Halogen-driven magnetic properties of two-dimensional binary and Janus Cr₂XYSe₂ (X, Y = F, Cl, Br, I) monolayers

Abstract

In this study, we investigate the structural, electronic, and magnetic properties of the orthorhombic phase of Cr₂XYSe₂ (X, Y = F, Cl, Br, I) two-dimensional monolayers using first-principles calculations based on density functional theory. The calculated exchange interaction parameters and magnetic anisotropies exhibit a strong dependence on both the chemical composition of the monolayers and the Hubbard parameter U. This finding indicates that the interplay between spin–orbit coupling and electron localization is crucial for stabilizing long-range magnetic order in these systems. Furthermore, the renormalized magnon spectrum derived from our calculations yields Curie temperatures of approximately 200 K for the iodine-based monolayers, the highest among the compositions studied, indicating the maximum thermal stability. The results of this investigation provide a comprehensive understanding of how chemical substitution, electron correlations, and symmetry reduction interact to control magnetism in two-dimensional materials. This study advances the understanding of magnetism in Cr₂XYSe₂ monolayers and establishes a distinct materials design framework for two-dimensional van der Waals magnetic systems.

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I. Introduction

The discovery of intrinsic long-range magnetic order in intrinsic 2D ferromagnets such as CrI₃ and Cr₂Ge₂Te₆ has established two-dimensional (2D) magnets as promising candidates for investigating spin-dependent phenomena in reduced dimensionality.^1–8 In these systems, magnetic anisotropy plays a critical role in stabilizing finite-temperature magnetic order by lifting the spin-rotational symmetry of the isotropic Heisenberg model, which results in a finite magnon gap that suppresses long-wavelength spin fluctuations, thereby enabling long-range magnetic order at finite temperature.^1–3,9–12

Recently, 2D van der Waals Cr–halogenide–chalcogenide, CrXY (X = F, Cl, Br, I; Y = S, Se) monolayers, typically characterized by hexagonal and triangular lattices, have been extensively investigated due to their distinctive magnetic and electronic properties in both theoretical and experimental studies.^13–24 Beyond high-symmetry lattices, orthorhombic structures lack three- or sixfold rotational symmetry and possess two inequivalent in-plane crystallographic directions, leading to distinct magnetic behavior and more complex spin and electronic phenomena.^{15–17,25–28} In these systems, substituting the top-layer chalcogen with a halogen of different electronegativity further lowers the symmetry and modifies the exchange interactions, thereby tuning the electronic structure and magnetic properties.²⁹

Additionally, considerable studies have been conducted to obtain CrXY monolayers with higher anisotropy energy and transition temperature.^17–24 The Curie temperature of the CrXY monolayer can be enhanced using various strategies, such as applying strain to 2D CrSBr,³⁰ replacing Cr atoms with other transition atoms in the CrSI monolayer,^19,20 or chemical doping in CrSBr.²¹ Furthermore, the low-symmetry Cr–halogenide–chalcogenide monolayer, belonging to the orthorhombic Pmmn space group, has been identified as a 2D long-range magnetic material through both experimental observations and theoretical calculations.^{8,15,17,31–33} The rectangular-phase CrSBr monolayer was synthesized using a modified chemical vapor transport technique, revealing A-type antiferromagnetic order with a Néel temperature of approximately 132 K.⁸ Subsequent research has demonstrated that 2D van der Waals CrSBr, fabricated through mechanically exfoliated flakes, exhibits stability in the orthorhombic structure, accompanied by notable in-plane anisotropy.³¹ From the results of this study, bulk and few-layer orthorhombic-CrSBr exhibit A-type antiferromagnetic order below approximately 132 K. In contrast, its monolayer counterpart exhibits intraplanar ferromagnetic order, with a transition temperature of approximately 145 K. A recent experimental investigation has reported the emergence of in-plane ferromagnetic order in monolayer CrSBr, with a Curie temperature of T_C ≈ 146 K.¹⁵ As reported in a theoretical study, the CrSI monolayer, which stabilized in the orthorhombic structure, exhibits in-plane ferromagnetism with a magnetic anisotropy energy of approximately 0.52 meV.¹⁷

In transition-metal (TM) compounds, the presence of partially filled d-orbitals makes it essential to include on-site Coulomb interactions, represented by the Hubbard parameter. The overlap between atomic orbitals partially screens the local d-electron repulsion, which in turn strongly influences the electronic and magnetic properties of TM monolayers.^32,33 The magnetic behavior of CrSBr monolayers can be strongly influenced by correlation effects, which are characterized by the Hubbard parameter U.^16,34 A theoretical study has shown that, in the orthorhombic CrSBr monolayer, both the Heisenberg exchange parameters and the magnetic anisotropy depend on the Hubbard parameter.¹⁶ The dependence of the magnetic parameters on U results in transition temperatures ranging from approximately ∼100 to 140 K for U values between 0 and 4 eV. A recent theoretical study of the orthorhombic CrSBr monolayer reports an intralayer ferromagnetic ground state with a transition temperature of approximately 145 K.³⁵

In this work, we investigate ten binary and Janus Cr₂XYSe₂ (X, Y = F, Cl, Br, I) monolayers via halogen substitution on the Cr–Se lattice, focusing on the orthorhombic phase, in which reduced symmetry introduces two inequivalent in-plane directions. Using DFT+U to account for electronic correlation in the Cr 3d orbitals and an anisotropic Heisenberg model combined with linear spin-wave theory, we establish how structural asymmetry and halogen chemistry modulate exchange interactions, magnetic anisotropy, and the easy-axis orientation. In particular, the Cr₂XYSe₂ family consists of orthorhombic 2D monolayers with tunable chalcogen and halogen atoms. However, how these substitutions, together with on-site electronic correlation, affect their magnetic properties remains to be comprehensively investigated. In this study, we clarify these effects and describe magnetism in this material family, offering a practical theoretical basis for the development of Cr-based magnetic and functional materials.

II. Model and method

A. First-principles calculations

Investigations of the electronic and magnetic properties of 2D Cr₂XYSe₂ were carried out using first-principles calculations based on density functional theory (DFT) with the projector augmented-wave (PAW) method,^36,37 as implemented in the Vienna ab initio simulation package.^38,39 The Perdew–Burke–Ernzerhof (PBE) formulation within the generalized gradient approximation (GGA) was employed for the exchange–correlation functional.⁴⁰ The DFT+U method was used to account for the strongly correlated 3d electrons of Cr atoms.^41,42 We considered Hubbard U values in the range 0–4 eV to evaluate the influence of on-site Coulomb interactions on the physical properties of Cr₂XYSe₂ monolayers. Within the DFT+U framework, the Hubbard U parameter represents the effective screened on-site Coulomb interaction of localized Cr 3d electrons rather than a directly tunable experimental variable, and its systematic variation is employed as a controlled theoretical approach to examine the influence of correlation strength—partially capturing changes in electronic screening induced by external environments such as dielectric substrates (e.g., hexagonal BN) or encapsulation—on the magnetic properties of Cr₂XYSe₂ monolayers. Additionally, the Heyd–Scuseria–Ernzerhof (HSE06) screened hybrid functional is employed to assess the appropriate value of the Hubbard parameter.^43,44 The plane-wave cutoff energy was set to 700 eV. For Brillouin zone (BZ) integration, a 16 × 16 × 1 Monkhorst–Pack k-point mesh⁴⁵ was utilized. A vacuum thickness of 30 Å was introduced to avoid spurious interactions between periodic layers along the z direction. The atomic positions and lattice parameters were relaxed using the conjugate-gradient method until the residual forces on each atom were less than 0.001 eV Å⁻¹, with an energy convergence criterion of 10⁻¹⁰ eV between successive electronic steps. To investigate the dynamical stability of the optimized structures, phonon dispersion calculations were performed for all ten chemical compositions of Cr₂XYSe₂. For this purpose, density functional perturbation theory (DFPT) calculations were carried out using the Phonopy code,⁴⁶ employing a 5 × 5 × 1 supercell. The thermodynamic stability of Cr₂XYSe₂ monolayers was investigated by using ab initio molecular dynamics (AIMD) simulations. The system has been regarded as a canonical ensemble (NVT), with a 4 × 4 × 1 supercell and the Andersen thermostat.⁴⁷ The temperature was set to be in the range of 110–180 K, depending on the calculated transition temperatures of the different chemical compositions, with a time step of 2 fs.

B. Anisotropic Heisenberg model and spin-wave excitations

Magnetic anisotropy plays a crucial role in stabilizing long-range magnetic order in 2D materials. In Sections II.B.1 and II.B.2, the anisotropic Heisenberg Hamiltonian was introduced for out-of-plane and in-plane magnetic configurations, respectively. The Curie temperature of two-dimensional ferromagnets is sensitive to magnetic anisotropy, which originates from spin–orbit coupling. Different theoretical approaches used to solve the Heisenberg Hamiltonian treat anisotropies at different levels of approximation, including Monte Carlo simulations, Green's function methods, and spin-wave theories. Consequently, the predicted Curie temperatures may vary depending on the computational framework employed.^2,48–50 As reported in ref. 51 and 52, the Monte Carlo and Green's function methods overestimate the Curie temperature compared to the spin-wave method. They also found that for CrCl₃, which has low magnetic anisotropy, the renormalized spin-wave and Green's function methods provide a more accurate estimation of the Curie temperature. To consider the thermal stability of the magnetic ground state, linear spin-wave theory was employed. For this purpose, the Holstein–Primakoff (HP) transformation was applied to map the spin Hamiltonian onto a bosonic representation.^2,50 Assuming a low magnon population, the first-order HP approximation is employed. In the following, we analytically derive the magnetic parameters and the associated magnon dispersions, enabling us to predict how the magnetic behavior varies with chemical composition and the strength of the on-site Coulomb interaction. To investigate the temperature dependence of magnetization, the magnon energy is renormalized using mean-field theory, as discussed in Section II.B.3.

1. Out-of-plane anisotropic Hamiltonian. The Hamiltonian for the anisotropic Heisenberg model for out-of-plane magnets can be written aswith S_i and S_j denoting the spin operators associated with the atomic lattice sites i and j, respectively, as shown in Fig. 1. In the orthorhombic phase configuration with lattice constants a and b (Table 1), three Cr–Cr interaction bonds are included, as each Cr atom is surrounded by two others in the first, four in the second, and two in the third nearest neighbors. Accordingly, in eqn (1), the Heisenberg exchange parameter J_τ is defined as, J₁ corresponds to the first-nearest-neighbor bonds, 〈ij〉₁, of length a, while J₂ describes the second-nearest-neighbor bonds, 〈ij〉₂, of length , connecting sites on the two different sublattices. The third-nearest-neighbor coupling, denoted by J₃, 〈ij〉₃, connects the sites separated by the lattice constant b. The second term of the Hamiltonian describes the single-ion magnetic anisotropy. For the van der Waals magnetic monolayer with uniaxial anisotropy, the sign of A determines the preferred orientation of the magnetic moments, where the positive and negative signs correspond to out-of-plane and in-plane easy axes of magnetization, respectively. Given that Cr₂XYSe₂ monolayers exhibit both out-of-plane and in-plane magnetic anisotropy depending on the chemical composition and the Hubbard parameter, the in-plane form of the Hamiltonian will be discussed in Section II.B.2. To derive the magnetic parameters of the Hamiltonian in eqn (1), four distinct magnetic states, one FM and three AFM, as illustrated in Fig. 1, are considered. By rewriting the Hamiltonian for each configuration and substituting the corresponding spin-orientation energies, the exchange and anisotropy parameters can be determined, as detailed in the SI.

Fig. 1 (a) Atomic structure of the Cr₂XYSe₂ monolayers in the orthorhombic phase, shown in the top and side views, with a and b indicating the lattice vectors. Blue and green colors represent Cr and Se atoms, and purple and brown ones denote the halogen atoms, X and Y. (b) Spin configurations employed to determine the magnetic parameters, including ferromagnetic alignments FM_x, FM_y, and FM_z, and antiferromagnetic configurations AFM1_x, AFM2_x, and AFM3_x. (Configurations with spins aligned along z for AFM states are provided in Fig. S2 of the SI.) The three exchange interactions J₁, J₂, and J₃ correspond to the first, second, and third nearest Cr–Cr pairs.

Table 1 Calculated in-plane lattice constants a and b (Å) as a function of the Hubbard parameter U (eV) for binary and Janus Cr₂XYSe₂ monolayers

	Lattice constant (Å)	U = 0	U = 1	U = 2	U = 3	U = 4
Cr2F2Se2	a	3.30	3.31	3.31	3.31	3.26
	b	5.09	5.11	5.14	5.19	5.28
Cr2Cl2Se2	a	3.53	3.55	3.56	3.57	3.58
	b	5.05	5.08	5.11	5.15	5.18
Cr2Br2Se2	a	3.64	3.65	3.66	3.68	3.69
	b	5.04	5.06	5.10	5.13	5.16
Cr2I2Se2	a	3.80	3.81	3.83	3.84	3.86
	b	5.01	5.04	5.08	5.11	5.15
Cr2BrFSe2	a	3.46	3.47	3.48	3.48	3.80
	b	5.05	5.08	5.11	5.15	5.19
Cr2BrClSe2	a	3.58	3.60	3.61	3.62	3.63
	b	5.04	5.07	5.10	5.14	5.17
Cr2ClISe2	a	3.66	3.68	3.69	3.70	3.70
	b	5.03	5.06	5.10	5.13	5.17
Cr2BrISe2	a	3.72	3.73	3.74	3.76	3.77
	b	5.02	5.05	5.08	5.12	5.16

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The first-order HP transformation⁵⁰ for the orthorhombic lattice with the out-of-plane easy axis of magnetization can be written as

for the sublattice in the a direction, andfor the sublattice in the b direction (a and b directions shown in Fig. 1). Here, S is the total spin moment (in units of ħ), and a_i, a^†_i, and b_i, b^†_i are the creation and annihilation operators in different sublattices that obey the bosonic commutation relations. Ladder spin operators are defined as S^±_i = S^x_i ± iS^y_i. To obtain the Hamiltonian in momentum space, the Fourier transformation is applied, . As a result, the Hamiltonian in reciprocal spacewhereIn eqn (5), the matrix indexes areIn the above equation,cos(k·ΔR)_τ is the structure factor for the first, second, and third nearest neighbors of Cr atoms, and ΔR is the vector that connects the nearest neighbors.In the equation, all structure factors are multiplied by two, as there are two Cr atoms in the unit cell. The eigenvalues of the Hamiltonian in eqn (5) can be derived as

ω^±_k = A_k ± B_k (9)

where ω⁺_k and ω⁻_k belong to the acoustic and optical magnon branches, respectively. More details of the derivation are available in the SI. The magnon energy gap in k → 0 for the acoustic branch is derived as AS/2. As a result, the magnon gap around the Γ point in the BZ strongly depends on the single-ion anisotropy. Although this anisotropy is small in magnitude, it plays a crucial role in opening a finite magnon gap and ensuring nonzero magnetization at finite temperature.⁵³

2. In-plane anisotropic Hamiltonian. In 2D magnets with an in-plane easy axis, the inter-site magnetic anisotropy within the αβ plane becomes crucial. Accordingly, the anisotropic Heisenberg Hamiltonian can be written asHere, the first two terms of the Hamiltonian are the same as those in eqn (1). The indices α, β denote the directions within the easy plane of magnetization and μ denote the axis normal to this plane in the Cartesian basis (x, y, z). The last terms in eqn (10) belong to the inter-site magnetic anisotropy within the easy plane of magnetization, αβ-plane. Given that the magnitude of δ is relatively small, it is sufficient to include only the sites closest to the first neighbor (〈ij〉₁). Taking into account the four different spin orientations (Fig. 1), the magnetic parameters of 2D Cr₂XYSe₂ for all possible in-plane easy axes α and β are derived, as reported in the SI.

For in-plane magnets, the HP transformations can be written for the sublattice in the a direction as

and for the sublattice in the b direction asThe conventional spin ladder operators are given by S^±_i = −S^μ_i ± iS^α,β_i. By applying the HP transformation to the Hamiltonian in eqn (1) and subsequently transforming to momentum space using the Fourier transformation,whereandWe can explicitly writeIn the above equation, the structure factor, cos(k·ΔR)_τ, is defined in eqn (8), and the corresponding expressions for the exchange parameters and anisotropy energies are provided in the SI. By solving the Hamiltonian of eqn (14), the magnon dispersion can be written asThe spin-wave gap around zero momentum can be derived for the acoustic band as . Here, the presence of the δ term is particularly critical for establishing a long-range magnetic order in 2D monolayers.^11,54–56

3. Magnetization renormalization due to magnon excitations. According to eqn (5) and (17), the magnon dispersion does not yet include the effect of temperature. To account for thermal effects, the magnon energy should be renormalized. For this purpose, using mean-field theory, the dependence of the number of excited magnons on temperature is investigated.⁵⁷ Since each magnon carries one unit of angular momentum, thermal effects lead to its excitation. This reduces the total spin of the system from S to S − 〈n〉, where n is the number operator that defines the number of magnons at each temperature. The renormalized magnon energy as a function of temperature is given byHere, M(T) and M_sat are the temperature-dependent magnetization function and the saturation magnetization, respectively. Employing the Bose–Einstein distribution, M(T) can be formulated asβ = 1/k_BT and k_B is the Boltzmann constant. Eqn (19) is solved iteratively by initially setting M(T) = M_sat and updating M(T) until convergence is reached. Finally, the Curie temperature is defined as the temperature at which the magnetization reaches M(T) = M_sat/2.

III. Results and discussion

A. Structural and electronic properties

To study the dynamical stability of the ten possible binary and Janus Cr₂XYSe₂ (X, Y = F, Cl, Br, I) monolayers in the orthorhombic structure with the space group Pmmn, DFPT phonon calculations were carried out along the high-symmetry path X → S → Y → Γ → S in the Brillouin zone. As shown in the SI, eight out of the ten considered compositions show no imaginary phonon modes, confirming their dynamical stability. On the other hand, the Cr₂FClSe₂ and Cr₂FISe₂ monolayers display imaginary phonon branches and are therefore dynamically unstable in the orthorhombic phase. In the subsequent sections, we conduct a thorough investigation into the electronic and magnetic properties of the eight stable compounds, which encompass the four binary systems denoted as Cr₂F₂Se₂, Cr₂Cl₂Se₂, Cr₂Br₂Se₂, and Cr₂I₂Se₂, and the four Janus monolayers Cr₂BrFSe₂, Cr₂BrClSe₂, Cr₂ClISe₂, and Cr₂BrISe₂.

The structural parameters are summarized in Table 1, where a and b denote the lattice constants along the crystallographic a and b axes, respectively, as shown in Fig. 1. For all chemical compositions, both a and b increase with U, except in Cr₂F₂Se₂, where a remains almost unchanged and decreases slightly at U = 4 eV. The Cr–Cr distances for the first-, second-, and third-nearest neighbors are reported in the SI. One can see that the lattice constants and Cr–Cr distances follow the same trend, as is well established from periodic trends, the electronegativity decreases from F to I (χ_F > χ_Cl > χ_Br > χ_I), whereas the atomic radius increases (r_F < r_Cl < r_Br < r_I).^58,59 Consequently, in the binary systems, the relevant structural parameters increase systematically from Cr₂F₂Se₂ to Cr₂I₂Se₂, while in the Janus systems, they increase from Cr₂BrFSe₂ to Cr₂BrISe₂. Additionally, the effect of halogen electronegativity is reflected in the charge redistribution, as confirmed by the Bader charge analysis. As the electronegativity decreases from F to I, the charge transferred from Cr to the halogen atoms decreases, indicating a gradual reduction in bond ionicity (see Tables III and IV in the SI).

Spin-polarized band structures at U = 0, 2, and 4 eV, as shown in Fig. S2 of the SI, reveal systematic trends across the Cr₂XYSe₂ family. The binary monolayers are semimetallic at U = 0 and 2 eV and become metallic at U = 4 eV as electronic correlations increase the band overlap. The Janus monolayers exhibit weak half-metallicity at U < 3 eV, arising from the broken mirror symmetry that opens a minority-spin gap. In contrast, this gap vanishes at U = 4 eV, and all Janus structures become metallic. Among all compositions, the F-containing compositions remain metallic for all U, consistent with stronger Cr–F hybridization. As presented in Fig. 2, SOC produces only minor modifications to the band dispersions and does not change the electronic classifications obtained from the collinear calculations. Moreover, the PBE and HSE06 band structures for the binary and Janus compositions, shown in Fig. S2 and S3 of the SI, are compared to estimate the choice of U = 2 eV.

Fig. 2 Electronic band structure in the presence of SOC for U = 0, 2, and 4 eV in 2D Cr₂XYSe₂.

Due to the lack of mirror symmetry, Janus exhibits characteristics distinct from those of a binary. One of these unique features is the intrinsic out-of-plane dipole moment p_z, which induces an internal electric field arising from the difference in electronegativity between X and Y in the up and down layers.^60,61 Investigation of binary Cr₂X₂Se₂ monolayers reveals nearly zero out-of-plane dipole moment, as required by their mirror symmetry. The structural asymmetry of the Janus Cr₂XYSe₂ monolayers results in a finite out-of-plane dipole moment, p_z, reported in Table 2. These systems show dipole moments along the z-direction. For most of the monolayers considered, the out-of-plane dipole p_z increases slightly with U. This can be attributed to reduced screening and to enhanced charge asymmetry between the two halogen atoms, X and Y, due to on-site Coulomb interactions. The maximum magnitude of p_z is predicted for Cr₂ClISe₂, which can be related to its larger electronegativity difference compared with Cr₂BrISe₂ and Cr₂BrClSe₂. This prediction is consistent with reports indicating that p_z increases with increasing electronegativity, as reported in ref. 62. Although Cr₂BrFSe₂ contains the element with the highest electronegativity, its p_z remains relatively small, suggesting that both structural and electronic effects, such as hybridization near the Fermi level, can partially screen out-of-plane polarization, as shown in Fig. 2.

Table 2 Calculated out-of-plane dipole moment p_z (in Debye) of Janus Cr₂XYSe₂ monolayers as a function of the Hubbard parameter U (in eV)

	pz (Debye)
	U = 0	U = 1	U = 2	U = 3	U = 4
The sign indicates the dipole's orientation along the z-axis.
Cr2BrFSe2	+0.19	+0.18	+0.18	+0.17	+0.15
Cr2BrClSe2	+0.13	+0.13	+0.14	+0.14	+0.14
Cr2ClISe2	−0.32	−0.34	−0.34	−0.36	−0.37
Cr2BrISe2	−0.22	−0.21	−0.22	−0.22	−0.23

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B. Magnetic properties and temperature-dependent magnetization

According to DFT+U calculations, all Cr₂XYSe₂ monolayers show the FM ground state at U ≤ 4 eV. The total magnetic moment per Cr atom is approximately 3μ_B for both binary and Janus systems, while it slightly increases to about 3.27μ_B for Cr₂F₂Se₂. The spin parameters J₁, J₂, J₃, A, and δ are derived from the total energy differences between various magnetic configurations. The total energy of each spin configuration was obtained from noncollinear DFT+U calculations.

Fig. 3 presents the variation of the exchange parameters as a function of the Hubbard U parameter for each Cr₂XYSe₂ composition. Notably, the magnitudes of the three different exchange interaction parameters are of the same order of magnitude. In all systems, the exchange interactions between the first- and second-nearest Cr atoms (J₁ and J₂) are positive, favoring ferromagnetic coupling. At the same time, J₃ becomes negative at higher U values in some compounds, indicating a preference toward antiferromagnetic alignment between the third-nearest Cr atoms. As shown in Fig. 3, the value of J₁ increases with U, ranging from approximately 0 to 3 meV, except for Cr₂F₂Se₂, which exhibits noticeably lower values of J₁ compared to the other compounds. This drop may be attributed to its smaller lattice parameter a, which further decreases with increasing U, as reported in Table 1. In addition, for Cr₂F₂Se₂, the magnitude of J₁ remains smaller than that of J₂ and J₃. The J₂ parameter gradually decreases from about 3 meV as U increases from 0 to 4 eV, indicating a weakening of the next-nearest ferromagnetic channel with enhanced electron localization. An exception is observed in Cr₂BrClSe₂, where for U > 3 eV the magnitude of J₁ decreases while J₂ increases, suggesting competition between the two exchange interactions. The magnitude of J₃ decreases as U increases in almost all chemical compositions. Besides, the investigation in J₃ shows that the sign is negative in some U values, in favor of the antiparallel spin configuration between the Cr atoms, which interact in the third nearest neighbor separated by distance b. This behavior occurs when the electronic structure of the system changes to a metallic phase (as depicted in Fig. 2). When the system becomes metallic, the exchange mechanism is no longer governed only by localized superexchange. Here, partially itinerant charge carriers at the Fermi level mediate an indirect exchange coupling between localized magnetic atoms via Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions between distant Cr moments.³² The spin polarization of these conduction electrons oscillates with the distance between the magnetic moments. Therefore, depending on the bond distance, it may be ferromagnetic or antiferromagnetic. Consistent with this theory, J₃ decreases with increasing U and becomes negative in the metallic phase, indicating that the RKKY interaction favors an antiparallel alignment between third-nearest-neighbour Cr ions separated by distance b. With decreasing halogen electronegativity from Cr₂F₂Se₂ to Cr₂I₂Se₂ in the binary systems and from Cr₂BrFSe₂ to Cr₂BrISe₂ in the Janus systems, J₂ and J₃ generally decrease as the lattice constant increases. This behavior is consistent with the reduction of the hopping integral (t) with increasing Cr–Cr distance, leading to a weaker exchange interaction: J ∝ −t²/U, where U is the Coulomb energy.³² However, this trend is not observed for J₁, which increases due to enhanced orbital hybridization arising from the reduced structural symmetry. Consequently, the competition among J₁, J₂, and J₃ determines the strength and stability of the ferromagnetic ordering.

Fig. 3 Variation of the magnetic exchange parameters J₁, J₂, and J₃ (in meV) as a function of the on-site Hubbard interaction U (in eV) for different Cr-based monolayer systems. Panels (a–d) correspond to the binary compounds Cr₂X₂Se₂ with X = F, Cl, Br, and I, respectively, while panels (e–h) show the Janus structures Cr₂BrXSe₂ with X = F, Cl, I, and Br, respectively.

The values of single-ion anisotropy (A) and inter-site magnetic anisotropy (δ) are studied as functions of the Hubbard parameter U for all monolayers, as shown in Fig. 4. The left vertical axis corresponds to A, where positive and negative values indicate the easy axis of magnetization out-of-plane and in-plane, respectively. Since the inter-site anisotropy δ is required only for the in-plane case (A < 0), its magnitudes are highlighted in green for the corresponding U values and are represented on the right axis.

Fig. 4 Variation of the magnetic anisotropy energies, A (left axis, red) and the δ (right axis, teal) as a function of the on-site Hubbard parameter U (in eV) for Cr-based monolayer systems. Panels (a–d) correspond to the binary compounds Cr₂X₂Se₂ with X = F, Cl, Br, and I, respectively, while panels (e–h) represent the Janus structures Cr₂BrXSe₂ with X = F, Cl, I, and Br, respectively.

For different values of U, the easy axis of magnetization in several monolayers reverses from in-plane to out-of-plane, indicating that the magnetic anisotropy is sensitive to the electron correlation strength. In systems containing heavy halogens such as iodine (Fig. 4(d)), strong spin–orbit coupling favors magnetocrystalline anisotropy, stabilizing an out-of-plane easy axis. Correspondingly, the largest single-ion anisotropy values occur in Cr₂I₂Se₂ (−100 to 150 µeV) and Cr₂ClISe₂ (−100 to 30 µeV), reflecting the significant spin–orbit interaction introduced by iodine. In Janus structures, the interplay between the out-of-plane asymmetry and spin–orbit coupling contributes to the magnitude of single-ion anisotropy.⁶² As seen from Table 2 and Fig. 4, Cr₂ClISe₂ exhibits the largest magnetic anisotropy energy, consistent with its enhanced out-of-plane dipole moment.

This behavior arises from changes in the localization of Cr 3d orbitals and the competition between crystal-field splitting and spin–orbit coupling, which together determine the preferred orientation of the magnetic moments,^2,10 as discussed in the following. To clarify the orbital origin of the easy-axis reorientation, we consider the perturbative contribution of the SOC Hamiltonian,²⁴ as depicted for Cr₂F₂Se₂, Cr₂I₂Se₂, Cr₂BrFSe₂, Cr₂ClISe₂, and Cr₂BrISe₂, as shown in Fig. S10–S14 of the SI. The U-induced reorientation of the magnetic easy axis originates from a redistribution of the orbital-resolved SOC matrix elements near the Fermi level. Changing U enhances the localization of the Cr 3d states and modifies their relative energy separation and hybridization with ligand p orbitals. As a representative case, in Cr₂F₂Se₂, the dominant contribution at U = 0 eV arises from the Cr d_x²−y²–d_xy/yz coupling together with Se p_x–p_y states, favoring in-plane anisotropy. When U increases to 1 eV, the spectral weight shifts toward the Cr d_xz–d_z² and d_xy–d_yz channels, accompanied by a change in the Se contribution from p_x–p_y to p_x–p_z, which enhances the matrix elements and drives the transition to out-of-plane magnetization. More details of each composition are available in the SI. To further understand the effect of crystal-field splitting, the partial density of states (PDOS) of the d orbital of Cr atoms is presented in Fig. S5 and S6 of the SI. Because of the low-symmetry orthorhombic structure and the significant hybridization between Cr d orbitals and ligand p states, the Cr 3d bands become fully nondegenerate. As a result, the conventional crystal-field picture with well-defined t_2g and e_g splitting is no longer practical.^24,63 Additionally, the largest orbital moments calculated along L_‖ and L_⊥ are reported in Tables VI and VII of the SI. It is well established that the magnetocrystalline anisotropy energy is determined by spin–orbit coupling interactions rather than by the magnitude of the orbital moments. Therefore, the magnetic easy axis can switch to the out-of-plane direction even in the absence of orbital moment reversal.

The magnitude of the inter-site anisotropy, δ, is considerably lower compared with the single-ion anisotropy A, but it plays an essential role in opening the magnon gap at the Γ point of the BZ and in stabilizing long-range magnetization at finite temperature for systems with in-plane magnetization. As shown in Fig. 4(a), the relatively large δ values in Cr₂F₂Se₂ (approximately 12 µeV) associated with lighter halogens indicate more substantial anisotropy within the plane, which can be attributed to the enhanced in-plane crystal-field anisotropy. The variations in magnetic parameters with U show the significant effect of electronic correlation on exchange interactions and magnetocrystalline anisotropy.^16,64

By using the HP transformation, the magnon dispersion is obtained using eqn (9) and (17) for the monolayers with out-of-plane and in-plane anisotropy, respectively. The corresponding magnon spectra for different materials over the U range are presented in Fig. 5, where acoustic and optical branches emerge due to the two sublattices of the orthorhombic structure. In almost all cases, the magnon energy exhibits a monotonic decrease with increasing Hubbard parameter U, approaching ∼100 meV, in good agreement with previous studies.¹⁶ Although the magnon dispersion differs along the Y−Γ and Γ−X directions, it exhibits a quadratic behavior near the Γ point, which is characteristic of ferromagnets. The magnon band of Cr₂F₂Se₂ at U = 4 eV differs from the systematic trend observed across all materials and U values. This variation may arise from the distinct behavior of its exchange parameters, illustrated in Fig. 3(a), and the comparatively large magnitude of J₃. The magnon gap at the Γ point is summarized in Table 3. The gap varies significantly with the Hubbard parameter, reflecting the strong U-dependence of the magnetic anisotropy parameters A and δ. Because the magnon dispersion changes with U in the Cr₂XYSe₂ monolayer, the results indicate that electron correlation can play a nontrivial role in their thermodynamic behavior.

Fig. 5 Magnon dispersion relations along the high-symmetry path Y–Γ–X–S for Cr-based monolayer systems at different on-site Hubbard interaction strengths U (0–4 eV). Panels (a–d) correspond to the binary compounds Cr₂X₂Se₂ (X = F, Cl, Br, I), while panels (e–h) show the Janus structures Cr₂BrXSe₂ (X = F, Cl, I, Br). Solid lines represent the acoustic magnon branches, whereas dashed lines denote optical branches.

Table 3 Calculated magnon gap at the Γ point (in meV) as a function of the Hubbard parameter U (eV) for binary and Janus Cr₂XYSe₂ monolayers

Material	U = 0	U = 1	U = 2	U = 3	U = 4
Cr2F2Se2	0.06	0.01	0.03	0.07	0.37
Cr2Cl2Se2	0.11	0.14	0.14	0.16	0.12
Cr2Br2Se2	0.09	0.10	0.08	0.11	0.04
Cr2I2Se2	0.23	0.11	0.16	0.22	0.26
Cr2BrFSe2	0.09	0.08	0.08	0.05	0.05
Cr2BrClSe2	0.10	0.11	0.14	0.12	0.10
Cr2ClISe2	0.12	0.14	0.16	0.01	0.03
Cr2BrISe2	0.13	0.01	0.03	0.05	0.09

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The thermal magnetization of the monolayers as a function of the Hubbard parameter U is shown in Fig. 6. As shown in the figure, the magnitude of T_C decreases with increasing U in almost all cases, consistent with the behavior reported in ref. 16. The highest T_C (approximately 200 K) is obtained for Cr₂I₂Se₂, consistent with Fig. 3(a), where this compound exhibits the strongest exchange interactions. In contrast, the minimum value of T_C belongs to Cr₂F₂Se₂, which shows the weakest exchange interactions and the smallest spin–orbit coupling due to the light fluorine atoms. To demonstrate the reliability of the above predictions, AIMD simulations show no structural reconstruction or bond instability in either binary or Janus Cr₂XYSe₂ monolayers, as shown in Fig. S8 and S9 of the SI. The total energy fluctuations remain below ∼40 meV per atom, confirming thermal stability at the transition temperature at the Hubbard parameter U = 2 eV.

Fig. 6 Temperature-dependent normalized magnetization and corresponding Curie temperatures for various chromium-based monolayers. (a–h) Normalized magnetization M(T)/M_sat as a function of temperature T (in K) for (a) Cr₂F₂Se₂, (b) Cr₂Cl₂Se₂, (c) Cr₂Br₂Se₂, (d) Cr₂I₂Se₂, (e) Cr₂BrFSe₂, (f) Cr₂BrClSe₂, (g) Cr₂ClISe₂, and (h) Cr₂BrISe₂. The different colored curves represent calculations performed with varying values of the Hubbard U parameter from 0 to 4 eV. The dashed horizontal line indicates M(T)/M_sat = 0.5. The insets show the extracted Curie temperature T_c (in K) as a function of the Hubbard U parameter for each corresponding system.

IV. Conclusion

We have investigated the structural, electronic, and magnetic properties of orthorhombic Cr₂XYSe₂ (X, Y = F, Cl, Br, I) monolayers using density functional theory and linear spin-wave analysis, focusing on halogen-engineered families. DFPT phonon calculations show that eight of the ten possible binary and Janus structures are dynamically stable in the Pmmn phase. For all stable compounds, DFT+U predicts ferromagnetic ground states with magnetic moments close to 3μ_B per Cr atom and electronic properties that vary with both chemical composition and the on-site Coulomb interaction. The calculated exchange interactions and magnetic anisotropies depend on the halogen atoms and U, leading to composition-dependent trends in their magnitudes and orientations. Linear spin-wave theory yields finite magnon gaps at the Γ point for all systems, demonstrating that magnetic anisotropy is required to stabilize long-range magnetic order in 2D Cr₂XYSe₂. The renormalized magnon spectrum causes Curie temperatures of up to approximately 200 K in iodine-based monolayers, representing the maximum thermal stability across the studied compositions.

This work establishes a comprehensive understanding of magnetism in Cr₂XYSe₂ monolayers, which are a tunable family of two-dimensional orthorhombic magnets. The symmetry reduction, chemical substitution, and electron correlation present in these monolayers provide effective control over exchange interactions, magnetic anisotropy, and thermal stability. These findings emphasize the potential of this material class for applications in 2D spintronics and magnonics.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data that support the findings of this study are available within the article and its supplementary information files. Additional data are available from the corresponding author upon reasonable request.

Supplementary information (SI) provides additional details on the following: (1) Phonon dispersion spectra of the stable and unstable  monolayers; (2) Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional band structure calculations to validate the Hubbard  parameter; (3) Different out-of-plane antiferromagnetic spin configurations; (4) Analytical formalism for the anisotropic Heisenberg model and spin-wave excitations; (5) Calculated Cr–Cr bond distances and Bader charges summarized in tables; (6) Full spin-polarized electronic band structures and partial density of states (PDOS); (7) Ab initio molecular dynamics (AIMD) simulation results; and (8) Analysis of orbital-projected spin–orbit coupling (SOC) and magnetocrystalline anisotropy energy (MAE). See DOI: https://doi.org/10.1039/d5cp04913j.

Acknowledgements

This work was supported by the FLAG-ERA grant MNEMOSYN and by the Scientific and Technological Research Council of Türkiye (TUBITAK) under project no. 221N400. The numerical calculations reported in this paper were partially performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRUBA resources). Y. Mogulkoc acknowledges The Turkish Academy of Sciences – Outstanding Young Scientists Award Program (TUBA-GEBIP) for partial funding of this research.

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