Halogen functionalization-induced modulation of ferromagnetism and electronic phases in CrXY monolayers

Abstract

Developing two-dimensional (2D) magnetic materials with tunable electronic and spintronic properties is essential for advancing next-generation quantum devices and low-power spintronic applications. Here, we systematically explore how halogen surface functionalization dramatically modulates the structural, electronic, and magnetic behaviors of Janus CrXY monolayers (X, Y = S, Se, and Te; X ≠ Y) using first-principles calculations. The functionalized CrXYT₂ (X, Y = S, Se, and Te; X ≠ Y; T = F, Cl, Br, and I) monolayers significantly enhance their magnetic performance and markedly increase the Curie temperatures (T_C) values compared to corresponding pristine CrXY monolayers. Most of the CrXY and CrXYT₂ monolayers possess strong perpendicular magnetic anisotropy, except for CrSSe, CrSTeCl₂, and CrSSeCl₂, which exhibit in-plane easy magnetization axes. While the pristine CrXY monolayers are half-metallic ferromagnets, functionalization induces magnetic phase transitions in several systems. Specifically, the CrSTeCl₂, CrSTeBr₂, CrSSeBr₂, and CrSSeI₂ monolayers transform into bipolar magnetic semiconductors, whereas the CrSeTeF₂ monolayer becomes a ferromagnetic metal. Additionally, spin-channel inversion is observed in CrSeTeI₂ and CrSTeI₂. The remaining CrXYT₂ monolayers retain their half-metallic ferromagnetic character, along with significantly enhanced T_C. The results show that surface functionalization is an effective strategy to regulate the CrXY monolayers. These findings broaden the application potential of two-dimensional Cr-based monolayers in spintronic devices and provide valuable theoretical guidance for future experimental synthesis.

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1. Introduction

Two-dimensional (2D) magnetic materials have garnered significant attention due to their potential for next-generation spintronic applications.^1–4 Since the experimental discovery of intrinsic ferromagnetism in 2D systems such as CrI₃,⁵ Cr₂Ge₂Te₆,⁶ and Fe₃GeTe₂,⁷ the exploration of low-dimensional magnetic systems has rapidly expanded.^8–10 2D magnetic transition metal dichalcogenides (TMDs) have received widespread attention owing to their well-defined layered structure and a range of desirable properties,^11,12 including strong spin–orbit coupling,¹³ Rashba spin splitting,¹⁴ valley polarization,¹⁵ piezoelectricity,¹⁶ and tunable electronic and magnetic characteristics. Within this context, Cr-based TMD monolayers, such as CrXY (X, Y = S, Se, and Te; X ≠ Y), have attracted significant interest due to their intrinsic ferromagnetism and moderate magnetic anisotropy.^17–19 However, similar to many other 2D magnetic materials, their Curie temperatures (T_C) remain far below room temperature, posing a critical challenge for practical implementation in spintronic devices.

To address this limitation, recent studies have focused on exploring and tuning the properties of individual CrXY systems. For instance, Y. Q. Liu et al. demonstrated that strain and doping could enhance the ferromagnetism, magnetic anisotropy, and spin polarization of CrSeTe, although the tunability under an external electric field remained limited.²⁰ Y. X. Wu et al. reported that applying a small biaxial tensile strain could elevate the T_C of CrSSe above room temperature.²¹ Similarly, Q. R. Cui et al. investigated strain-tunable ferromagnetism and chiral spin textures in CrXTe (X = S and Se) monolayers, showing that tensile strain significantly enhanced ferromagnetic exchange interactions and perpendicular magnetic anisotropy, while reducing the Dzyaloshinskii–Moriya interaction (DMI).²² In another study, Q. Q. Li et al. examined the magnetic properties of bilayer CrSTe with AB- and AC-stacking configurations, highlighting their potential for applications in double spin-filter devices.²³ Additionally, Z. H. Guan et al. employed atomic substitution to systematically explore the electronic and topological magnetic properties of CrXY monolayers (X, Y = S, Se, Te, Cl, Br, and I; X ≠ Y).²⁴ Strain engineering, doping, electric field modulation, bilayer stacking, and atomic substitution have been shown to influence exchange interactions and magnetic anisotropy. Yet, these approaches often offer limited tunability or require external fields. Surface functionalization has been demonstrated to be an effective approach to modulate the properties of 2D materials, which can induce many excellent properties, such as wide bandgap and high specific capacity,²⁵ valley polarizations and enhanced magnetism,^26,27 high Curie temperatures and carrier mobility.^28,29 Thus, surface functionalization serves as another crucial and effective modulation method.

In this work, we conduct a comprehensive first-principles investigation of Janus CrXY (X, Y = S, Se, and Te; X ≠ Y) monolayers and their functionalized CrXYT₂ (T = F, Cl, Br, I). We assess their thermodynamic and dynamic stability, investigate their electronic structures and magnetic properties, and estimate the T_C using the Ising model within the Monte Carlo simulations. Through atom-resolved and orbital-resolved analyses, we examine the microscopic mechanisms governing the magnetic anisotropy energies (MAEs). Additionally, we identify key functionalized systems that exhibit desirable spintronic characteristics, such as enhanced magnetic moments, strong perpendicular magnetic anisotropy (PMA), elevated T_C, and emergent half-metallic ferromagnet or bipolar magnetic semiconductor behavior. This work provides a solid theoretical basis for the design and potential experimental realization of high-performance 2D spintronic materials based on Cr-based Janus monolayers.

2. Calculation methods

Density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP)³⁰ with the projector augmented wave (PAW) method.³¹ The generalized gradient approximation (GGA)³² with the Perdew–Burke–Ernzerhof (PBE) functional was employed for exchange–correlation interactions, and a PBE + U method (U_eff = 3.0 eV)^18,20 was applied to the Cr 3d orbitals. The cutoff energy was set to 520 eV. The Monkhorst–Pack k-points mesh utilized for the structural relaxations was set to 9 × 9 × 1 within the Brillouin zone,³³ while an enhanced grid of 18 × 18 × 1 k-points was employed for the electronic structure calculations. Structural optimization continued until atomic forces were below 10⁻² eV Å⁻¹, with a total energy convergence criterion of 10⁻⁶ eV. A vacuum spacing of 20 Å was introduced to eliminate interlayer interactions. Ab initio molecular dynamics simulations³⁴ were conducted at 300 K in the NVT ensemble using a Nosé–Hoover thermostat, with a time step of 1 fs over 10 ps, to verify thermodynamic stability. The spin–orbit coupling (SOC) effect was considered in calculations of magnetic anisotropy energy. Curie temperatures were estimated using Monte Carlo simulations³⁵ in a 50 × 50 × 1 supercell, with exchange parameters (J) determined from ferromagnetic and antiferromagnetic state energies. The phonon spectra were calculated using the finite displacement method as implemented in the Phonopy package, interfaced with VASP. A 4 × 4 × 1 supercell was adopted for these calculations.^36,37

3. Results and discussion

The structures of Janus CrXY (X, Y = S, Se, and Te; X ≠ Y) adopt a hexagonal lattice, derived from the 1T-CrX₂ (X = S, Se, and Te) monolayers.^38,39 By substituting one of the chalcogen layers in 1T-CrX₂ with a different chalcogen atom, the original mirror symmetry of D_3d is broken, resulting in a lower C_3v symmetry and the formation of a Janus structure. Owing to the unique asymmetric structure, we investigate the electronic and magnetic properties of the CrXY monolayers, and further explore the effects of symmetric functional group modifications.

The top and side views of the CrXY monolayers are shown in Fig. 1(a) and (b). The Cr atoms are sandwiched between two distinct chalcogen atom layers, where the Y atom has a higher atomic number than the X atom. Two magnetic configurations are considered based on a 2 × 2 × 1 supercell: a ferromagnetic (FM) state and a stripy antiferromagnetic (s-AFM) state, as illustrated in Fig. 1(c) and (d). To further explore the structural and magnetic tunability, we symmetrically modified the surfaces of the CrXY monolayers by introducing fluorine, chlorine, bromine, and iodine functional groups, resulting in a series of functionalized structures denoted as CrXYT₂ (X, Y = S, Se, and Te; X ≠ Y; T = F, Cl, Br, and I). A total of five surface adsorption configurations (Models I–V) are investigated, corresponding to various available adsorption sites, as shown in Fig. 2.

Fig. 1 (a) Top and (b) side views of the CrXY (X, Y = S, Se, and Te; X ≠ Y) monolayers. (c) FM state, and (d) s-AFM state. The blue, yellow, and green spheres represent Cr, X, and Y atoms, respectively. The red and blue arrows on Cr atoms represent spin up and spin down directions, respectively.

Fig. 2 Top and side views of the CrXYT₂ (X, Y = S, Se, and Te; X≠Y; T = F, Cl, Br, and I) monolayers. (a) Model I, (b) Model II, (c) Model III, (d) Model IV, and (e) Model V. The blue, yellow, green, and light purple spheres represent Cr, X, Y, and T atoms, respectively.

Ab initio molecular dynamics (AIMD) simulations are performed at 300 K to evaluate the dynamic stability of CrXY. As shown in Fig. 3, the structural model of CrXY remains intact after 10 000 fs, with no evident bond breaking or structural collapse. In addition, the energy versus time curve shows only minor fluctuations and remains stable throughout the simulation, indicating that the CrXY monolayers exhibit good thermal stability. Furthermore, we have calculated the phonon spectra of CrXY monolayers. As illustrated in Fig. 4, the results show no imaginary frequencies in the Brillouin zone, further confirming the structural stability of these materials. This provides the basis for further tuning of CrXY via functionalization. We further perform AIMD simulations and calculate the phonon dispersion of the CrSSeF₂ monolayer, as presented in Fig. S1. These results indicate that the CrSSeF₂ monolayer also possesses dynamic and thermal stability.

Fig. 3 Temperature evolution as a function of step at 300 K for the (a) CrSeTe, (b) CrSTe, and (c) CrSSe monolayers.

Fig. 4 Phonon dispersion spectra of the (a) CrSeTe, (b) CrSTe, and (c) CrSSe monolayers.

To further evaluate the feasibility of forming these functionalized configurations, we calculate the formation energies of CrXYT₂ using the equation:

E_f-CrXYT2 = E_CrXYT2 − E_CrXY − 2μ_T (1)

where E_CrXYT2 and E_CrXY are the total energies of functionalized and pristine CrXY monolayers, respectively, and μ_T is the chemical potential of the T atoms. A negative formation energy indicates that the functionalized structure is thermodynamically stable and potentially synthesizable, with more negative values corresponding to higher stability. For the five functionalized structures of CrXY monolayers, the results shown in Fig. 5 indicate that Model V is the most stable structure, exhibiting the lowest formation energy. This suggests that Model V possesses the highest thermodynamic stability among the considered configurations. Therefore, subsequent calculations are carried out based on the most stable adsorption configuration for each functionalized structure.

Fig. 5 The formation energies of the five adsorption models (I–V) for the CrXYT₂ monolayers.

Furthermore, the magnetic ground states of the CrXY and CrXYT₂ monolayers are determined by comparing the total energies of the FM and s-AFM configurations. As illustrated in Fig. 6, the total energies of s-AFM states are higher than those of the FM state. These results confirm that FM ordering is the magnetic ground state for both CrXY and CrXYT₂ monolayers.

Fig. 6 The total energies of the two magnetic configurations (FM, s-AFM) of the CrXY and CrXYT₂ monolayers.

The optimized lattice constants (a₀), bond lengths and bond angles for the CrXY and CrXYT₂ monolayers are summarized in Table 1. The a₀ values of monolayers CrSeTe, CrSTe and CrSSe are 3.57 Å, 3.48 Å and 3.39 Å, respectively, consistent with previous simulation results reported in the literature. Moreover, the bond lengths between the Cr atom and the chalcogen atoms (d_Cr–X and d_Cr–Y) in these Janus structures clearly correlate with the atomic numbers of X and Y in the periodic table. A similar trend is observed for the functionalized CrXYT₂ monolayers. Additionally, the a₀ of the CrXYT₂ monolayers exhibits a significant increase compared to their pristine CrXY counterparts. For example, the a₀ of pristine CrSeTe is 3.57 Å, whereas the a₀ values for CrSeTeF₂, CrSeTeCl₂, CrSeTeBr₂, and CrSeTeI₂ increase to 3.78 Å, 4.09 Å, 4.15 Å, and 4.18 Å, respectively. This clearly demonstrates that the lattice constants increase with the atomic radius of the adsorbed halogen elements. Such an increase upon functionalization can be attributed to the weakened bonding interactions between the Cr atom and the chalcogen atoms X/Y. For the CrSeTeT₂ monolayers, the bond lengths d_Te–T are larger than d_Se–T, and both bond lengths increase with the atomic number of the halogen atoms T. The valence electrons of heavier elements are typically situated in higher-energy orbitals, which are more spatially diffuse. This diffuseness reduces orbital overlap efficiency during covalent bond formation, leading to longer bond lengths and subsequently affecting the magnetic properties of the materials. A similar trend is observed in CrSTeT₂ and CrSSeT₂. Additionally, in CrXY monolayers, the X–Cr–Y bond angle is 90°, and this angle remains approximately 90° in the CrXYT₂ monolayers. According to the Goodenough–Kanamori–Anderson (GKA) rules,^43–45 the super-exchange interaction between Cr atoms favors FM coupling, which aligns with the results presented in Fig. 6.

Table 1 The optimized lattice constant (a₀), the bond lengths of Cr and X atoms (d_Cr–X), Cr and Y atoms (d_Cr–Y), X/Y and T atoms (d_X–T/d_Y–T), and X–Cr–Y bond angle (θ) of the CrXY and CrXYT₂ monolayers

	Model	a 0 (Å)	d Cr–X (Å)	d Cr–Y (Å)	d X–T (Å)	d Y–T (Å)	θ (deg.)
CrSeTe	—	3.57^40,41	2.50	2.85	—	—	90.00
CrSeTeF2	V	3.78	2.51	2.71	1.82	1.97	86.64
CrSeTeCl2	V	4.09	2.62	2.79	2.23	2.41	80.84
CrSeTeBr2	V	4.15	2.63	2.80	2.40	2.56	80.08
CrSeTeI2	V	4.18	2.65	2.81	2.59	2.77	79.76
CrSTe	—	3.48^41,42	2.35	2.60	—	—	90.00
CrSTeF2	V	3.88	2.46	2.75	1.67	1.97	82.48
CrSTeCl2	V	4.13	2.58	2.83	2.08	2.39	79.38
CrSTeBr2	V	4.14	2.58	2.84	2.26	2.56	79.15
CrSTeI2	V	4.23	2.63	2.86	2.46	2.77	78.09
CrSSe	—	3.39^21	2.36	2.87	—	—	90.00
CrSSeF2	V	3.73	2.42	2.57	1.67	1.77	85.33
CrSSeCl2	V	4.00	2.55	2.66	2.07	2.22	79.26
CrSSeBr2	V	4.01	2.55	2.66	2.26	2.39	79.17
CrSSeI2	V	4.12	2.60	2.72	2.45	2.59	77.73

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To gain deeper insight into the magnetic properties, the total magnetic moments (M_tot) and individual atomic magnetic moments of the CrXY and CrXYT₂ monolayers are summarized in Table 2. The calculated total magnetic moments for monolayers CrSeTe, CrSTe and CrSSe are 2.83, 2.86, and 2.57µ_B, respectively. The calculated magnetic moment per Cr atom is approximately 3µ_B, whereas the magnetic moments of X and Y atoms are about one-tenth of that value or even smaller. This indicates that localized magnetism originates primarily from the Cr atoms, while X and Y atoms exhibit minimal induced magnetization. This is visually confirmed by the projected density of states (PDOS), as shown in Fig. 7. The PDOS analysis demonstrates that Cr 3d orbitals are almost fully spin-polarized, exhibiting a pronounced asymmetry between the spin-up and spin-down channels near the Fermi level, which is the primary origin of magnetism in the system. Conversely, the spin polarization on X and Y atoms is notably weak; their p orbitals show minimal and nearly symmetric spin-up and spin-down densities, resulting in only minor antiparallel induced magnetic moments.

Table 2 The magnetic moments of unit cell M_tot, Cr atoms M_Cr, upper X atoms M_X, lower Y atoms M_Y, and T atoms M_T, for the CrXY and CrXYT₂ monolayers

	M Cr (µB)	M X (µB)	M Y (µB)	M T (µB)	M tot (µB)
CrSeTe	3.29	−0.29	−0.16	—	2.83
CrSeTeF2	3.45	−0.16	−0.16	−0.04/−0.01	3.08
CrSeTeCl2	3.87	−0.11	−0.16	0.04/0.03	3.66
CrSeTeBr2	3.85	−0.12	−0.16	0.04/0.04	3.64
CrSeTeI2	3.82	−0.13	−0.17	0.03/0.03	3.57
CrSTe	3.21	−0.23	−0.12	—	2.86
CrSTeF2	3.42	−0.14	−0.14	−0.03/0.01	3.13
CrSTeCl2	3.79	−0.10	−0.13	0.03/0.05	3.65
CrSTeBr2	3.76	−0.11	−0.14	0.03/0.05	3.59
CrSTeI2	3.82	−0.12	−0.15	0.02/0.03	3.60
CrSSe	3.06	−0.24	−0.25	—	2.57
CrSSeF2	3.56	−0.12	−0.17	−0.02/0.01	3.27
CrSSeCl2	3.84	−0.09	−0.16	0.03/0.05	3.66
CrSSeBr2	3.82	−0.10	−0.17	0.04/0.05	3.63
CrSSeI2	3.86	−0.11	−0.18	0.03/0.03	3.64

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Fig. 7 The PDOS of the CrXY and CrXYT₂ monolayers. (a) CrSeTe, (b) CrSTe, (c) CrSSe, (a1) CrSeTeF₂, (b1) CrSTeF₂, (c1) CrSSeF₂, (a2) CrSeTeCl₂, (b2) CrSTeCl₂, (c2) CrSSeCl₂, (a3) CrSeTeBr₂, (b3) CrSTeBr₂, (c3) CrSSeBr₂, (a4) CrSeTeI₂, (b4) CrSTeI₂ and (c4) CrSSeI₂ monolayers.

Upon functionalization, the total magnetic moments of the CrXYT₂ monolayers significantly increase compared with those of the pristine CrXY counterparts, with the most pronounced increase occurring in systems modified with Cl atoms. This enhancement primarily originates from the increased magnetic moment localized on the Cr atoms. For instance, in the case of CrSeTeCl₂, the magnetic moment of the Cr atom increases by 0.58µ_B, while the Se atom exhibits an increase of 0.18µ_B. The contributions from Te and Cl atoms are negligible. Similar trends are observed across other CrXYT₂ systems, where the Cr atoms dominate the overall magnetic behavior. This phenomenon can be attributed to the formation of covalent bonds between the functional groups (halogen atoms) and the chalcogen atoms (S, Se, Te), weakening the hybridization between the Cr atoms and their neighboring chalcogen atoms. Consequently, the pairing tendency for Cr 3d electrons is suppressed, leading to enhanced spin polarization and stronger localized magnetic moments on the Cr atoms. These results demonstrate that functionalization effectively enhances the ferromagnetism of CrXY monolayers, offering a viable route for magnetic tuning.

To further investigate the electronic properties of the CrXY and CrXYT₂ monolayers, we calculate their band structures using the PBE + U method (U = 3 eV), as illustrated in Fig. 8. The CrSeTe, CrSTe and CrSSe monolayers display half-metallic ferromagnetic (HMF) behavior, characterized by a direct band gap semiconductor in the spin-up channel and metallic conductivity in the spin-down channel. Specifically, their band gaps in spin-up channel are 1.32 eV, 1.11 eV, and 2.13 eV, respectively, the schematic illustration of the HMF band structures is shown in Fig. 9(a).

Fig. 8 The electronic band structures of the CrXY and CrXYT₂ monolayers. (a) CrSeTe, (b) CrSTe, (c) CrSSe, (a1) CrSeTeF₂, (b1) CrSTeF₂, (c1) CrSSeF₂, (a2) CrSeTeCl₂, (b2) CrSTeCl₂, (c2) CrSSeCl₂, (a3) CrSeTeBr₂, (b3) CrSTeBr₂, (c3) CrSSeBr₂, (a4) CrSeTeI₂, (b4) CrSTeI₂ and (c4) CrSSeI₂ monolayers.

Fig. 9 The schematic diagrams of band structures for (a) a half metallic ferromagnet and (b) a bipolar magnetic semiconductor.

After functionalization, the CrXY monolayers undergo significant phase transitions. Specifically, the CrSeTeF₂ monolayer transforms into a ferromagnetic metal, while CrSTeCl₂, CrSTeBr₂, CrSSeBr₂, and CrSSeI₂ monolayers become bipolar magnetic semiconductors (BMS). The schematic illustration of the BMS band structures is provided in Fig. 9(b). As shown, when the Fermi level is shifted downward into Δ1 (or upward into Δ3) by applying an appropriate gate voltage, the BMS can exhibit half metallic behavior with the fully spin-polarized carriers in the spin-up (or spin-down) direction.^46,47 Furthermore, the relatively large characteristic energy gaps (Δ1, Δ2, and Δ3) facilitate easier modulation of the Fermi level, enhance error tolerance, and thus improve the feasibility of experimental realization. As summarized in Table 3, the three characteristic energy gaps exhibit suitable values, which could be beneficial for practical device applications. The band gaps for the spin-up channel (Gap↑) and the band gaps of the spin-down channel (Gap↓) in these BMS (CrSTeCl₂, CrSTeBr₂, CrSSeBr₂, and CrSSeI₂) are 1.23 and 1.48 eV, 1.23 and 1.57 eV, 1.57 and 1.73 eV, 1.42 and 1.84 eV, respectively. Such wide band gaps are beneficial for enhancing the functionality and electronic tunability in spintronic devices, thereby broadening the potential application of CrXY-based monolayers in the spintronic field. In addition, the total density of states and band structures of the CrSSeBr₂ monolayer are calculated using the HSE06 method, as shown in Fig. 10. The HSE06 results further confirm that the CrSSeBr₂ monolayer is a bipolar magnetic semiconductor, consistent with the conclusions obtained from the PBE calculations, and reveal larger characteristic energy gaps (Δ1 and Δ3). Additionally, the CrSeTeCl₂, CrSeTeBr₂, CrSeTeI₂, CrSTeF₂, CrSTeI₂, CrSSeF₂, and CrSSeCl₂ monolayers retain HMF character. HMFs inherently possess 100% spin polarization and allow electrons to conduct through a single spin channel, making them highly suitable for generating and injecting pure spin currents, which is advantageous for the development of practical spintronic devices. It is worth noting that, in the CrSeTeI₂ and CrSTeI₂ monolayers, the metallic and semiconducting spin channels are reversed compared to the pristine CrXY systems. Besides, in the band structures of the CrSSe monolayer, a flat band is observed, which primarily originates from the significant contribution of the Cr outermost 3d⁵4s¹ orbital electron in the near this energy level, as revealed by the PDOS in Fig. S2. The same mechanism accounts for the appearance of flat bands in the CrSeTeF₂, CrSTeF₂ and CrSSeF₂ monolayers.

Table 3 The property, the band gap of spin-up (Gap↑), the band gap of spin-down (Gap↓), and the nearest-neighbor exchange coupling parameter (J) of the CrXY and CrXYT₂ monolayers. The three characteristic energy gaps (the spin–flip gap Δ1 in the valence band, band gap Δ2, and spin–flip gap Δ3 in the conduction band) of bipolar magnetic semiconductors

	Property	Gap↑ (eV)	Gap↓ (eV)	Gap Δ1 (eV)	Gap Δ2 (eV)	Gap Δ3 (eV)	J (meV)
CrSeTe	HMF	1.32	0	—	—	—	28.70
CrSeTeF2	FM	0	0	—	—	—	50.98
CrSeTeCl2	HMF	1.33	0	—	—	—	98.42
CrSeTeBr2	HMF	1.42	0	—	—	—	98.90
CrSeTeI2	HMF	0	1.44	—	—	—	81.90
CrSTe	HMF	1.11	0	—	—	—	51.91
CrSTeF2	HMF	1.57	0	—	—	—	87.80
CrSTeCl2	BMS	1.23	1.48	0.63	0.86	0.37	91.69
CrSTeBr2	BMS	1.23	1.57	0.67	0.90	0.33	90.82
CrSTeI2	HMF	0	1.63				86.71
CrSSe	HMF	2.13	0	—	—	—	20.56
CrSSeF2	HMF	2.08	0	—	—	—	79.28
CrSSeCl2	HMF	1.46	0	—	—	—	86.44
CrSSeBr2	BMS	1.57	1.73	0.69	1.04	0.53	89.33
CrSSeI2	BMS	1.42	1.84	0.65	1.19	0.23	86.42

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Fig. 10 (a) Total density of states and (b) band structure of the CrSSeBr₂ monolayer calculated using the HSE06 method.

In addition, we take SOC into account to calculate the total MAE of these systems utilizing the formula:

MAE = E₁₀₀ − E₀₀₁ (2)

where E₁₀₀ and E₀₀₁ denote the energies corresponding to in-plane and out-of-plane directions, respectively. The positive (negative) value of MAE indicates the out-of-plane (in-plane) orientation of the magnetization axis. The calculated results are illustrated in Fig. 11. These materials exhibit perpendicular magnetic anisotropy (PMA), except for CrSSe, CrSTeCl₂, and CrSSeCl₂ with an in-plane magnetic easy axis. It is worth noting that the enhancement of the PMA is most pronounced in CrXY systems functionalized with F and Br groups. To further elucidate the origin of magnetic anisotropy, we further calculate the atom-resolved MAE and the corresponding energy contribution percentages, as detailed below.where E_i denotes the atom-resolved MAE for atom i, and represents the sum of the absolute values of the atom-resolved MAE. The detailed results are clearly presented in Fig. 12, which provides the foundation for the subsequent analysis of the orbital-resolved MAE.

Fig. 11 Total magnetic anisotropy energies per unit cell of the CrXY and CrXYT₂ monolayers.

Fig. 12 Atom-resolved MAE and percentage contributions of individual atomic components in CrXY and CrXYT₂ monolayers. (a) CrSeTe, (b) CrSTe, (c) CrSSe, (a1) CrSeTeF₂, (b1) CrSTeF₂, (c1) CrSSeF₂, (a2) CrSeTeCl₂, (b2) CrSTeCl₂, (c2) CrSSeCl₂, (a3) CrSeTeBr₂, (b3) CrSTeBr₂, (c3) CrSSeBr₂, (a4) CrSeTeI₂, (b4) CrSTeI₂ and (c4) CrSSeI₂ monolayers. Black text indicates out-of-plane MAE contributions, while white text indicates in-plane contributions.

Therefore, we further investigate the mechanism of MAE changes arising from the hybridization of p orbitals from chalcogen atoms and d orbitals from the Cr atom. The corresponding orbital-resolved MAE and PDOS are shown in Fig. 13 and 14. According to the second-order perturbation theory,⁴⁸ the MAE can be expressed as follows:⁴⁹

where ξ denotes the SOC constant; o and u refer to occupied and unoccupied states, respectively; α and β represent the spin components (+ and − represent the spin-up and spin-down channels), and δ_αβ is the Kronecker delta, which equals 1 when α = β and 0 otherwise. The term |〈o^α|L_z|u^β〉|² − |〈o^α|L_x|u^β〉|² describes the difference between the spin–orbit angular momentum matrix elements.⁵⁰ Detailed matrix-element differences arising from the p orbital hybridization are provided in Table S1, while those associated with the d orbital hybridization are summarized in Table S2. As shown in the tables, the MAE contribution from hybridization between occupied and unoccupied states with the same spin orientation differs by a minus sign compared to that from states with opposite spin orientations.

Fig. 13 Orbital-resolved MAE of the CrXY and CrXYT₂ monolayers. The dark cylindrical object represents PMA, and the light cylindrical object indicates IMA.

Fig. 14 The PDOS of the CrXY and CrXYT₂ monolayers. Solid lines denote spin-up states and dashed lines denote spin-down states. The curves of the chalcogen atoms' p_y and p_x orbitals overlap, as do those of the Cr d_xy and d_x²−y² orbitals, and the Cr d_yz and d_xz orbitals.

According to the atom-resolved MAE results, the MAE of the CrSTeBr₂ monolayer is primarily determined by the d orbitals of the Cr atom, in contrast to other CrXYT₂ monolayers, where the MAE is mainly contributed by the chalcogen elements. Therefore, we analyze the orbital-resolved MAE of the S and Te atoms in the CrSTeBr₂ monolayer, originating from p orbital hybridization, as illustrated in Fig. S3. The in-plane magnetic anisotropy (IMA) contribution from the p_x and p_y hybridization nearly offsets the PMA contribution from the p_z and p_y hybridization, leading to the Te atom not being the dominant contributor to the atom-resolved MAE in the CrSTeBr₂ monolayer. Additionally, upon functionalization, both the IMA contribution from the p_x and p_y hybridization and the PMA contribution from the p_z and p_y orbital hybridization become negligible. As a result, the S atom also ceases to be a major contributor to the MAE. We further analyze the contribution of d orbital hybridization of the Cr atom to the atom-resolved MAE in the CrSTeBr₂ monolayer. As shown in Fig. 13(b3) and 14(b3), the IMA contribution primarily originates from the hybridization between spin-up occupied states and spin-down unoccupied states. In contrast, the PMA contribution arises from hybridization between spin-up occupied and spin-down unoccupied states, states, states, as well as states.

Moreover, we also analyze the mechanisms responsible for driving MAE variations in the other materials. For the CrSeTe, CrSeTeCl₂, CrSeTeI₂ and CrSTeI₂ monolayers, the PMA contribution arising from the hybridization between p_y & p_z states is stronger than the IMA contribution, which originates from the hybridization between p_y & p_x states. Conversely, the CrSSe and CrSTeCl₂ monolayers exhibit the opposite trend, with the IMA contribution dominating. For the CrSSeCl₂ monolayer, orbital hybridization involving both p_y & p_z and p_y & p_x contributes to the IMA. In contrast, in the CrSTe, CrSeTeF₂, CrSTeF₂, CrSSeF₂, CrSeTeBr₂, CrSSeBr₂ and CrSSeI₂ monolayers, such orbital hybridization mainly contributes to the PMA. Thus, the atom-resolved magnetic anisotropy direction is primarily determined by the competition between and orbital hybridization of the chalcogen atoms.

Beyond magnetic anisotropy, the thermal stability of ferromagnetic ordering is another critical factor for spintronic applications. To assess this, we employ the Ising model combined with Monte Carlo simulations in a 50 × 50 × 1 supercell to determine their Curie temperatures. The Ising Hamiltonian is expressed as follows:⁵¹

where S represents the atomic spin vector, J is the nearest-neighbor exchange coupling parameter, and h denotes the external magnetic field. The total energies obtained from DFT calculations are substituted into eqn (5) to derive the following results:

E_FM = E₀ − 6J|S|² − h|S|² (6)

E_AFM = E₀ + 2J|S|² − h|S|² (7)

The nearest-neighbor exchange coupling parameter J is determined from the energy difference between the FM and AFM states as follows:

The positive (negative) value of J reflects the preference for ferromagnetic (antiferromagnetic) ordering. The computed J values for all CrXY and CrXYT₂ monolayers are summarized in Table 3. The results are greater than 0, suggesting that FM coupling is energetically favored in these systems. Generally, a stronger magnetic exchange interaction typically leads to a higher T_C, thereby enhancing the thermal stability of magnetic ordering. The T_C is a crucial property for practical spintronic applications, which often require room-temperature or above-room-temperature operation. Based on the obtained J values, the T_C values are calculated using the Ising model by Monte Carlo simulations.⁴⁰ The temperature-dependent magnetic moment (M) and specific heat capacity (C_v) are shown in Fig. S4. The T_C values of the CrXY and CrXYT₂ monolayers are identified by the peak position in C_v or the inflection points in the magnetic moment curves. A summary of the calculated T_C values for all monolayers is provided in Fig. 15. For the pristine CrXY monolayers, the T_C values of CrSeTe, CrSTe and CrSSe monolayers are 217 K, 129 K, and 171 K, respectively, all of which fall well below room temperature. This severely limits their practical applications in spintronic devices. In contrast, the functionalized CrXYT₂ monolayers exhibit a remarkable enhancement in magnetic thermal stability. The incorporation of terminal functional groups substantially increases the exchange coupling, leading to much higher Curie temperatures. Notably, halogen functionalization with Cl and Br yields the most pronounced improvements. The corresponding T_C values for CrSeTeCl₂, CrSeTeBr₂, CrSTeCl₂, CrSTeBr₂, CrSSeCl₂, and CrSSeBr₂ monolayers are 816 K, 806 K, 758 K, 745 K, 715 K, and 722 K, respectively. These results indicate that halogen-functionalized CrXYT₂ monolayers, particularly those modified with Cl and Br, are promising candidates for spintronic applications.

Fig. 15 The Curie temperature T_C of the CrXY and CrXYT₂ monolayers.

4. Conclusions

In summary, we have systematically investigated the structural, electronic and magnetic properties of Janus CrXY (X, Y = S, Se, and Te; X ≠ Y) monolayers and their functionalized CrXYT₂ (X, Y = S, Se, and Te; X ≠ Y; T = F, Cl, Br, and I) monolayers using first-principles calculations. The results show that both CrXY and CrXYT₂ monolayers are dynamically and thermodynamically stable. The pristine CrXY monolayers are half-metallic ferromagnets. Upon functionalization, magnetic phase transitions occur in several systems; CrSTeCl₂, CrSTeBr₂, CrSSeBr₂, and CrSSeI₂ monolayers transform into bipolar magnetic semiconductors, which can lead to reversible high spin polarization in spintronics. The CrSeTeF₂ monolayer becomes a ferromagnetic metal. Moreover, spin-channel inversion is observed in CrSeTeI₂ and CrSTeI₂, where the spin character of the metallic and semiconducting bands is reversed compared to their pristine counterparts. Additionally, the functionalized CrXYT₂ monolayers significantly enhance their magnetic properties. Specifically, their magnetic moments increase from a minimum of 2.57µ_B (CrSSe) and a maximum of 2.83µ_B (CrSeTe) in pristine CrXY, to a range of 3.08µ_B to 3.66µ_B after functionalization, with Cl functional groups exhibiting the most pronounced improvement. Furthermore, functionalization markedly elevates the T_C above room temperature, particularly with Cl and Br groups, whereas the pristine CrXY monolayers exhibit significantly lower T_C. Magnetic anisotropy analysis reveals that most monolayers possess perpendicular magnetic anisotropy (PMA), except for CrSSe, CrSTeCl₂ and CrSSeCl₂ with an in-plane magnetic easy axis. Notably, the CrXYF₂ monolayers display strong PMA, highlighting their potential for magnetic memory applications. Orbital-resolved MAE analysis shows that the atom-resolved magnetic anisotropy direction is primarily governed by the competition between the orbital hybridization between and of chalcogen atoms. Overall, our study demonstrates that surface functionalization effectively enhances ferromagnetism and Curie temperature, induces spin-channel inversion, and opens energy gaps in Janus CrXY monolayers. These findings extend the potential of Cr-based monolayers in spintronic applications and provide valuable theoretical reference for future experimental synthesis.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

The data that support the findings of this study are available within the article and its supplementary information (SI). Supplementary information is available. See DOI: https://doi.org/10.1039/d5tc02548f.

Acknowledgements

This work was supported by Guizhou Provincial Science and Technology Foundation (Grant No. ZK[2023]247).

References

1. N. Miao and Z. Sun, Computational design of two-dimensional magnetic materials, Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2021, 12, e1545.
2. W. Q. Liu, P. K. J. Wong and Y. B. Xu, Hybrid spintronic materials: Growth, structure and properties, Prog. Mater. Sci., 2019, 99, 27–105.
3. B. Dieny, I. L. Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. Ebels, S. O. Demokritov, J. Akerman, A. Deac, P. Pirro, C. Adelmann, A. Anane, A. V. Chumak, A. Hirohata, S. Mangin, S. O. Valenzuela, M. C. Onbasli, M. D'Aquino, G. Prenat, G. Finocchio, L. Lopez-Diaz, R. Chantrell, O. Chubykalo-Fesenko and P. Bortolotti, Opportunities and challenges for spintronics in the microelectronics industry, Nat. Electron., 2020, 3, 446–459.
4. J. Xiao, M. He, B. Zhan, H. Guo, J.-L. Yang, Y. Zhang, X. Qi and J. Gu, Multifunctional microwave absorption materials: construction strategies and functional applications, Mater. Horiz., 2024, 11, 5874–5894.
5. B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature, 2017, 546, 270–273.
6. C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia and X. Zhang, Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nature, 2017, 546, 265–269.
7. Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu, D. H. Cobden, J.-H. Chu and X. Xu, Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2, Nat. Mater., 2018, 17, 778–782.
8. L. Yao, J. Dang, J. Xiao, Y. Chen, J. Ding, Y. Qu, Q. Peng, X. Qi and W. Zhong, Metal chelate-derived and catalytical strategy to produce CoFe/C@bamboo-like carbon nanotubes for microwave absorption, hydrophobicity, and corrosion resistance, J. Mater. Sci. Technol., 2026, 240, 190–200.
9. J. Xiao, B. Zhan, M. He, X. Qi, Y. Zhang, H. Guo, Y. Qu, W. Zhong and J. Gu, Mechanically Robust and Thermal Insulating Nanofiber Elastomer for Hydrophobic, Corrosion-Resistant, and Flexible Multifunctional Electromagnetic Wave Absorbers, Adv. Funct. Mater., 2024, 35, 2419266.
10. J. Xiao, B. Zhan, M. He, X. Qi, X. Gong, J. L. Yang, Y. Qu, J. Ding, W. Zhong and J. Gu, Interfacial Polarization Loss Improvement Induced by the Hollow Engineering of Necklace-like PAN/Carbon Nanofibers for Boosted Microwave Absorption, Adv. Funct. Mater., 2024, 35, 2316722.
11. B. Zhan, Y. Hao, X. Qi, Y. Qu, J. Ding, J.-L. Yang, X. Gong, Y. Chen, Q. Peng and W. Zhong, Multifunctional cellular carbon foams derived from chitosan toward self-cleaning, thermal insulation, and highly efficient microwave absorption properties, Nano Res., 2023, 17, 927–938.
12. Q. Liang, M. He, B. Zhan, H. Guo, X. Qi, Y. Qu, Y. Zhang, W. Zhong and J. Gu, Yolk–Shell CoNi@N-Doped Carbon-CoNi@CNTs for Enhanced Microwave Absorption, Photothermal, Anti-Corrosion, and Antimicrobial Properties, Nano-Micro Lett., 2025, 17, 167.
13. A. Y. Lu, H. Zhu, J. Xiao, C. P. Chuu, Y. Han, M. H. Chiu, C. C. Cheng, C. W. Yang, K. H. Wei, Y. Yang, Y. Wang, D. Sokaras, D. Nordlund, P. Yang, D. A. Muller, M. Y. Chou, X. Zhang and L. J. Li, Janus monolayers of transition metal dichalcogenides, Nat. Nanotechnol., 2017, 12, 744–749.
14. J. Chen, K. Wu, H. Ma, W. Hu and J. Yang, Tunable Rashba spin splitting in Janus transition-metal dichalcogenide monolayers via charge doping, RSC Adv., 2020, 10, 6388–6394.
15. E. Bruyer, D. Di Sante, P. Barone, A. Stroppa, M. H. Whangbo and S. Picozzi, Possibility of combining ferroelectricity and Rashba-like spin splitting in monolayers of the1T-type transition-metal dichalcogenides MX₂ (M = Mo, W; X = S, Se, Te), Phys. Rev. B, 2016, 94, 195402.
16. M. K. Mohanta, A. Arora and A. De Sarkar, Conflux of tunable Rashba effect and piezoelectricity in flexible magnesium monochalcogenide monolayers for next-generation spintronic devices, Nanoscale, 2021, 13, 8210–8223.
17. W. Shi and Z. Wang, Mechanical and electronic properties of Janus monolayer transition metal dichalcogenides, J. Phys.: Condens. Matter, 2018, 30, 215301.
18. J. J. He and S. Li, Two-dimensional Janus transition-metal dichalcogenides with intrinsic ferromagnetism and half-metallicity, Comput. Mater. Sci., 2018, 152, 151–157.
19. A. González-García, C. Bacaksiz, T. Frauenheim and M. V. Milošević, Strong spin-lattice coupling and high-temperature magnetic ordering in monolayer chromium dichalcogenides, Phys. Rev. Mater., 2024, 8, 064001.
20. Y. Q. Liu, L. Zhang, X. M. Wu and G. Y. Gao, Enhanced ferromagnetism, magnetic anisotropy, and spin polarization in Janus CrSeTe via strain and doping, Appl. Phys. Lett., 2023, 123, 192407.
21. Y. Wu, Q. Liu, P. Shi, J. Su, Y. Zhang and B. Wang, High temperature ferromagnetic metal: a Janus CrSSe monolayer, Phys. Chem. Chem. Phys., 2023, 25, 9958–9964.
22. Q. R. Cui, J. H. Liang, Z. J. Shao, P. Cui and H. X. Yang, Strain-tunable ferromagnetism and chiral spin textures in two-dimensional Janus chromium dichalcogenides, Phys. Rev. B, 2020, 102, 094425.
23. Q. Q. Li, W. W. Liu, Z. K. Ding, H. Pan, X. H. Cao, W. H. Xiao, N. N. Luo, J. Zeng, L. M. Tang, B. Li, K. Q. Chen and X. D. Duan, Stacking- and strain-dependent magnetism in Janus CrSTe bilayer, Appl. Phys. Lett., 2023, 122, 121902.
24. Z. Guan, Z. Shen, Y. Xue, T. Zhong, X. Wu and C. Song, Electronic properties, skyrmions and bimerons in Janus CrXY (X, Y = S, Se, Te, Cl, Br, I, and X not equal Y) monolayers, Phys. Chem. Chem. Phys., 2023, 25, 24968–24975.
25. W. Feng, P. Long, Y. Feng and Y. Li, Two-Dimensional Fluorinated Graphene: Synthesis, Structures, Properties and Applications, Adv. Sci., 2016, 3, 1500413.
26. S. Li, J. J. He, L. Grajciar and P. Nachtigall, Intrinsic valley polarization in 2D magnetic MXenes: surface engineering induced spin-valley coupling, J. Mater. Chem. C, 2021, 9, 11132–11141.
27. X. Gong, L. Xiang, X. Qi, X. Gong, Y. Chen, Q. Peng, Y. Qu, F. Wu, K. Sun and W. Zhong, Improved polarization loss and impedance matching induced by carbon paper-based magnetic heterostructured composites for lightweight and strong microwave absorption, Adv. Compos. Hybrid Mater., 2024, 7, 216.
28. Y. Hu, X. Y. Liu, Z. H. Shen, Z. F. Luo, Z. G. Chen and X. L. Fan, High Curie temperature and carrier mobility of novel Fe, Co and Ni carbide MXenes, Nanoscale, 2020, 12, 11627–11637.
29. B. Zhan, Y. Qu, X. Qi, J. Ding, Jj Shao, X. Gong, J. L. Yang, Y. Chen, Q. Peng, W. Zhong and H. Lv, Mixed-Dimensional Assembly Strategy to Construct Reduced Graphene Oxide/Carbon Foams Heterostructures for Microwave Absorption, Anti-Corrosion and Thermal Insulation, Nano-Micro Lett., 2024, 16, 221.
30. G. Kresse and J. Furthmuller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
31. P. E. Blochl, Projector augmented-wave method, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953–17979.
32. J. P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett., 1996, 77, 3865–3868.
33. H. J. Monkhorst and J. D. Pack, Special Points for Brillouin-Zone Integrations, Phys. Rev. B: Condens. Matter Mater. Phys., 1976, 13, 5188–5192.
34. S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys., 1984, 81, 511–519.
35. N. Metropolis and S. Ulam, The Monte Carlo method, J. Am. Stat. Assoc., 1949, 44, 335–341.
36. A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater., 2015, 108, 1–5.
37. A. Togo, F. Oba and I. Tanaka, First-principles calculations of the ferroelastic transition between rutile-type and CaCl₂-type SiO₂ at high pressures, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 134106.
38. D. C. Freitas, R. Weht, A. Sulpice, G. Remenyi, P. Strobel, F. Gay, J. Marcus and M. Nunez-Regueiro, Ferromagnetism in layered metastable 1T-CrTe₂, J. Phys.: Condens. Matter, 2015, 27, 176002.
39. M. R. Habib, S. Wang, W. Wang, H. Xiao, S. M. Obaidulla, A. Gayen, Y. Khan, H. Chen and M. Xu, Electronic properties of polymorphic two-dimensional layered chromium disulphide, Nanoscale, 2019, 11, 20123–20132.
40. Y. Liu, L. Zhang, X. Wu and G. Gao, Enhanced ferromagnetism, magnetic anisotropy, and spin polarization in Janus CrSeTe via strain and doping, Appl. Phys. Lett., 2023, 123, 192407.
41. Q. Cui, J. Liang, Z. Shao, P. Cui and H. Yang, Strain-tunable ferromagnetism and chiral spin textures in two-dimensional Janus chromium dichalcogenides, Phys. Rev. B, 2020, 102, 094425.
42. Q. Q. Li, W. W. Liu, Z. K. Ding, H. Pan, X. H. Cao, W. H. Xiao, N. N. Luo, J. Zeng, L. M. Tang, B. Li, K. Q. Chen and X. D. Duan, Stacking- and strain-dependent magnetism in Janus CrSTe bilayer, Appl. Phys. Lett., 2023, 122, 121902.
43. J. B. Goodenough, Theory of the Role of Covalence in the Perovskite-Type Manganites [La, M(II)] MnO₃, Phys. Rev., 1955, 100, 564–573.
44. J. Kanamori, Superexchange interaction and symmetry properties of electron orbitals, J. Phys. Chem. Solids, 1959, 10, 87–98.
45. S. Jana, P. Aich, P. A. Kumar, O. K. Forslund, E. Nocerino, V. Pomjakushin, M. Mansson, Y. Sassa, P. Svedlindh, O. Karis, V. Siruguri and S. Ray, Revisiting Goodenough-Kanamori rules in a new series of double perovskites LaSr_1–xCa_xNiReO₆, Sci. Rep., 2019, 9, 18296.
46. X. X. Li and J. L. Yang, First-principles design of spintronics materials, Natl. Sci. Rev., 2016, 3, 365–381.
47. L. Zhang, Y. Q. Liu, M. H. Wu and G. Y. Gao, Electric-Field- and Stacking-Tuned Antiferromagnetic FeClF Bilayer: The Coexistence of Bipolar Magnetic Semiconductor and Anomalous Valley Hall Effect, Adv. Funct. Mater., 2025, 35, 2570097.
48. D. Wang, R. Wu and A. J. Freeman, First-principles theory of surface magnetocrystalline anisotropy and the diatomic-pair model, Phys. Rev. B: Condens. Matter Mater. Phys., 1993, 47, 14932–14947.
49. R. Han, Z. Jiang and Y. Yan, Prediction of Novel 2D Intrinsic Ferromagnetic Materials with High Curie Temperature and Large Perpendicular Magnetic Anisotropy, J. Phys. Chem. C, 2020, 124, 7956–7964.
50. L. Zhang, Y. Zhao, Y. Liu and G. Gao, High spin polarization, large perpendicular magnetic anisotropy and room-temperature ferromagnetism by biaxial strain and carrier doping in Janus MnSeTe and MnSTe, Nanoscale, 2023, 15, 18910–18919.
51. J. Xie, D. Wu, Y. Liao, X. Cao and S. Zhou, Charge doping and electric field tunable ferromagnetism and Curie temperature of the MnS₂ monolayer, Phys. Chem. Chem. Phys., 2023, 26, 267–277.

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