Janus S-XSSe: a new family of 2D polar half-metals with stacking-engineered magnetism and antiferromagnetic spin splitting

Abstract

The quest for materials uniting polarity with metallicity or half-metallicity is pivotal for next-generation multifunctional devices, yet their realization, particularly with robust magnetism, remains a formidable challenge. Here, we unveil a new family of two-dimensional (2D) Janus S-XSSe (X = Re, Os, V, Cr and Mo) monolayers as intrinsically polar metals/half-metals, exhibiting substantial out-of-plane polarization (2.56–4.00 pC m⁻¹) and excellent structural stability. We demonstrate that S-VSSe, S-CrSSe and S-MoSSe are polar half-metals, where the distinct origins of polarity (S–Se electronegativity difference) and half-metallicity (transition metal d-orbitals) enable their robust coexistence. Furthermore, intriguing anisotropic Rashba effects have been observed in polarized metallic S-ReSSe and S-OsSSe monolayers. In addition, we identify a viable polarization switching pathway, whose energy barrier can be effectively tuned by biaxial strain. Moreover, bilayer S-XSSe systems exhibit enhanced magnetic transition temperature and stacking-dependent magnetism, elucidated via spin Hamiltonian analysis and interlayer electron hopping. Remarkably, an interlayer antiferromagnetic S-VSSe bilayer with parallel polar stacking exhibits significant spin splitting alongside nontrivial topological characteristics, a direct consequence of the built-in electric field breaking inversion symmetry. This discovery of a novel class of polar metals/half-metals, particularly the emergent antiferromagnetic spintronic phenomena in bilayers, paves the way for innovative spintronic and multifunctional electronic applications.

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1 Introduction

Traditionally, polarity has been considered incompatible with metallicity due to the screening effect of free carriers on long-range dipole–dipole interactions. However, based on symmetry considerations, Anderson and Blount proposed that metallic systems could coexist with polarity, suggesting that structural phase transitions analogous to ferroelectricity (FE) could occur in metals, leading to the emergence of polar metals.¹ This concept has recently regained attention due to the discovery of numerous novel physical properties in polar metals.^2–15 Experimentally, various approaches have been successfully employed to fabricate polar metals, including intrinsic materials such as LiOsO₃ and NdNiO₃ thin films.^2,4 Both systems share the common feature of decoupling (weak coupling principle) between polar electrons induced by displacive ions and states at the Fermi level. Extrinsic methods include introducing metallicity into polar structures through chemical doping^6,7 or achieving the coexistence of polarity and metallicity at interfaces by constructing heterostructures.^11,12

Moreover, inspired by the promising magnetoelectric coupling observed in multiferroic materials (i.e., materials exhibiting coexistence of two or more ferroic orders, such as ferromagnetic (FM), FE and ferroelastic (FA)), it is highly meaningful to introduce magnetism into polar metals and further explore the exotic physical properties.^16–23 Experimentally, electric-field-tunable ferromagnetism and superconductivity have been achieved in a non-centrosymmetric system generated by doping at the LaAlO₃/SrTiO₃ interface.²⁴ Coexistence of polarity, magnetism and metallicity has been realized in Co-doped non-centrosymmetric (Fe_0.5Co_0.5)₅GeTe₂ and quasi-two-dimensional Ca₃Co₃O₈.^25,26 Theoretically, the coexistence of FE and FM has been predicted in layered metal Bi₅Mn₅O₁₇,²⁷ monolayer CrN²⁸ and monolayer Os₂Se₃.²⁹ Interestingly, polarity has been combined with half-metallicity in doped α-In₂Se₃³⁰ and monolayer Co₂Se₃.³¹ Such works show that the search for new polar half-metals with excellent properties has the potential to generate new functional devices such as information storage and processing.^32–34

However, achieving the coexistence of polarity, FM and metallicity remains highly challenging, primarily due to the need to combine the conflict between polar displacements and metallicity while incorporating appropriate electronic correlations to form FM states.²⁶ A widely utilized strategy involves introducing polar displacements and metallicity in different sublattices, as demonstrated in experimentally synthesized LiOsO₃ and NdNiO₃.^2,4 Theoretical studies have also predicted such coexistence in (PbMnO₃)₁/(SrMnO₃)₁ superlattices, where polar displacements arise from the ionic displacements of Pb and Sr, while the half-metallicity originates from the strong hybridization of Mn 3d and O 2p states.³⁵ Additionally, studies have shown that two-dimensional Janus transition metal dichalcogenides (TMDCs) can break out-of-plane symmetry, thereby introducing polarization.^36–41 For instance, Janus XMnY (X, Y = S, Se and Te) are spin-splitting antiferromagnetic (AFM) systems;⁴² Janus FeXY (X/Y = Cl/Br/I, X ≠ Y) monolayers are gapless ferromagnets with tunable magnetism;⁴³ Janus AsSn₂Bi, PSn₂Bi and PSn₂Sb monolayers,⁴⁴ as well as Janus MoSi₂N_xZ_4−x monolayers, have been reported as polar metals;⁴⁵ Guo et al. successfully predicted an intriguing piezoelectric quantum anomalous Hall insulator (PQAHI) in Janus Fe₂IX (X = Cl and Br) monolayers.⁴⁶ Notably, the intrinsic symmetry breaking of Janus materials, combined with strong spin–orbit coupling (SOC), provides an ideal platform for the study of the Rashba effect. It has been found that, under this effect, electron spins, rather than merely charges, can be manipulated by an electric field, which has significant implications for the field of spintronics.⁴⁷ In summary, an increasing number of theoretical and experimental studies on polar metals/half-metals continue to emerge, revealing novel physical properties worthy of in-depth exploration. It is worth noting that five metallic and half-metallic Janus S-XSSe (X = Re, Os, V, Cr and Mo) monolayers with a square lattice have been theoretically designed.⁴⁸ Among them, S-ReSSe and S-OsSSe are nonmagnetic metals, while S-VSSe, S-CrSSe and S-MoSSe are half-metals. However, systematic investigations into their polar properties remain largely unexplored.

In this work, based on first-principles calculations, we have discovered a novel class of two-dimensional polar metals/half-metals, Janus S-XSSe (X = Re, Os, V, Cr and Mo), which exhibit robust out-of-plane polarization above room temperature. Meanwhile, we also proposed a possible pathway for polarization switching and demonstrated that biaxial strain can further reduce the switching barrier. Interestingly, S-VSSe, S-CrSSe and S-MoSSe are polar half-metals, and we have demonstrated that the distinct origins of polarity and half-metallicity enable their coexistence in these materials. Meanwhile, S-ReSSe and S-OsSSe are polar metals with a sizable anisotropic Rashba effect. Additionally, through different bilayer stacking configurations, we revealed the microscopic mechanisms of rich magnetic properties using spin Hamiltonian and interlayer electron hopping, and found that the interlayer coupling enhances the magnetic transition temperature. Furthermore, we uncover rich electronic structures in bilayer stacking configurations. Notably, spin splitting is observed in the antiferromagnetic S-VSSe bilayer with parallel polarization configuration, confirming its topologically nontrivial nature. The discovery of novel polar metals/half-metals, coupled with the emergence of rich magnetic and electronic structures in bilayer stacking configurations, provides a theoretical foundation for the development and application of future multifunctional electronic devices.

2 Computational methods

We carried out first-principles calculations based on density functional theory (DFT) with the projector augmented wave (PAW) method,⁴⁹ as implemented in the Vienna ab initio simulation package (VASP).⁵⁰ The generalized gradient approximation (GGA) in the form of the Perdew–Burke–Ernzerhof (PBE) function was used for the exchange correlation potential.⁵¹ The kinetic energy cut-off for plane wave expansion was set to 500 eV and Γ-centered k-meshes of 12 × 12 × 1 and 21 × 21 × 1 were used to sample the Brillouin zone (BZ) of the structure optimization and self-consistent calculations, respectively. The thickness of the vacuum layer along the z-axis were set to 15 and 40 Å for S-XSSe (X = Re, Os, V, Cr and Mo) monolayers and bilayers, respectively. The DFT-D3 method of Grimme with BJ damping was applied to consider the vdW interaction for bilayers.⁵² Lattice parameters and atomic positions are fully relaxed, and the convergence criteria for the energy and force are set to 10⁻⁶ eV and 10⁻³ eV Å⁻¹, respectively. To better describe the strongly correlated d electrons of transition atoms, the GGA+U approach was employed, and the U_eff values are consistent with previous work,⁴⁸i.e., 3 eV for Cr and Mo and 4 eV for V. The dipole correction was applied in electrostatic potential calculations.⁵³ An FE-switching pathway was obtained with the climbing-image nudged elastic band (CI-NEB) method.⁵⁴ The ab initio molecular dynamics (AIMD) simulations were implemented using the VASP code, and a supercell of 6 × 6 × 1 was used. We consider SOC effects in the S-VSSe bilayer band calculations.⁵⁵ The WANNIER90 and WannierTools software packages are utilized to generate the maximal localized Wannier functional, edge state, Berry curvature, and the anomalous Hall conductivity.^56,57

Due to the disappearance of the P uncertainty caused by the absence of periodicity in the out-of-plane direction, the vertical P of 2D systems is well defined by the classical dipole method.^31,45 In the case that the vacuum slab is large enough to eliminate interlayer interactions and neglect periodicity, the out-of-plane P can be easily defined by classical electrodynamics and estimated as P = q × d, where q is the total number of valence charges and d is the vector of the dipole moment from the negative charge center (NCC) to the positive charge center (PCC). The z coordinates of NCC and PCC can be calculated as follows:

where ρ, z and n are the charge density, coordinates along the out-of-plane z direction, and the number of valence electrons in each ion, respectively.

Our Monte Carlo (MC) simulations of the Heisenberg model were carried out using the Metropolis algorithm.⁵⁸ To minimize finite-size effects, we employed a 25 × 25 supercell containing 1250 sites. Statistical convergence was ensured by performing over 10⁶ MC sweeps for each simulation.

3 Results and discussion

3.1 Structure and polarity of metallic/half-metallic Janus S-XSSe monolayers

The atomic configuration of the two-dimensional Janus S-XSSe monolayers consists of a transition metal atomic layer sandwiched between S and Se layers. Fig. 1(a) shows the top and side views of S-XSSe (X = Re, Os, V, Cr and Mo) monolayers. The S-XSSe monolayers adopt an orthorhombic structure with the space group Pmm2 (no. 25), where each X atom is located at the center of a tetrahedron formed by two S and two Se atoms. The lattice constants (a and b) of the five S-XSSe monolayers are positively correlated with the radius of X (see Table 1). As shown in Fig. 1(b), by analyzing the symmetry operations of the point group, it can be observed that the polarization direction of S-XSSe aligns with the z-axis. According to the Pauling scale, the electronegativity of S is stronger than that of Se,⁵⁹ leading to an uneven charge distribution along the z-direction.

Fig. 1 (a) Top and side views of the 4 × 4 × 1 supercell of the S-XSSe monolayer (the primitive cell is marked by the red rectangle dashed line, while the yellow, green and blue balls represent S, Se and X atoms, respectively). (b) Schematic representation of the symmetric operation of S-XSSe under the mm2 point group, with the red dashed line indicating the direction of P. (c) Planar-average charge density along the z-direction of the S-CrSSe monolayer. The gray dashed line denotes the center of the S-CrSSe monolayer. The light blue arrow indicates the direction of P.

Table 1 The lattice constants (a, b), distance between S and Se atoms (h), spontaneous polarization (P), polarization switching energy barrier (E_b) and melting temperature (T_m) of the five S-XSSe monolayers

S-XSSe	a/b (Å)	h (Å)	P (pC m^−1)	E b (eV)	T m (K)
S-VSSe	3.70/3.81	2.80	2.89	1.19	400
S-CrSSe	3.69/3.83	2.69	3.81	0.87	500
S-MoSSe	3.86/3.91	2.78	2.56	1.28	700
S-ReSSe	3.69/3.73	2.72	4.00	2.16	600
S-OsSSe	3.71/3.87	2.54	3.68	1.94	900

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In Table 1, the polarization (P) of the five S-XSSe monolayers ranges from 2.56 to 4.00 pC m⁻¹, which is higher than or comparable to those of many two-dimensional polar materials. Examples include group IVA binary honeycomb ferroelectrics (1–4 pC m⁻¹),⁶⁰ the polar half-metal Co₂Se₃ (4 pC m⁻¹),³¹ bimetallic phosphates (1–10 pC m⁻¹)⁶¹ and MXenes (3–17 pC m⁻¹).⁶² The relatively high polarization strength makes the polar metal/half-metal S-XSSe monolayers highly promising for practical applications. To gain deeper insight into the polarization, we calculated the planar-averaged charge density along the z-direction, as shown in Fig. 1(c) (taking S-CrSSe as an example; the remaining four S-XSSe monolayers are shown in Fig. S1 of the SI). We conclude that the electronegativity difference between S and Se atoms leads to a non-centrosymmetric charge distribution, with electrons preferentially distributed on the S side, resulting in polarization directed from S to Se.

Additionally, using the CI-NEB method, we identified a possible polarization switching path for the S-XSSe system (taking S-CrSSe in Fig. 2(a) as a reference, the results for the remaining S-XSSe monolayers are shown in Fig. S2 of the SI). The polarization switching energy barriers (E_b) for the five S-XSSe monolayers range from 0.87 to 2.16 eV (see Table 1). Except for S-CrSSe, which has an E_b of 0.87 eV (slightly lower than the 0.88 eV of the polar half-metal monolayer Co₂Se₃³¹), the E_b values of the other four S-XSSe materials exceed 1 eV. This is primarily due to the high lattice distortion energy associated with the transition through a purely planar paraelectric (PE) phase during polarization switching. However, such a large E_b in practical applications would require a relatively high electric field for switching, which could be advantageous for information storage.⁶³

Fig. 2 (a) P switching path and the energy profile as a function of the switching step number in the CI-NEB computations for S-CrSSe. (b) Biaxial strain dependence of the E_b of the S-CrSSe monolayer. (c) Temperature dependence of the average ionic displacement dz the of S-CrSSe monolayer; the blue and orange circles represent the temperatures before and after melting, respectively, and the respective structures are schematically shown within the rounded rectangles.

To further reduce the polarization switching energy barrier, we considered that the S-XSSe system undergoes a purely planar PE phase during polarization switching, accompanied by lattice expansion. Therefore, applying biaxial tensile strain to the FE phase could facilitate polarization switching. We applied biaxial strain using ε = a/a₀ − 1 × 100%, where a and a₀ are the lattice constants along the biaxial direction for the strained and unstrained structures, respectively. To determine the maximum biaxial strain the materials can withstand, we calculated the energy–strain relationship, as shown in Fig. S3 of the SI. We found that the energy changes smoothly for the S-XSSe system under biaxial strains ranging from 0% to 12%, indicating no irreversible structural deformation. Further CI-NEB calculations confirmed our expectations, showing that E_b decreases with increasing biaxial tensile strain. Specifically, the E_b values for S-VSSe, S-CrSSe and S-MoSSe decrease to 0.79, 0.69 and 0.91 eV, respectively, under 12% biaxial strain, while S-ReSSe and S-OsSSe remain above 1 eV (see Fig. 2(b) and Fig. S4 of the SI). In summary, for the polar metal/half-metal S-XSSe monolayers, we confirmed their considerable polarization strength, identified a possible polarization switching path, and discovered a method to reduce E_b. However, given that the polarization switching in this system involves lattice expansion and overcomes substantial energy barriers (E_b), its practical switchability requires further investigation. This behavior fundamentally distinguishes it from conventional ferroelectric materials, leading us to categorize it as a polar metal/half-metal system.

3.2 Thermal stability of Janus S-XSSe monolayers

Thermal stability is an important property in evaluating the utility of polar materials. To investigate the temperature effects in S-XSSe monolayers, we employ AIMD simulations at various temperatures for 5 ps at a 6 × 6 × 1 supercell and calculate the average ionic displacement using , where z_Xn − 1/2(z_Sn + z_Sen) denotes the coordinate differences along the z direction between the nearest neighbor X atom and the midpoint of S and Se atoms, and N is the total number of X–S–Se structural units in the supercell. As the temperature increases, the dz value oscillates with respect to 0 K, until at a certain temperature the dz value changes drastically, indicating that the degree of structural distortion in the z-direction is large and the material is melting. In the case of S-CrSSe, as shown in Fig. 2(c), as the temperature increases from 0 K to 500 K, the dz value oscillates around −0.09 Å and the structure is still stable. When the temperature reaches 600 K, there is a dramatic change in the dz value (−0.13 Å), at which point the structure melts. The AIMD results for the remaining 4 S-XSSe monolayers and the T_m values are shown in Fig. S5 of the SI and Table 1. Notably, the T_m values of S-XSSe systems are all above room temperature, up to 900 K for S-OsSSe, indicating their excellent working temperatures.

3.3 Coexistence mechanism of polarity and half-metallicity in Janus S-XSSe

We recognize that achieving the coexistence of polarity, FM and metallicity is highly challenging, primarily due to the need to reconcile the conflict between polar displacements and metallicity while incorporating appropriate electronic correlations to form FM states. In the S-XSSe system, S-VSSe, S-CrSSe and S-MoSSe have been confirmed to exhibit FM ground states, with Curie temperatures reaching 210 K, 810 K and 390 K, respectively,⁴⁸ enabling their application in higher-temperature environments. Therefore, it is highly meaningful to further investigate the coexistence mechanisms of polarity, FM and metallicity in the Janus S-XSSe system.

Under the GGA+U approach, the projected density of states (PDOS) of S-VSSe, S-CrSSe and S-MoSSe is shown in Fig. 3(a)–(c). It is clear that the three materials exhibit similar features near the Fermi level. The spin-down channels (semiconducting) are dominated by the d orbitals of transition metal atoms and the p orbitals of Se atoms at the conduction band minimum (CBM) and the valence band maximum (VBM), respectively; the spin-up channels (metallic) are mainly dominated by the d orbitals of transition metal atoms. Since the S-VSSe monolayer (the outermost electron arrangement of V is 3d⁴4s¹) has a Dirac point at the Fermi level of the spin-up channel, the density of states at this position is close to zero, as shown in Fig. 3(a). Moreover, for Cr and Mo with 3d⁵4s¹ and 4d⁵5s¹ arrangements, the density of states at the Fermi level in the spin-up band increases significantly due to one more outermost electron. Thus, the central transition metal atom contributes mainly to the metallicity as well as the FM properties; in addition, the difference in the electronegativity of the S and Se atoms leads to out-of-plane polarization, and thus polarity, magnetism and metallicity can coexist. It can be observed that our system is analogous to type-I multiferroic materials such as BiFeO₃,⁶⁴ where polarity and magnetism originate from different atoms.

Fig. 3 Projected density of states (PDOS) of (a) S-VSSe, (b) S-CrSSe, and (c) S-MoSSe. The red dashed line represents the Fermi level. Red and blue arrows represent the spin-down and spin-up channels, respectively.

3.4 Rashba spin splitting

The Rashba effect emerges as a fascinating physical phenomenon in systems with broken inversion symmetry and strong SOC.⁶⁵ The intrinsic inversion asymmetry of the Janus configuration has made them an ideal platform for studying this effect, as demonstrated in various 2D Janus materials, including transition-metal dichalcogenide monolayers⁶⁶ and Janus SeMoSiN₂.⁶⁷ The Rashba effect holds significant promise for spintronic applications, enabling efficient spin manipulation through spin-to-charge interconversion and electrically driven spin torque.⁶⁸ In the S-XSSe (X = Re, Os, V, Cr and Mo) monolayers, the combined effect of surface inversion symmetry breaking (induced by the asymmetric S/Se termination) and strong SOC from the central transition-metal atoms is expected to give rise to prominent Rashba spin splitting.

As shown in Fig. S6 of the SI, comparative analysis of band structures with and without SOC reveals distinct behavior depending on the central X atom. Systems with V, Cr or Mo as the central atom exhibit relatively weak SOC effects, showing no observable Rashba splitting. In striking contrast, when X is a 5d transition metal (Re or Os), the SOC effect becomes markedly enhanced. Notably, both S-ReSSe and S-OsSSe monolayers display pronounced Rashba splitting near the Fermi level, particularly around the S point in the Brillouin zone (Fig. 4). The strength of the Rashba effect can be quantitatively characterized by three key parameters:⁶⁹ Rashba energy E_R, Rashba momentum offset k_R and Rashba constant α_R = 2E_R/k_R. Due to the structural anisotropy of the S-XSSe system along the x and y directions, we distinguish between the corresponding Rashba constants (α_Rx and α_Ry). For S-ReSSe, we obtain E_Rx/E_Ry = 0.0639/0.0129 eV, k_Rx/k_Ry = 0.0648/0.0218 Å⁻¹ and α_Rx/α_Ry = 1.97/1.18 eV Å. The S-OsSSe monolayer shows even more pronounced anisotropy with E_Rx/E_Ry = 0.1159/0.0180 eV and k_Rx/k_Ry = 0.0625/0.0435 Å⁻¹, resulting in α_Rx/α_Ry = 3.71/0.83 eV Å (see Table S1 of the SI). Note that theses α_R values compare favorably with other prominent 2D Rashba systems, such as BiSb (2.3 eV Å),⁷⁰ WSeTe (0.52 eV Å)⁶⁶ and AlBi (2.77 eV Å).⁷¹ The large Rashba constants, particularly in S-OsSSe, suggest these materials could enable shorter spin channel lengths, making them promising candidates for spin field-effect transistors.⁷²

Fig. 4 A fraction of the band structures of (a) S-ReSSe and (b) S-OsSSe monolayers near the S-point.

3.5 Bilayer stacking

Bilayer stacking is an effective method to realize the coexistence of multiple physical properties, e.g., the coexistence of FE and ferrovalley in the YI₂ bilayer⁷³ and the coupling of FE, ferrovalley and magnetism in the GdI₂ bilayer.⁷⁴ Thus, we investigate further by double stacking. As shown in Fig. 5(a), we consider three stacking methods including parallel polarization (PP), antiparallel polarization (APP)-SS and APP-SeSe. Specifically, PP refers to stacking two monolayers with the same polarization direction; APP-SS or APP-SeSe refers to stacking two monolayers with opposite polarization directions, with the interlayer being S or Se, respectively. Meanwhile, considering the FM materials S-VSSe, S-CrSSe and S-MoSSe, as shown in Fig. 5(b), we constructed interlayer FM and AFM orders to confirm the magnetic ground state. In addition, four types of bilayer stacking are shown in Fig. 5(c): type AA is to translate the second layer by the first layer directly in the z direction. Furthermore, if the second layer is shifted in the xy plane by (a/2, 0), (0, b/2) and (a/2, b/2), we can obtain three other types of stacking, named AB, AC and AD, respectively. After optimizing the interlayer distance and atomic positions (including van der Waals interactions), we obtained the interlayer binding energy in different stacking types (see Tables S2–S4 of the SI).

Fig. 5 In the z-direction, schematic diagrams of (a) three polarities, including PP, APP-SS and APP-SeSe, (b) two magnetic orders, including interlayer FM and AFM, and (c) four stacking types in the xy plane, including AA, AB, AC and AD. The blue and red arrows in the figure indicate the direction of P and the direction of the magnetic moment of the magnetic atoms, respectively.

Binding energy calculations show that all 5 S-XSSe (X = Re, Os, V, Cr and Mo) bilayers tend to be AA-type under PP stacking, with S-VSSe showing interlayer AFM and S-CrSSe and S-MoSSe showing interlayer FM characteristics. To understand the microscopic mechanism of magnetic coupling, at the 2 × 2 × 1 supercell, we consider the simple spin Hamiltonian:

where H₀ is the ground state energy independent of the spin configurations. S_i (S_j, S_k) and S_i′ (S_j′, S_k′) denote the magnetic moments at the top and bottom layer sites, respectively. J_1‖, J_2‖, J_1⊥, J_2⊥ and J_3⊥ represent the nearest-neighbor (NN) intralayer, second-NN intralayer, NN interlayer, second-NN interlayer and third-NN interlayer X–X (X = V, Cr and Mo) exchange interactions, respectively (see Fig. S7 of the SI). The details of the calculation of Heisenberg exchange parameters based on the above spin Hamiltonian and total energy calculations employing DFT are shown in the SI and Table 2. For S-VSSe, S-CrSSe and S-MoSSe, the intralayer exchange is strong FM coupling (J_1‖, J_2‖ > 0), while the interlayer exchange interaction is jointly determined by J_1⊥, J_2⊥ and J_3⊥. Taking the S-VSSe bilayer as an example, although the NN interlayer is FM coupling (J_1⊥ = 4.07 meV), the second- and third-NN interlayer dominate the AFM coupling (J_2⊥ = −2.54 meV, J_3⊥ = −2.94 meV), resulting in an interlayer AFM state. Similarly, S-CrSSe and S-MoSSe can be judged to be interlayer FM coupling, which is consistent with our binding energy calculations.

Table 2 The calculated Heisenberg exchange parameters for the different stacking configurations of S-VSSe, S-CrSSe and S-MoSSe bilayers

S-XSSe	PP					APP-SS			APP-SeSe
	AA type					AD type			AD type
J (meV)	J 1‖	J 2‖	J 1⊥	J 2⊥	J 3⊥	J 1‖	J 2‖	J 1⊥	J 1‖	J 2‖	J 1⊥
S-VSSe	126.62	104.25	4.07	−2.54	−2.94	128.91	102.97	−1.73	126.86	109.41	−4.96
S-CrSSe	127.38	108.64	1.73	−0.70	1.43	121.62	109.88	2.54	141.18	96.41	−5.07
S-MoSSe	38.65	55.46	−4.18	2.97	3.27	33.72	58.51	1.44	43.05	52.27	0.68

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In addition, under APP (whether APP-SS or APP-SeSe) stacking, the 5 S-XSSe (X = Re, Os, V, Cr and Mo) bilayers preferred the AD-type. As for the interlayer magnetic coupling mode, we can likewise use the calculation of the exchange parameter to understand, at the 2 × 2 × 1 supercell (at this stacking type there exists only the NN interlayer, J_1⊥, see Fig. S8 of the SI), we consider the spin Hamiltonian:

The details of the calculation of Heisenberg exchange parameters are shown in the SI and Table 2. Obviously, under APP stacking, S-VSSe, S-CrSSe and S-MoSSe are strongly intralayer FM coupling (J_1‖, J_2‖ > 0), while the magnetic coupling mode of the interlayer depends mainly on the NN interlayer exchange parameter J_1⊥, whose positive and negative values represent the FM and AFM couplings, respectively. It is clear that, under both APP-SS and APP-SeSe stacking, S-VSSe exhibits interlayer AFM coupling, S-MoSSe exhibits interlayer FM coupling, and S-CrSSe exhibits FM and AFM coupling, respectively. This is consistent with the binding energy calculations in the SI. In addition, we calculated the specific heat of S-XSSe (X = V, Cr and Mo) bilayers for different stacking configurations using Monte Carlo simulations. As illustrated in Fig. S9 (SI), the magnetic transition temperatures (T_c) of the S-VSSe bilayer are 272 K, 258 K and 292 K under PP, APP-SS and APP-SeSe stacking, respectively. Similarly, the S-CrSSe bilayer exhibits T_c values of 1035 K, 1070 K and 1151 K, while the S-MoSSe bilayer shows T_c values of 478 K, 445 K and 446 K for the corresponding stacking orders. Notably, the magnetic transition temperatures in all bilayer systems are significantly enhanced compared to their monolayer counterparts, suggesting that weak interlayer coupling helps suppress thermal fluctuations, which is consistent with previous observations in bilayer CrI₃.⁷⁵

Furthermore, by GGA+U band calculations, we confirm the rich electronic nature of the optimal configuration of the S-XSSe bilayer system (see Fig. S10 and Table S5 of the SI). Under PP stacking, the out-of-plane polar electric field leads to charge transfer between the top and bottom layers, thus affecting the magnetic properties. By Bader charge analysis (see Table S6 of the SI), it was shown that a top to bottom layer charge transfer occurs in the S-VSSe, S-CrSSe and S-MoSSe bilayers. In general, the AFM material exhibits no spin splitting. However, our S-VSSe bilayer shows obvious spin splitting in the band structure, and the formation mechanism is similar to that of Janus AFM Mn₂ClF.⁷⁶ The magnetic moments of the bottom and top V atoms are 1.690 and −1.669 μ_B, and the net magnetic moment per unit cell is 0.013 μ_B, the system changes to a ferrimagnetic (FiM) state, and the intrinsic spin splitting facilitates practical device applications. Moreover, S-CrSSe and S-MoSSe are interlayer FM coupled, and the out-of-plane polar electric field leads to a slightly larger magnetic moment for the magnetic atoms in the bottom layer than in the top layer (3.103/3.092 and 2.233/2.232 μ_B for the bottom/top layer of S-CrSSe and S-MoSSe bilayers), and the band structure exhibits half-metallicity (spin-up is metallic and spin-down has band gaps of 1.79 and 1.24 eV for S-CrSSe and S-MoSSe bilayers, respectively). In addition, under APP-SS and APP-SeSe stacking, the out-of-plane polar electric field disappears, the S-VSSe bilayer is metallic, S-MoSSe is half-metallic (spin-up is metallic and spin-down has band gaps of 1.33 and 1.10 eV for APP-SS and APP-SeSe stacking, respectively), while S-CrSSe presents half-metallic (spin-up is metallic and spin-down has a band gap of 1.98 eV) and metallic properties, respectively. In addition, S-ReSSe and S-OsSSe remain metallic under three stacking modes.

Notably, for the S-VSSe bilayer, under APP-SS and APP-SeSe stacking, there is a Dirac cone at the Fermi level on the Γ–X path (see Fig. 6(b) and (c)), which implies high carrier mobility. Interestingly, as shown in Fig. 6(a), the out-of-plane polar electric field generated by the PP stacking results in the splitting of the spin-up and spin-down bands, which were shifted below and above the Fermi level, respectively, and leads to the creation of two Dirac cones (D1 and D2 in Fig. 6(a)). When SOC is considered, D1 and D2 under PP stacking open a gap of 47.0 and 55.4 meV, respectively (see Fig. 6d). Similarly, gaps of 46.8 and 52.9 meV are opened under APP-SS and APP-SeSe stacking, respectively (see Fig. 6(e) and (f)). By calculating the GGA+SOC band structures for the three stacking modes of the S-VSSe bilayer, we find that the energy bands do not show a large gap opening at U = 0 eV, as shown in Fig. S11 of the SI. Therefore, we believe that the band gap in the system is caused by the enhanced SOC effect induced by the correlated electrons.⁷⁷ From the band structures shown in Fig. 6, it can be observed that all three structures possess two pairs of degenerate points along the Γ–X path in the Brillouin zone. Therefore, its topological properties are worth further exploration.^78,79 However, as shown in Fig. S12 of the SI, our systematic calculations and analyses conclusively demonstrate that only the PP stacking S-VSSe bilayer exhibits nontrivial topological characteristics, while the APP-SS and APP-SeSe stacking are confirmed to be topologically trivial. Detailed calculations and the corresponding discussions are provided in the SI. In short, the S-XSSe bilayer system is rich in electronic structure and magnetism, combined with its spontaneous polarization, and thus it serve as a candidate for multifunctional electronic devices.

Fig. 6 Enlarged view of the band structure of the S-VSSe bilayer in the Γ–X path without SOC under (a) PP, (b) APP-SS and (c) APP-SeSe stacking and with SOC under (d) PP, (e) APP-SS and (f) APP-SeSe stacking. The blue and green lines represent spin-up and spin-down bands, respectively. The Fermi level was set to 0 eV. The arrow in the figure indicates the direction of the magnetic moment, and the size of the arrow indicates the magnitude of the magnetic moment.

4 Conclusion

Using first-principles calculations, we discovered a new class of two-dimensional Janus polar metals/half-metals, S-XSSe (X = Re, Os, V, Cr and Mo), with spontaneous polarization values ranging from 2.56 to 4.00 pC m⁻¹, all of which remain structurally stable above room temperature. Notably, S-VSSe, S-CrSSe and S-MoSSe are polar half-metals, with S-CrSSe exhibiting the potential for polarization switching (E_b = 0.87 eV), which can be further reduced by biaxial strain. We further investigated the coexistence mechanisms of polarity, FM and metallicity in three polar half-metals. We found that polarity arises from the electronegativity difference between S and Se, while metallicity and FM are provided by the central transition metal X atoms, indicating that the origins of polarity and magnetism are fundamentally distinct. Significant anisotropic Rashba effects are observed in polar metal S-ReSSe and S-OsSSe monolayers, with Rashba constants reaching up to 1.97 and 3.71 eV Å along the x-direction, and 1.18 and 0.83 eV Å along the y-direction, respectively. Additionally, after bilayer stacking, we revealed the microscopic mechanisms of rich magnetic properties through spin Hamiltonian and interlayer electron hopping. Meanwhile, the weak interlayer coupling enhances the magnetic transition temperature. Specifically, the interlayer AFM S-VSSe exhibits intriguing electronic structures: Dirac points appear at the Fermi level in both APP-SS and APP-SeSe stacking configurations, where the built-in electric fields cancel each other out. In contrast, under PP stacking, the built-in electric field induces pronounced spin splitting in the S-VSSe bilayer. This leads to the emergence of Dirac points both above and below the Fermi level, accompanied by nontrivial topological characteristics. Our work identifies a new class of polar metals/half-metals, which can serve as promising candidates for novel spintronic and multifunctional electronic devices.

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information: Planar-average charge density, polarization switching path and the energy profile, biaxial strain dependence of energy change and energy barrier, temperature dependence of the average ionic displacement, band structures with and without SOC of S-XSSe monolayers; Magnetic configurations of S-XSSe bilayers under AA and AD stacking; Monte Carlo simulation results of S-VSSe, S-CrSSe and S-MoSSe bilayers; GGA + U band structure of S-XSSe bilayers; Enlarged view of the GGA+SOC band structure of S-VSSe bilayer in the Γ-X path at U = 0 eV; Discussion on the topological properties of S-VSSe bilayer; Rashba constants of S-ReSSe and S-OsSSe monolayers; Binding energies and bandgap values of S-XSSe bilayers; Bader charge analysis and spin Hamiltonian model of S-XSSe (X = V, Cr and Mo) bilayer. See DOI: https://doi.org/10.1039/d5tc01948f.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant No. 12304165 and 12364038), the Natural Science Foundation of Inner Mongolia Autonomous Region (Grant No. 2023QN01003 and 2025MS01017), the Industrial Technology Innovation Projects of Inner Mongolia Academy of Science and Technology of China (Grant No. 2023JSYD01002), the Science and Technology Plan Projects of Inner Mongolia Autonomous Region of China (Grant No. 2023KYPT0012), the “Grass-land Talents” project of the Inner Mongolia Autonomous Region (Grant No. 21200-242920), the young science and technology talents cultivation project of Inner Mongolia University (21200-5223708), and the Startup Project of Inner Mongolia University (Grant No. 21200-5223733).

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