fully microscopic level, Kashin et al.^{15} have recently shown, using the MFT, that the total FM exchange interaction results from an interplay between an AFM coupling channel between Cr t_{2g} orbitals and an FM channel between t_{2g} and e_{g} orbitals. In both, LSDA and LSDA + U calculations this interplay is dominated by the FM t_{2g}–e_{g} channel so that the total exchange interaction is also FM. To understand how this microscopic picture is affected by different choices of the target space, the different orbital channels of the Coulomb tensor, and by environmental screening effects, we follow Kashin et al.^{15} and analyze the results of our HF calculation by means of the MFT^{49}, which allows us to calculate the orbitally resolved exchange interaction matrix elements via the second variations of the total energy with respect to infinitesimal rotations of the magnetic moments, leading to the expression^{50,51}
Here, Latin (Greek) indices denote atomic (orbital) indices, respectively, E_{F} is the Fermi energy, and \({{{\Delta }}}_{i}^{\alpha \beta }={H}_{ii}^{\alpha \beta \uparrow }{H}_{ii}^{\alpha \beta \downarrow }\) is the exchange splitting matrix defined in the orbital space. In Eq. (15), \({G}_{ij}^{\alpha \beta \sigma }(\varepsilon )={\sum }_{{{{\bf{k}}}}}{G}_{{{{\bf{k}}}}}^{\alpha \beta \sigma }(\varepsilon ){e}^{i{{{\bf{k}}}}\cdot {{{{\bf{R}}}}}_{ij}}\) is the realspace Green’s function, whose kspace matrix representation reads
where \({{{\mathcal{I}}}}\) is the unity matrix, η → 0^{+} is a numerical smearing parameter, R_{ij} is the translation vector connecting atoms i and j, and \({{{{\mathcal{H}}}}}_{{{{\bf{k}}}}}^{\sigma }\) is the reciprocalspace Hamiltonian for spin σ = ↑, ↓, whose matrix elements are obtained in the basis of Wannier functions from our HF calculations. Positive and negative J_{ij} correspond here to FM and AFM couplings, respectively.
Magnetic properties
The spin Hamiltonian describing the exchange interactions between Cr atoms in CrI_{3} can be written as:
where the first term A describes singleion anisotropy, J_{ij} is the isotropic Heisenberg exchange, and λ_{ij} is the anisotropic symmetric exchange. To calculate the spinwave spectrum, we transform Eq. (17) into a bosonic Hamiltonian using the linearized HolsteinPrimakoff transformation with A = 0 and λ = 0.09 meV (see ref. ^{13})
where J and \(J^{\prime}\) correspond to nearest and nextnearest neighbor isotropic exchange couplings, and λ is the nearest neighbor anisotropic symmetric exchange. In kspace, the Hamiltonian Eq. (18) for a honeycomb lattice takes the form
where \({\epsilon }_{0}=3J+6J^{\prime} +3\lambda\), and f_{1}(q) = ∑_{R}e^{−iq⋅R} and \({f}_{2}({{{\bf{q}}}})={\sum }_{{{{\bf{R}}}}^{\prime} }{e}^{i{{{\bf{q}}}}\cdot {{{\bf{R}}}}^{\prime} }\) are form factors with R and \({{{\bf{R}}}}^{\prime}\) running over cells with nearest and nextnearest neighbor atoms, respectively. The diagonalization of this Hamiltonian yields the magnon spectrum, which reads
To calculate the magnetic T_{c}, we use Tyablikov’s decoupling approximation (also known as RPA)^{52} through the expression
Substrate screening
Next to the CrI_{3} intrinsic properties, we aim to also understand the role of external screening properties such as resulting for substrate materials or capping dielectrics. To this end, we utilize our WFCE approach^{19}, which realistically modifies the CrI_{3} Coulomb interaction tensor according to dielectric environmental screening and which has been shown before to reliably describe the environmental screening impact to layered materials^{23,48}. In this way we will be able to understand how the electronic band structure and the microscopic magnetic properties are affected, e.g., by encapsulating CrI_{3} with hBN or under the influence of bulk substrates.
We start with the nonlocal bare Coulomb interaction of the CrI_{3} monolayer as obtained from our cRPA calculations in momentum space. Within a matrix representation v_{αβ}(q) using a product basis α, β = {n, m} we can diagonalize the Coulomb tensor
with v_{ν}(q) and \(\left{v}_{\nu }(q)\right\rangle\) being the corresponding eigenvalues and eigenvectors of the Coulomb tensors and q = ∣q∣. Assuming that the eigenbasis does not drastically change upon the effects of the cRPA screening, we can thus represent the full cRPA Coulomb tensor as
where ε_{ν}(q) is the corresponding pseduoeigenvalues of the dielectric tensor describing the different screening channels. In Fig. 6 we show the first three ε_{ν}(q) as a function of momentum q and find that only one shows a significant dispersion, which we refer to as the “leading” eigenvalue in the following. This behavior becomes clear upon investigating the corresponding eigenvectors on the basis of the two Cr atoms in the longwavelength limit, i.e., for q → 0
The leading eigenvalue v_{1}(q) thus renders Coulomb penalties for monopolelike perturbations (all orbitally resolved electronic densities on both Cr atoms are inphase), while the second eigenvalue v_{2}(q) corresponds to CrCr dipolelike Coulomb penalties. ε_{1}(q) and ε_{2}(q) thus correspondingly describe mono and Crdipolelike screening. While the dipolelike screening from the environment is negligible, the monopolelike screening as rendered by ε_{1}(q) is strongly affected. This classical electrostatic screening can be modeled by solving the Poisson equation for a dielectric slab of height h embedded in some different dielectric environment^{19,53,54,55} yielding
with
For \({\varepsilon }_{\,{{\mbox{sub}}}}^{{{\mbox{above}}}}={\varepsilon }_{{{\mbox{sub}}}}^{{{\mbox{below}}}\,}=1\) this describes the leading dielectric function of a freestanding monolayer, which we can fit perfectly to the cRPA data, as shown in Fig. 6 and yielding h ≈ 5.2 Å and \({\varepsilon }_{1}^{(0)}\approx 8.7\) (for the case of the (d+p)model). With these fitting parameters, we can now modify the full cRPA Coulomb tensor to describe environmental screening rendered by finite \({\varepsilon }_{\,{{\mbox{sub}}}}^{{{\mbox{above}}}\,}\) and \({\varepsilon }_{\,{{\mbox{sub}}}}^{{{\mbox{below}}}\,}\). Due to the monopolelike character of this environmental screening, we only affect densitydensity Coulomb matrix elements in the very same way. Therefore Coulomb exchange elements, such as J^{H} are not affected by the substrate screening. In Supplementary Methods 4 we benchmark this approach to full cRPA calculations taking the screening from additional (strained) hBN layers into account.
Data availability
The data that support the findings of this study is available from the corresponding author upon reasonable request.
Code availability
All noncommercial numerical codes to reproduce the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank M. Kaltak for sharing his cRPA implementation with us. M.I.K. acknowledges support by European Research Council via Synergy Grant 854843—FASTCORR. D.S. thanks financial support from EU through the MSCA project Nr. 796795 SOT2DvdW. ANR acknowledges partial support from the Russian Science Foundation, Grant No. 217210136. Part of this work was carried out on the Dutch national einfrastructure with the support of SURF Cooperative.
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D.S. and M.R. conceived and designed the project. Calculations have been performed by D.S., A.N.R. and M.R. All authors contributed to the paper writing.
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Soriano, D., Rudenko, A.N., Katsnelson, M.I. et al. Environmental screening and ligandfield effects to magnetism in CrI_{3} monolayer. npj Comput Mater 7, 162 (2021). https://doi.org/10.1038/s41524021006314
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DOI: https://doi.org/10.1038/s41524021006314
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