Electronic excitations and spin interactions in chromium trihalides from embedded many-body wavefunctions

Abstract

Although chromium trihalides are widely regarded as a promising class of two-dimensional magnets for next-generation devices, an accurate description of their electronic structure and magnetic interactions has proven challenging to achieve. Here, we quantify electronic excitations and spin interactions in CrX₃ (X = Cl, Br, I) using embedded many-body wavefunction calculations and fully generalized spin Hamiltonians. We find that the three trihalides feature comparable d-shell excitations, consisting of a high-spin ⁴A₂$\left(t_{2g}^3{e}_g^0\right)$ ground state lying 1.5–1.7 eV below the first excited state ⁴T₂ ($t_{2g}^2{e}_g^1$). CrCl₃ exhibits a single-ion anisotropy A_sia = − 0.02 meV, while the Cr spin-3/2 moments are ferromagnetically coupled through bilinear and biquadratic exchange interactions of J₁ = − 0.97 meV and J₂ = − 0.05 meV, respectively. The corresponding values for CrBr₃ and CrI₃ increase to A_sia = −0.08 meV and A_sia= − 0.12 meV for the single-ion anisotropy, J₁ = −1.21 meV, J₂ = −0.05 meV and J₁ = −1.38 meV, J₂ = −0.06 meV for the exchange couplings, respectively. We find that the overall magnetic anisotropy is defined by the interplay between A_sia and A_dip due to magnetic dipole–dipole interaction that favors in-plane orientation of magnetic moments in ferromagnetic monolayers and bulk layered magnets. The competition between the two contributions sets CrCl₃ and CrI₃ as the easy-plane (A_sia + A_dip >0) and easy-axis (A_sia + A_dip <0) ferromagnets, respectively. The differences between the magnets trace back to the atomic radii of the halogen ligands and the magnitude of spin–orbit coupling. Our findings are in excellent agreement with recent experiments, thus providing reference values for the fundamental interactions in chromium trihalides.

Introduction

The advent of two-dimensional (2D) magnets has created new opportunities to explore and manipulate spin interactions in the ultimate limit of atomic thickness, holding promise for an array of nanoscale applications ranging from magneto-optoelectronics to quantum information^1–4. In these 2D magnets, the non-vanishing critical temperature originates from the magnetic anisotropy, which is key to preserving long-ranged magnetic order against thermal fluctuations, as explained by the theorems of Mermin–Wagner⁵ and Hohenberg⁶. Within the rapidly expanding class of ultrathin magnets, layered chromium trihalides of general formula CrX₃ (X = Cl, Br, I)⁷ are among the most discussed members by virtue of their versatile magnetic properties that can be engineered via, e.g., atomic thickness⁸, stacking configuration^9,10, electrostatic gating^11,12, lattice deformations^13–15, defects incorporation^16,17, or light irradiation¹⁸. Such tunability is key for the construction of new energy-efficient spin-based devices.

In the recent years, a flurry of theoretical investigations on this family of materials established the important role played by t_2g–e_g interactions¹⁹, ligand p orbitals^20,21 and stacking sequence²² in stabilizing the ferromagnetic order. The insulating chromium trihalides realize highly localized magnetic moments close to 3 μ_B, corresponding to an S = 3/2 system²³, with short-range superexchange interactions, as well as local magnetic anisotropy. The latter allows this family of 2D materials to overcome the Mermin–Wagner theorem and to establish long-range magnetic order²⁴. Inelastic neutron scattering experiments on CrI₃ showed ≈4 meV gap opening between the two magnon branches²⁵. Chen et al.²⁵ argued that the gap originates from the Dzyaloshinskii–Moriya (DM) interaction^26–28 and predicted an unusually large dominant DM interaction from the fitting of magnon spectra. However, the nearest-neighbor (NN) DM vector in CrI₃ is not allowed by symmetry and only the next NN DM interaction is finite. Alternatively, such a gap was argued to result from significant Kitaev interactions (symmetric anisotropic exchange)²⁹. The Kitaev term is allowed by symmetry, but similar to DM interaction, it also originates from spin–orbit coupling (SOC) and is usually quite small in 3d materials. The heavier ligands do contribute to SOC, but the presence of such dominant anisotropic interaction is still under debate as the magnetic interaction parameters reported from various experiments and theoretical calculations differ^7,29–31. Further studies have also discussed higher-order interactions (biquadratic exchange) and lattice defects as the origin of the gap^32,33.

Despite the rapidly growing interest in CrX₃ magnets, an accurate description of magnetic interactions as well as electronic excitations is still needed. On the one hand, high-resolution measurements based on soft X-ray spectroscopies, which are instrumental to accessing electronic excitations, are challenging to accomplish and tedious to interpret³⁴. On the other hand, recent experiments assessing spin interactions are mainly limited to isotropic Heisenberg exchange couplings, the strength of which remains debated since reported values differ by upto one order of magnitude^7,29,35. From the theoretical point of view, earlier studies primarily resort to density-functional methods, which are inherently inadequate to cope with both excited and correlated electronic states. Such a drawback was shown to be particularly severe when dealing with chromium trihalides and other 2D magnets, where, depending on the adopted exchange-correlation approximation and computational scheme, the strength of the calculated magnetic interactions spreads over a wide energy window^7,36,37. This calls for a rigorous and unambiguous account of electron-electron effects in order to achieve a detailed comprehension of CrX₃ magnets.

In this work, we present a comparative investigation of electronic excitations and spin interactions in two-dimensional CrX₃ (X = Cl, Br, I) using post-Hartree-Fock methods, namely the complete-active-space self-consistent-field (CASSCF) and multireference configuration interaction (MRCI) methodologies (see “Methods”), that are arguably the most accurate and rigorous approach to treat electronic correlations. First, we determine the d-shell multiplet structures, which allows us to offer an unambiguous interpretation to recent X-ray spectroscopic measurements. Next, we quantify the strength of dominant magnetic interactions, including single-ion anisotropy, g-factors, and exchange couplings, using a fully generalized spin Hamiltonian. Altogether, our findings establish a theoretical ground to recent spectroscopic observations, thus providing a quantitative understanding of the microscopic physics in chromium trihalides from a quantum chemical perspective.

Results

We begin our work by establishing the Cr³⁺ ground state and d-shell excitations in CrX₃. In Table 1, we give the multiplet structure, as obtained at the CASSCF and MRCI levels using the one-site model shown in Fig. 1b. Additional details of our calculations are provided in Supplementary Note 1. According to the crystal-field theory, the octahedral environment encaging each Cr³⁺ ion lifts the energy degeneracy of the d-orbital manifold, giving rise to a lower-lying triply degenerate t_2g and a higher-lying doubly degenerate e_g group of levels. The ground state of the halides is the high-spin singlet state ⁴A₂ dominated by the ($t_{2g}^3{e}_g^0$) configuration, in compliance with Hund’s rule of maximum multiplicity. The half-filled t_2g subshell renders each Cr³⁺ ion a spin-3/2 center, in line with the magnetization saturation of 3 μ_B observed in SQUID magnetometry measurements⁸. The first excited state ⁴T₂ consists of dominant contributions from the configuration that comprises a pair of electrons in the t_2g orbitals and a single electron in the e_g orbitals. In the case of CrCl₃, this leads to a splitting between the t_2g and e_g orbitals as large as ~1.6 eV at CASSCF level, which slightly increases to ~1.7 eV at the MRCI level. Consistently with the order of the ligand strength subsumed in the spectrochemical series, such a splitting decreases in energy as the size of the ligand increases (that is, moving from CrCl₃ to CrI₃). In both CrBr₃ and CrI₃ the calculated splitting is ~1.5 eV at the MRCI level of theory. The reason for this traces back to the more extended electron density in the heavier halogen atom and the consequently longer transition metal-to-ligand distances. Indeed, the t_2g–e_g splitting is sensitive to structural deformations of the octahedral cage and increases (decreases) when the crystal is subjected to moderate tensile (compressive) in-plain lattice strain, as we show for the representative case of CrCl₃ in Supplementary Table 1.

Table 1.

Relative energies of the 3d³ multiplet structure of Cr³⁺ ions in CrX₃ (X = Cl, Br, I), as obtained from CASSCF and MRCI calculations

	CrCl3		CrBr3		CrI3
	CASSCF	MRCI	CASSCF	MRCI	CASSCF	MRCI
4A2 ($t_{2g}^3{e}_g^0$)	0.00	0.00	0.00	0.00	0.00	0.00
4T2 ($t_{2g}^2{e}_g^1$)	1.57, 1.58, 1.61	1.67, 1.68, 1.72	1.38, 1.59, 2.00	1.46, 1.66, 2.05	1.44, 1.56, 1.58	1.48, 1.59, 1.61
2E ($t_{2g}^3{e}_g^0$)	2.36, 2.36	2.22, 2.22	1.92, 2.19	1.77, 2.10	2.32, 2.33	2.16, 2.17
4T1 ($t_{2g}^2{e}_g^1$)	2.51, 2.53, 2.60	2.50, 2.52, 2.59	2.51, 2.59, 2.95	2.49, 2.62, 2.96	2.48, 2.50, 2.52	2.39, 2.49, 2.51
2T1 ($t_{2g}^3{e}_g^0$)	2.45, 2.47, 2.48	2.31, 2.33, 2.34	2.36, 2.45, 2.49	2.22, 2.32, 2.37	2.39, 2.39, 2.42	2.26, 2.30, 2.31
2T1 ($t_{2g}^2{e}_g^1$)	3.30, 3.31, 3.33	3.03, 3.05	3.32, 3.41, 3.45	3.23, 3.25, 3.26	3.23, 3.29, 3.31	2.95, 3.02, 3.02
2A1 ($t_{2g}^2{e}_g^1$)	3.56	3.47	3.56	3.45	3.47	3.39
2T1 ($t_{2g}^2{e}_g^1$)	3.83, 3.84, 3.85	3.76, 3.77, 3.78	4.06, 4.32, 4.80	3.94, 4.24, 4.73	3.68, 3.70, 3.71	3.59, 3.61, 3.63
4T1 ($t_{2g}^1{e}_g^2$)	4.13, 4.15, 4.16	4.05, 4.07, 4.08	3.71, 3.85, 3.89	3.56, 3.74, 3.78	3.90, 4.01, 4.04	3.80, 4.10, 4.12

Energies are given in eV and referenced to the ground state. Occupations in parentheses indicate the dominant configuration.

Fig. 1. Crystal structure and embedding models of chromium trihalides.

a Crystal structure of monolayer CrX₃ (X = Cl, Br, I). Blue and orange spheres represent chromium and halogen atoms, respectively. b Embedding model used in the determination of the multiplet structures and intra-site magnetic interactions. c Embedding model used in the determination of the inter-site magnetic interactions. In panels (b, c), dark (light) colors indicate the atoms that are treated with many-body (Hartree–Fock) wavefunctions. The models are embedded in an array of points charges (not shown) to reproduce the crystalline environment and ensure charge neutrality.

We gain theoretical insight into the recent X-ray spectroscopy experiments³⁴ probing electronic excitations in CrX₃. To that end, we simulate the Cr L_3,2-edge X-ray absorption spectra (XAS) and L₃-edge resonant inelastic X-ray scattering (RIXS) spectra of CrCl₃ and CrI₃ using CASSCF calculations, and the minimal finite-size model shown in Supplementary Fig. 1a. The resulting d-shell excitation energies of the Cr³⁺ ion are listed in Supplementary Tables 2 and 3. We consider the same scattering geometry used in recent experiments by Shao et al.³⁴ and given in Supplementary Fig. 1b. Additional details of these calculations are given in Supplementary Note 2. We remark that similar computational schemes have been validated against experimental measurements for diverse magnetic compounds^38–40. The simulated RIXS and XAS spectra of CrCl₃ and CrI₃ are shown in Fig. 2. The excellent agreement between the theoretical and experimental spectra allows us to assign the electronic excitations observed. In Fig. 2a, b, the L-edge absorption process involves a Cr 2p electron that is promoted to the empty 3d orbitals (2p → 3p), leading to the final 2p⁵3d⁴ configuration (2p⁶3d³ → 2p⁵3d⁴). This transition results in the two main features in the XAS spectra located at incident photon energies of 574–580 eV and 582–588 eV, which correspond to the Cr L₃- and L₂-edges, respectively. These two features are separated by ~9 eV due to spin–orbit interactions in the presence of the 2p core-hole in the XAS final states⁴¹. On the other hand, spin–orbit interactions weakly affect the Cr 3d³ valence states; see Supplementary Tables 2 and 3.

Fig. 2. X-ray spectra of chromium trihalides.

Simulated and experimental X-ray absorption spectra (XAS) of a CrCl₃ and b CrI₃. Simulated and experimental resonant X-ray inelastic scattering (RIXS) spectra of c CrCl₃ and d CrI₃. Vertical lines in the simulated spectra mark the excitation energies. The experimental spectra are digitized from ref. ³⁴. The simulated XAS spectra were shifted to match the position in energy of the highest intensity peak in the experimental spectra. Notice the different order of intensity between theory and experiment observed for the L₃- and L₂-edges is due to undesired self-absorption effects in experimental measurements.

Due to the core-hole broadening in L-edge XAS, the on-site d-d excitations are not well-resolved or determined by comparing our calculations with the experimental XAS spectra. We further simulate the L₃-edge RIXS spectra to resolve the otherwise elusive d-shell excitations experimentally observed. This is accomplished by setting the incident energy, E_in, at the maximum intensity of L₃-edge XAS, i.e., E_in = 576.6 eV for CrCl₃ and E_in= 576.0 eV for CrI₃, and determining the RIXS intensities of π-polarization at incident angle 50°, similarly to experiments. As we restrict ourselves to on-site d-shell excitations and neglect ligand-metal charge transfer states, we focus on energy losses lower than 4 eV. The simulated RIXS spectra are shown in Fig. 2c, d and, in analogy with experiments, exhibit three main features. For CrCl₃, the first excitation peak at ~1.6 eV corresponds to the t_2g−e_g transitions associated with spin quartet states ⁴T₂($t_{2g}^2{e}_g^1$). The second excitation peak occurring in the 2.2–2.7 eV energy range is dominated by the ⁴T₁ ($t_{2g}^2{e}_g^1$) states, with a slight spectral weight contribution from the ²E ($t_{2g}^3{e}_g^0$) and ²T₁ ($t_{2g}^3{e}_g^0$) excited states at 2.3–2.4 eV. The feature arising at ~3.2 eV is related to the ²T₂ ($t_{2g}^3$) states. The remaining higher energy loss feature from 3.5 to 4.0 eV is attributed to the t_2g − e_g transitions associated with spin doublet states (²A, ²T₁, and ²T₂). Similarly, for CrI₃ the first excitation peak at ~1.5 eV is assigned to the t_2g − e_g transitions associated with spin quartet states ⁴T₂ ($t_{2g}^2{e}_g^1$). As compared to CrCl₃, however, the first excitation peak shift to lower energy loss by ~0.2 eV, both in the simulated and experimental spectra. The second feature at ~2.3 eV is contributed by the ⁴T₁ ($t_{2g}^2{e}_g^1$) and ²T₁ ($t_{2g}^3$) states. At energy losses larger than 2.5 eV, the feature becomes broad and the t_2g–e_g transitions associated with spin doublet states are dominated.

Next, we focus on magnetic interactions in CrX₃ materials. We write the effective spin Hamiltonian $\mathcal{H}$ as

$$\mathcal{H}={\mathcal{H}}^1+{\mathcal{H}}^2,$$ 1

where ${\mathcal{H}}^1$ encodes the contribution of the interactions within sites of spin ${\overrightarrow{S}}_i$, while ${\mathcal{H}}^2$ describes the coupling between the ith and jth nearest-neighbor sites bearing spins ${\overrightarrow{S}}_i$ and ${\overrightarrow{S}}_j$, respectively. The intra-site contribution ${\mathcal{H}}^1$ takes the form

$${\mathcal{H}}^1=\sum _i{\overrightarrow{S}}_i\cdot {\overline{\overline{D}}}_{\mathrm{sia}}\cdot {\overrightarrow{S}}_i+\sum _i{\mu }_{\mathrm{B}}\overrightarrow{B}\cdot \overline{ḡ}\cdot {\overrightarrow{S}}_i,$$ 2

where ${\overline{\overline{D}}}_{\mathrm{sia}}$ is the single-ion anisotropy tensor and $\overline{ḡ}$ is the g-tensor in the Zeeman term that emerges upon the application of an external magnetic field $\overrightarrow{B}$. The inter-site contribution ${\mathcal{H}}^2$ reads

$${\mathcal{H}}^2=J_1\sum _{i<j}\left({\overrightarrow{S}}_i\cdot {\overrightarrow{S}}_j\right)+J_2\sum _{i<j}{\left({\overrightarrow{S}}_i\cdot {\overrightarrow{S}}_j\right)}^2+\sum _{i<j}{\overrightarrow{S}}_i\cdot \overline{\overline{\mathrm{\Gamma }}}\cdot {\overrightarrow{S}}_j,$$ 3

with J₁ and J₂ being the bilinear and biquadratic isotropic exchange couplings, respectively, and $\overline{\overline{\mathrm{\Gamma }}}$ the symmetric anisotropic tensor. We remark that the nearest-neighbor Dzyaloshinskii–Moriya interaction vanishes due to the centrosymmetric crystal structure of CrX₃²⁷. Assuming a local Kitaev frame where the z axis is perpendicular to the Cr₂ X₂ plaquette for each Cr–Cr bond^42,43, $\overline{\overline{\mathrm{\Gamma }}}$ can be expressed as

$$\overline{\overline{\mathrm{\Gamma }}}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill {\mathrm{\Gamma }}_{xy}\hfill & \hfill -{\mathrm{\Gamma }}_{yz}\hfill \\ \hfill {\mathrm{\Gamma }}_{xy}\hfill & \hfill 0\hfill & \hfill {\mathrm{\Gamma }}_{yz}\hfill \\ \hfill -{\mathrm{\Gamma }}_{yz}\hfill & \hfill {\mathrm{\Gamma }}_{yz}\hfill & \hfill K\hfill \end{array}\right),$$ 4

where K is the Kitaev interaction parameter.

We determine the intra-site interactions appearing in ${\mathcal{H}}^1$, i.e., the single-ion anisotropy and the g-tensor, relying on the one-site model shown in Fig. 1b. Further details of our calculations are given in Supplementary Note 1. Our results are presented in Table 2. The single-ion anisotropy quantifies the zero-field splitting of the ground state that stems from spin–orbit and crystal-field effects. To assess this quantity, we rely on the methodology developed in ref. ⁴⁴ taking advantage of the multiplet structures listed in Table 1 and corresponding wavefunctions. In brief, the mixing of the low-lying ⁴A₂ states with the higher-lying states is treated perturbatively and the spin–orbit wavefunctions related to the high-spin $t_{2g}^3$ configuration are projected onto the space spanned by the $\genfrac{}{}{0ex}{}{4}{}{A}_2\mid S,M_s⟩$ states. The orthonormalized projections of the low-lying quartet wavefunctions ${\tilde{\psi }}_k$ and the corresponding eigenvalues E_k are used to construct the effective Hamiltonian ${\tilde{H}}_{\mathrm{eff}}={\mathrm{\Sigma }}_k{E}_k\mid {\tilde{\psi }}_k⟩⟨{\tilde{\psi }}_k\mid$. A one-to-one correspondence between ${\tilde{H}}_{\mathrm{eff}}$ and the model Hamiltonian ${\tilde{H}}_{\mathrm{mod}}={\overrightarrow{S}}_i\cdot {\overline{\overline{D}}}_{\mathrm{sia}}\cdot {\overrightarrow{S}}_i$ leads to the ${\overline{\overline{D}}}_{\mathrm{sia}}$ tensor (listed in Supplementary Table 4), which is then diagonalized and the axial parameter A_sia obtained as $A_{\mathrm{sia}}=D_{\mathrm{sia}}^{zz}-\left(D_{\mathrm{sia}}^{xx}+D_{\mathrm{sia}}^{yy}\right)/2$. At the MRCI level of theory, we find A_sia = −0.03 meV for CrCl₃, −0.08 meV for CrBr₃ and −0.12 meV for CrI₃. The negative sign of the axial parameter corresponds an easy axis of magnetization. In all the three cases, we find an easy axis pointing perpendicular to the honeycomb plane. The considerably large magnitude of the single-ion anisotropy in CrI₃ compared to CrCl₃ reflects the strong spin–orbit interactions inherent to the heavier ligands⁴⁵.

Table 2.

Intra-site magnetic interactions in CrX₃ (X = Cl, Br, I), i.e., single-ion anisotropy axial parameter A_sia, along with the in-plane (g_xx, g_yy) and out-of-plane (g_zz) components of the g-tensor, as obtained from CASSCF and MRCI calculations

	CrCl3		CrBr3		CrI3
	CASSCF	MRCI	CASSCF	MRCI	CASSCF	MRCI
Asia (meV)	−0.02	−0.03	−0.07	−0.08	−0.11	−0.12
gxx = gyy	1.44	1.45	1.86	1.86	1.91	1.92
gzz	1.46	1.47	1.87	1.88	1.92	1.93

Another important contribution to the anisotropy originates from magnetic dipole–dipole interactions that are not accounted for in the electronic structure calculations^46,47. This contribution, however, should not be neglected for monolayer as well as layered bulk magnetic materials as a result of magnetic shape anisotropy. The magnetic dipolar anisotropy for any given lattice can be determined as:

$${\mathcal{H}}_{\mathrm{dip}}={\left(g_e{\mu }_{\mathrm{B}}\right)}^2\sum _{i<j}\left[\frac{\left(\right(\mid {\overrightarrow{r}}_{ij}{\mid }^2\left({\overrightarrow{S}}_i\cdot {\overrightarrow{S}}_j\right)-3\left({\overrightarrow{S}}_i\cdot {\overrightarrow{r}}_{ij}\right)\left({\overrightarrow{S}}_j\cdot {\overrightarrow{r}}_{ij}\right)}{\mid {\overrightarrow{r}}_{ij}{\mid }^5}\right],$$ 5

where ${\overrightarrow{S}}_{i\left(j\right)}$ is spin at site i(j) and $\overrightarrow{r_{ij}}$ is the vector connecting sites i and j. This expression can be rewritten in a form that is similar to Eq. (2):

$${\mathcal{H}}_{\mathrm{dip}}=\sum _{i<j}{\overrightarrow{S}}_i\cdot {\overline{\overline{D}}}_{\mathrm{dip}}\cdot {\overrightarrow{S}}_j,$$ 6

where ${\overline{\overline{D}}}_{\mathrm{dip}}$ is a tensor. The matrix elements $D_{\mathrm{dip}}^{\alpha \beta }$ (α, β = x, y, z) are computed as

$$D_{\mathrm{dip}}^{\alpha \beta }={\left(g_e{\mu }_{\mathrm{B}}\right)}^2\sum _{i<j}\left[\frac{\mid {\overrightarrow{r}}_{ij}{\mid }^2\left({\widehat{e}}_i^{\alpha }\cdot {\widehat{e}}_j^{\beta }\right)-3\left({\widehat{e}}_i^{\alpha }\cdot {\overrightarrow{r}}_{ij}\right)\left({\widehat{e}}_j^{\beta }\cdot {\overrightarrow{r}}_{ij}\right)}{\mid {\overrightarrow{r}}_{ij}{\mid }^5}\right],$$ 7

with ${\widehat{e}}_i^{\alpha }$ being the unit vector at site i pointing along α = x, y, z. In order to avoid summing over an infinite number of pairs of spins we used the approach described in ref. ⁴⁸, which is based on performing summation upto a certain cutoff distance between the spins and carefully evaluating the convergence of the matrix elements of ${\overline{\overline{D}}}_{\mathrm{dip}}$ with respect to this cutoff. Supplementary Fig. 2 presents the converged values of $D_{\mathrm{dip}}^{\alpha \alpha }$ for the x, y, z orientation of magnetic moments in CrCl₃, CrBr₃ and CrI₃. Similar to the single-ion anisotropy axial parameter, we obtain the axial parameter for the magnetic dipolar anisotropy $A_{\mathrm{dip}}=D_{\mathrm{dip}}^{zz}-\left(D_{\mathrm{dip}}^{xx}+D_{\mathrm{dip}}^{yy}\right)/2$. The A_dip parameters for all three chromium trihalides are listed in Table 3. The magnitudes of A_dip decrease along the series as a result of a slight increase of the in-plane lattice constants and are larger for monolayers as compared to bulk materials. Most importantly, these values are positive, which is consistent with the expectation of the demagnetization energy being lower for the in-plane magnetization in slab geometries, and are of magnitude comparable to that of single-ion anisotropy. This allows us to conclude that the overall magnetic anisotropy in chromium trihalides is dictated by the interplay between these two competing contributions. In the heavier CrI₃ single-ion anisotropy dominates (A_sia + A_dip = −67 μeV) establishing as the (out-of-plane) easy-axis ferromagnet, while the opposite is true for CrCl₃ (A_sia + A_dip = 54 μeV) that we conclude to be an easy-plane ferromagnet. The case of CrBr₃ is borderline with the marginally larger contribution from single-ion anisotropy, which establishes this material as an easy-axis magnet with weak anisotropy. Our findings are in full agreement with recent experiments^8,23,49,50.

Table 3.

Comparison of the dipolar anisotropy parameter A_dip and the single-ion anisotropy parameter A_sia obtained from MRCI calculations for bulk CrX₃ (X = Cl, Br, I)

	CrCl3	CrBr3	CrI3
Asia (μeV)	−31.4	−81.1	−124.2
Adip (μeV)	85.2	75.5	57.3

The negative sign indicates that the out-of-plane spin moments are favored. The shape anisotropy parameter A_dip for monolayer CrX₃ (X = Cl, Br, I) approximately doubles in magnitude as compared to the bulk, and attains values of 144 μeV, 121 μeV, 94 μeV for chloride, bromide and iodide, respectively.

There is, however, one subtle point⁵¹. The transition from Eqs. (5) to (6) is rigorous only for homogeneous distribution of magnetization plus ellipsoidal shape of the sample⁴⁷. The spin-wave spectrum for ferromagnets with an easy-plane anisotropy and for those with dipole–dipole interactions are essentially different, and for the latter the spin-wave energy tends to zero at zero wave vector much slower^46,51. Also, due to the long-range character of dipole–dipole interactions the formal conditions of applicability of Mermin–Wagner–Hohenberg theorem are not fulfilled. Maleev suggested⁵¹ that in 2D ferromagnets with dipole–dipole interactions one has the true long-range order at low enough temperatures, contrary to the “almost broken symmetry” of Berezinskii–Kosterlitz–Thouless type^52,53 for the easy-plane ferromagnets. Later this suggestion was confirmed within the framework of self-consistent spin-wave theory⁵⁴. This means that for all three CrX₃ (X = Cl, Br, I) ferromagnets, one can expect the true long-range ordered state below the Curie temperature.

Of crucial importance to understand the response of CrX₃ to external magnetic fields is the g-tensor. We determine this quantity following the scheme devised in ref. ⁵⁵. Many-body wavefunction calculations enable to extract the matrix elements of the total magnetic moment operator $\widehat{\mathit{\mu }}$ in the basis of the spin–orbit coupled eigenstates $\widehat{\mathit{\mu }}=-{\mu }_B\left(g_e\widehat{S}+\widehat{L}\right)$, where $\widehat{L}$ is the angular momentum operator, $\widehat{S}$ the spin operator, and g_e the free-electron Landé factor for a given magnetic site. The Zeeman Hamiltonian in terms of total magnetic moment, $H_{\mathrm{Z}}=-\widehat{\mu }\cdot \overrightarrow{B}$, is then mapped onto the Zeeman contribution to ${\mathcal{H}}^1$ given in Eq. (2). To obtain the g-factors, we use an active space that includes all five d orbitals of the Cr³⁺ ion averaged over five quartets and seven doublets. The g-factors at both CASSCF and MRCI level of theory are listed in Table 2. These values match those determined in recent electron spin resonance experiments¹⁸, that is 1.49 and 1.96 for CrCl₃ and CrI₃, respectively. The slight anisotropy (~1%) of the calculated g-tensors arises from the mild trigonal distortion of the octahedral cage occurring in CrX₃ ferromagnets.

Finally, we determine the inter-site interactions appearing in the spin Hamiltonian ${\mathcal{H}}^2$, i.e., the bilinear and biquadratic isotropic exchange couplings J₁ and J₂, respectively, together with the symmetric anisotropic tensor $\overline{\overline{\mathrm{\Gamma }}}$. We rely on the two-site model shown in Fig. 1c and give further technicalities of our calculations in Supplementary Note 3. We map the resulting ab initio Hamiltonian onto the anisotropic biquadratic model Hamiltonian given in Eq. (3), which involves sixteen spin–orbit states corresponding to one septet, one quintet, one triplet, and a singlet. The mapping is accomplished according to the procedure described in ref. ⁵⁶. The resulting magnetic interactions are listed in Table 4. Additional results obtained on a simpler isotropic bilinear Hamiltonian are provided in Supplementary Table 5.

Table 4.

Inter-site magnetic interactions in CrX₃ (X = Cl, Br, I), i.e., bilinear (J₁) and biquadratic (J₂) exchange couplings, along with the off-diagonal (Γ_xy, Γ_yz, Γ_zx) and Kitaev (K) parameters comprised in the symmetric anisotropic tensors, as obtained from the CASSCF and MRCI calculations

	CrCl3		CrBr3		CrI3
	CASSCF	MRCI	CASSCF	MRCI	CASSCF	MRCI
J1 (meV)	−0.64	−0.97	−0.61	−1.21	−0.60	−1.38
J2 (meV)	−0.04	−0.05	−0.05	−0.05	−0.06	−0.06
Γxy (meV)	−1.2 × 10−4	−2.1 × 10−4	−6.9 × 10−4	−0.8 × 10−3	−1.2 × 10−3	−4.2 × 10−4
Γyz = − Γzx (meV)	−1.3 × 10−5	−0.8 × 10−4	−6.9 × 10−4	−0.9 × 10−3	−1.1 × 10−4	−3.1 × 10−4
K (meV)	−1.7 × 10−4	−1.1 × 10−4	−8.2 × 10−3	−0.01	−9.3 × 10−3	−0.05

The dominant term among the inter-site interactions is the bilinear isotropic exchange coupling J₁, the negative sign of which indicates a parallel spin interaction between the spin-3/2 centers. This is fully consistent with the observations of an intra-layer ferromagnetic order probed by the magneto-optical Kerr effect in CrI₃⁸ and temperature-dependence transport measurements in CrCl₃⁴⁹. The strength of J₁ is largely dependent on whether the CASSCF or MRCI level of theory is adopted due to the different types of exchange mechanisms involved in these approaches. At the CASSCF level, the main mechanism is the direct exchange between the transition metal atoms. As a result, J₁ is slightly larger in CrCl₃ and CrBr₃ as compared to CrI₃ as a consequence of the shorter distance between the Cr³⁺ ions in the former trihalides. At the MRCI level, however, the superexchange pathway through the non-magnetic ligands becomes operative, significantly strengthening J₁ which attains the value of −0.97 meV in CrCl₃, −1.21 meV in CrBr₃ and −1.38 meV in CrI₃. This increase in the magnitude of J₁ quantifies the superexchange contribution to the bilinear exchange interaction, which is greater in CrI₃ (0.78 meV) as compared to CrCl₃ (0.33 meV) owing to the more extended p valence orbitals of the iodine atoms compared to the chlorine atoms. Hence, the sign of the bilinear exchange interaction can be qualitatively rationalized in terms of the rule devised by Goodenough⁵⁷ and Kanamori⁵⁸, which points to a ferromagnetic coupling when the Cr–X–Cr bond angle approaches 90^o. The calculated values of the bilinear exchange interaction obtained at the MRCI level are in excellent agreement with certain recent experimental estimates, e.g., J₁ = −0.92 meV for CrCl₃⁵⁰ and −1.42 meV for CrI₃³⁵, and their trend is consistent with the significantly lower critical temperature observed in the former magnet (~16 K⁵⁰) than the latter (45 K⁸).

Importantly, we reveal a relatively sizable biquadratic exchange couplings J₂, thus indicating that ferromagnetic two-electron hopping processes are active in chromium trihalides^32,33,59. The strength of J₂ appears to be weakly dependent on the nature of the halogen atom and on whether or not superexchange effects are described. This possibly suggests that, contrary to bilinear interactions, multispin interactions occur via the direct exchange between the Cr³⁺ ions. Both the isotropic bilinear and biquadratic exchange coupling largely exceed the anisotropic exchange interactions. Among the anisotropic exchange couplings $\overline{\overline{\mathrm{\Gamma }}}$, only the Kitaev interaction in CrI₃ is non-vanishing (−0.05 meV), with all the off-diagonal terms being negligible. Irrespective of the nature of the halogen ligands, however, our results indicate a very small Kitaev-to-Heisenberg ratio, lending support to earlier findings on CrI₃^30,45,60, and casting doubts on others^29,61.

In particular, we find that the magnitudes of isotropic Heisenberg and Kitaev terms agree well with the couplings reported in ref. ⁶². At the same time, our results indicate a weaker quadratic exchange in all three chromium halides. A dominant isotropic and a weak anisotropic magnetic coupling is a common trend observed in various studies on 2D materials based on 3d transition metals^62–64. In addition to the nearest-neighbor coupling, sizable second and third neighbor couplings have also been reported in chromium trihalides^20,60. Also relevant are the interlayer exchange interactions in bulk halides, which are known to be ferromagnetic for CrI₃ and CrBr₃, but antiferromagnetic in the case of CrCl₃. It has also been shown that the interlayer exchange is different for thin few layers as compared to bulk systems^7,10,65. However, obtaining longer-range couplings from quantum chemistry calculations is not feasible at the moment due to very rapid growth of the computational cost with the cluster size.

Discussion

In conclusion, by using multiconfigurational and multireference wavefunctions, we have achieved a description of reference accuracy for the electronic excitations and the magnetic Hamiltonian parameters in chromium trihalides, as demonstrated by the excellent agreement with a variety of spectroscopy experiments. Our work serves as a reference for future computational studies of CrX₃ and establishes many-body wavefunction-based calculations on embedded clusters as an effective approach to elucidating the nature of electronic and spin interactions in two-dimensional semiconducting magnets. Our calculations show that the magnetization direction which is determined by the interplay of out-of-plane magnetocrystalline anisotropy and magnetic dipole–dipole interaction favoring in-plane magnetization is clearly out-of-plane for CrI₃ and in-plane for CrCl₃ whereas for CrBr₃ these two contributions almost exactly compensate one another, with a small out-of-plane resulting anisotropy. Note however that since out-of-plane direction in CrCl₃ is provided by long-range dipole–dipole interactions Mermin–Wagner–Hohenberg theorem is not applicable in this case and one case expect true long-range order at low temperatures^51,54.

Methods

We perform many-body wavefunction calculations on electrostatically embedded finite-size models constructed from the experimental bulk crystal structures of CrX₃ (X = Cl, Br, I) materials^23,66,67. As shown in Fig. 1a, the crystal structure of CrX₃ magnets consists of layers in which Cr atoms form a honeycomb network and are sixfold coordinated by halogen atoms. The finite-size models adopted in our calculations comprise a central unit hosting either one (Fig. 1b and Supplementary Fig. 1a) or two (Fig. 1c) octahedra treated with multireference or multiconfigurational wavefunctions, surrounded by the nearest-neighbor octahedra. These latter octahedra account for the charge distribution in the vicinity of the central unit and are treated at the Hartree–Fock level. The crystalline environment is restored by embedding each model in an array of point charges fitted to recreate the long-range Madelung electrostatic potential⁶⁸. Electron correlation effects in the central units are described at the complete-active-space self-consistent-field (CASSCF) and multireference configuration interaction (MRCI) levels of theory⁶⁹, including spin–orbit interactions. Technical details of our ab initio calculations are discussed at length in Supplementary Notes 1–3. CASSCF approach is well suited to study systems involving degenerate or nearly degenerate configurations, where static correlation is important. In addition to static correlation, MRCI calculations also take into account dynamic correlation by considering single and double excitations from the 3d (t_2g) valence shells of Cr³⁺ ions and the p valence shells of the bridging halogen ligands. Therefore, the MRCI level of theory is more accurate and should be used when comparing with the experimental results. We present both CASSCF and MRCI results only to assess the relative importance of the two types of correlation effects. We remark that embedded finite-size model approaches have proven effective to describe electronic excitations and magnetic interactions in a rich variety of strongly correlated insulators^70,71, including d-electron lattices^56,72–74, owing to their localized nature.

Author contributions

R.Y., L.X., and M.P. performed the calculations and analyzed the data. J.v.d.B. and M.I.K. contributed in the form of discussions. O.V.Y. directed the project. All authors contributed to the manuscript writing.

Data availability

The data that support the findings of this study can be obtained from authors (R.Y. and O.V.Y.) upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41699-024-00494-5.