Janus Mn₂OS monolayers with piezoelectric altermagnetism and their application in photocatalytic water splitting

Abstract

Janus monolayers exhibit versatility due to structural symmetry breaks. Using a first-principles design approach, we construct a Janus monolayer of Mn₂OS, which crystallizes in the P_4mm space group (No. 99) with broken spatial inversion symmetry. The constructed Mn₂OS monolayer was identified as an altermagnetic semiconductor with good energetic, dynamical, and mechanical stability. It exhibits zero net magnetization but shows momentum-dependent spin-splitting. The estimated Néel temperature (T_N) surpasses room temperature. Strikingly, unidirectional magnetic anisotropy was rarely observed in a two-dimensional altermagnetic single crystal with a magnetic anisotropy energy (MAE) of 215 µeV per Mn. Horizontal mirror symmetry breaking results in an inherent electric field and piezoelectricity, with an out-of-plane piezoelectric coefficient of e₃₁ of −4.1 pC m⁻¹ and d₃₁ of −0.074 pm V⁻¹. These results demonstrate the high potential of the Janus monolayer for applications in spintronics and piezoelectrics. Furthermore, the absorption spectra reveal that the monolayer exhibits outstanding optical absorption across the entire visible spectrum. The alignment of the CBM and VBM energies with the redox potentials of water indicates that Mn₂OS has the potential to be used as a photocatalyst in water splitting reactions.

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1. Introduction

Magnetic semiconductors, in which both the charge and the spin freedom of electrons are manipulated for faster, simultaneous information processing and storage of data, have been considered as the ideal materials for spintronic devices.^1–3 Among them, ferromagnetic (FM) semiconductors are the most attractive spintronic materials for the next generation of spintronics applications due to their advantages of macroscopic spontaneous magnetization and spin-polarized electronic structures. However, some inherent disadvantages of ferromagnets limit their application in spintronic devices. For example, ferromagnets usually trigger strong stray fields in devices and are very vulnerable to external magnetic fields, which can adversely affect the stability of data storage.⁴ Additionally, the response speed of magnetic domain reversal in FM materials cannot keep up with the frequency of external fields, especially when the frequency reaches up to THz levels. Unlike ferromagnets, although antiferromagnets have zero net magnetization, which makes it difficult for antiferromagnets to work in integrated devices,⁵ antiferromagnets are free from parasitic stray fields and magnetic field perturbation^4,6 and show ultrafast dynamics. Recently, a new magnetic phase known as altermagnetism^7–12 has emerged. Unlike the conventional antiferromagnets, altermagnets are characterized by nonrelativistic spin-split band structures in reciprocal space with a zero net magnetization. The altermagnets therefore combine both the advantages of ferromagnetic and antiferromagnetic materials. The unique physical properties of the altermagnets allow them to be used in a wide range of applications.

Following the experimental observation of intrinsic ferromagnetism in the two-dimensional (2D) CrI₃¹³ and Cr₂Ge₂Te₆¹⁴ materials, the scientific community has been working tirelessly to discover new, high-temperature-stable, 2D magnetic materials with novel properties, due to their potential applications in spintronic nanodevices.^15,16 Although several 2D altermagnets^17–22 have been predicted so far, none have yet been experimentally verified. In magnetic space group theory, opposite-spin sublattices in an altermagnet must be connected by a rotation or mirror operation, rather than by a translation or inversion operation, as is the case for opposite-spin sublattices in conventional antiferromagnets. The guiding principle for designing altermagnets is to break the joint symmetry of space inversion and time-reversal or time-reversal symmetry.²³ One conventional method for achieving altermagnets is to construct 2D Janus materials.²⁴ For example, in the case of the monolayer Mn₂S₂ (D_4h point group, Fig. 1(a)), if one of the two S layers is fully replaced by an O atom, the Janus monolayer Mn₂OS (D_4v point group) is formed. Consequently, inversion and mirror symmetry are broken and the opposite-spin sublattices are connected only through a 90° rotation. Symmetry-breaking can create many new chemical and physical properties in addition to the transition of an antiferromagnet into an altermagnet. 2D Janus materials exhibit broken out-of-plane symmetry, enabling significant out-of-plane piezoelectricity, piezovalley, and piezomagnetism.^18,25,26

Fig. 1 (a) Top and side views of the Mn₂OS monolayer. (b) Planar distributions of the electron localization function in the [100]/[010] plane. Three possible anti-magnetic states: (c) AFM1, (d) AFM2, and (e) AFM3, and a ferrimagnetic state (f) FIM. The yellow and green contours of spin charge densities represent the directions of the spin-up (↑) and spin-down (↓) states, respectively.

In this study, we construct a Janus monolayer of Mn₂OS with broken spatial inversion symmetry. Systematic investigations of its geometric structure, stability, electronic structure, magnetism, piezoelectricity, and optical properties reveal that the monolayer is a polar altermagnetic system with a T_N of 312 K. The suitable band-edge positions and appreciable optical absorption coefficient in the visible region make it promising for photocatalytic water splitting.

2. Computational methodology

In this work, spin-polarized first-principles calculations are carried out based on density functional theory (DFT), as implemented in the Vienna Ab initio Simulation Package (VASP)^27,28 using the projector augmented wave²⁹ method with a cutoff energy of 500 eV. The exchange–correlation functional within the formation of the generalized gradient approximation (GGA) parameterized by Perdew, Burke, and Ernzerhof (PBE)³⁰ was used. The convergence criteria were set at 10⁻⁶ eV and 0.01 eV Å⁻¹, respectively, for energy and force. The vacuum thickness was set to 20 Å. The Brillouin zone was sampled using a gamma-centered Monkhorst–Pack³¹k-mesh with a density of 15 × 15 × 1. The GGA+U method introduced by Dudarev et al.³² is applied to the d orbitals of Mn atoms, with the Hubbard term U = 4.1 eV and the exchange parameter J = 1.0 eV.³³ Phonon dispersions were calculated by the finite displacement method embedded in the Phonopy code.³⁴ The ab initio molecular dynamics (AIMD) simulations were conducted in the NVT ensemble and lasted for 10 ps with a time step of 1.0 fs. The temperatures were controlled using a Nosé–Hoover³⁵ thermostat during the simulations. The T_N was estimated using Monte Carlo (MC) simulations on a 50 × 50 × 1 supercell containing 5000 Mn atoms, as implemented in the ESpinS package.³⁶ The VASPKIT program was used to process the VASP data.³⁷ The optical properties were calculated within the random-phase approximation (RPA) and the Bethe−Salpeter equation (BSE)³⁸ on top of single-shot GW (G₀W₀) eigenvalues and wave functions. The BSE calculations were performed by using the ten lowest conduction bands and the ten highest valence bands.

3. Results and discussion

3.1. Structure and stability

Fig. 1(a) shows the fully optimized geometric structures of the primitive cell of the Mn₂OS monolayer, which contains two manganese, one oxygen and one sulfur atoms. Similar to the recently identified 2D Tm₂XX' altermagnets,²¹ the Janus Mn₂OS monolayer is built up with three atomic layers as O–Mn–S, in which one Mn layer is sandwiched with O and S layers, exhibiting a tetragonal structure with in-plane C_4v symmetry and P_4mm space group (no. 99). As a consequence, Mn₂OS breaks the horizontal mirror and inversion symmetries, implying a built-in dipole moment and piezoelectricity. Each Mn is tetra-coordinated with four S atoms, forming a tetrahedral structure. The calculated lattice parameter for Mn₂OS is a = b = 4.048 Å at the PBE level. The Mn–O and Mn–S bonds are of 2.167 and 2.454 Å, respectively. The electron localization functions in Fig. 1(b) indicate that both the Mn–O and Mn–S bonds are mainly ionic in nature. The bonding strength can be evaluated from the cohesive energy E_coh = (E_Mn2OS − 2E_Mn – E₀ − E_S)/4, where E_Mn2OS represents the total energy of a Mn₂OS unit cell, and E_O and E_S are the energies of the isolated Mn, O, and S atoms, respectively. The estimated E_coh = −3.92 eV per atom, which is much lower than zero and thus indicates a strong bonding network.

The calculated phonon dispersion curves in Fig. 2(a) exhibit no imaginary frequencies within the Brillouin zone, suggesting that the Mn₂OS monolayer is dynamically stable. To determine the thermal stability, AIMD simulation at 350 K with a time scale of 10 fs is carried out, and Fig. 2(b) shows the variation of the total energy as a function of dynamic steps with the corresponding screenshot taken at the end of the dynamic simulations. Although the monolayer exhibits slight structural distortion, no significant bond breakage is observed, indicating that the Janus monolayer is thermally stable at room temperature. Following the energy–strain relationship ³⁹ for a 2D orthogonal lattice, the three independent linear elastic constants C₁₁, C₁₂, and C₆₆ are calculated by parabolic curve fitting. The obtained C_ij are 39.18, 16.00 and 25.21 N m⁻¹ for C₁₁, C₁₂, and C₆₆, respectively, which satisfy the Born–Huang criteria C₁₁ > 0, C₆₆ > 0, and C₁₁ > |C₁₂|⁴⁰ for a 2D orthorhombic crystal system, confirming the mechanical stability of the Mn₂OS monolayer.

Fig. 2 (a) Phonon spectrum of the Mn₂OS monolayer. (b) The fluctuation of the total energy during the AIMD simulation at a temperature of 350 K. The inset shows a snapshot of the Mn₂OS monolayer at the end of the AIMD simulation.

3.2. Electronic structure and magnetic properties

Before we can discuss the electronic structure, we must first clarify the magnetic ground state of the material. To this end, we consider several possible magnetic configurations, as shown in Fig. 1. Table 1 presents the calculated energies relative to the AFM1 state, based on GGA+U and HSE06 hybrid functions. Both methods reveal that the AFM1 configuration has the lowest energy and the FM configuration has the highest energy. In the AFM1 configuration, the spins at each Mn site are antiparallel to those at its four nearest Mn sites, suggesting that Mn₂OS has G-type antiferromagnetic (AFM) ordering. Fig. 3 shows the spin-polarized electronic band structures for the Mn₂OS monolayer. Both the GGA+U and HSE06 schemes predicted that the monolayer is a magnetic semiconductor with indirect band gaps of 2.07 and 3.07 eV at the GGA+U and HSE06 levels, respectively. A comparison of Fig. 3(a) and (c) shows that the spin–orbit coupling (SOC) effect does not significantly affect the electronic structure and that there is no spin splitting caused by the SOC effect. According to Kramers’ theorem, in a conventional AFM material with PT symmetry, the spin-up and spin-down energy bands overlap at each momentum point in the Brillouin zone. Fig. 3(a) and (c) display that the band dispersions along the high-symmetry path Γ–M remain spin degenerate. In contrast, the bands along Γ–X–M and Γ–Y–M show identical dispersion but spin-splitting, exhibiting altermagnet behavior. The spin splitting at the high-symmetry X/Y point is significantly large at 0.23 eV. The momentum-dependent spin-splitting band arises from the anisotropic crystal fields of the ligand O and S atoms. As shown in Fig. 1(c), the Mn atoms are located on two sets of opposite-spin sublattices, and the two corresponding crystal fields can only be connected by a 90° rotation. In other words, breaking of space inversion symmetry (P) lifts the Kramers degeneracy, resulting in momentum-dependent spin-splitting.

Table 1 The summarized values of the energies (eV per unit cell) relative to the AFM1 ground state, exchange coupling parameters J_i (meV), and T_N (K) obtained from GGA+U and HSE06 schemes

Method	ΔEFM	ΔEAFM2	ΔEAFM3	ΔEAFM4	J 1	J ^O2	J ^S2	T N (K)
GGA+U	0.476	0.142	0.232	0.204	−59.5	−16.8	−7.2	320
HSE06	0.604	0.174	0.449	0.222	−75.5	−39.8	7.9	410

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Fig. 3 Electronic band structures of the Mn₂OS monolayer at (a) PBE, (b) HSE06, and (c) GGA+U + SOC levels, respectively. The red and black lines indicate the up-spin and down-spin channels, respectively.

In the G-type AFM state, the local magnetic moments on the two sublattices of Mn sites are 4.402 and −4.402µ_B, respectively, resulting in zero net magnetization. The local magnetic moments mainly originated from the unpaired Mn:3d electrons, as illustrated by the projected density of states (PDOS) shown in Fig. 4a. In order to qualitatively analyze the origin of the magnetic moment, we may simply assume that Mn ions reside in the center of the tetrahedron constructed by O and S. The tetrahedron crystal field splits the Mn:3d states into a set of lower-lying triply degenerate states (t_2g: d_xy + d_yz + d_zx) and one set of higher-lying doubly degenerate states (e_g: d_z² + d_x²−y²). According to the Hund rule, the electrons at the Mn site adopt a high-spin configuration because the spin splitting of Mn:3d is much larger than the crystal field splitting. Thus, the nominal Mn²⁺ ion carries a magnetic moment of 5.0µ_B (Fig. 4(a)). The in-plane electron orbital Mn:d_xy hybridizes with the anion's p_x and p_y orbitals (Fig. 4(b)), resulting in a decrease in the local magnetic moment from 5.0 to 4.4µ_B.

Fig. 4 Orbital projected DOS for (a) e_g of Mn-3d, (b) T_2g of Mn-3d, (c) O-2p, and (d) S-3p. The inset shows the local magnetic moments on the Mn site.

The Mermin–Wagner theorem⁴¹ suggests that long-range isotropic magnetism in two dimensions is fragile and is often destroyed by thermal fluctuations. Robust magnetic anisotropy in 2D materials stabilizes intrinsic long-range magnetic ordering by removing the Mermin–Wagner restriction. As demonstrated in Fig. 5, the spatial distribution of magnetic anisotropy energy (MAE) indicates that the monolayer exhibits the lowest energy along the z-axis, thereby suggesting that the easy axis corresponds to the out-of-plane magnetization direction. It is noteworthy that the positive direction along the x-axis corresponds to the secondary easy axis, while the opposite direction along the x-axis is the hard axis. This observation demonstrates a distinctive unidirectional magnetic anisotropy in the x-axis direction, which now predominantly occurs at the interface between FM and AFM systems,^42,43 and rarely in two-dimensional materials.⁴⁴ However, this phenomenon is not an isolated incident. The present study demonstrates that this phenomenon also exists in Mn₂XI (X = P, As, Sb, Bi) with a Mn₂OS-like structure. Such uniaxial magnetic anisotropy can be attributed to exchange anisotropy.⁴⁵ The MAE, which is defined as the energy difference between magnetization along the (100)/(1̄00) and (001) directions, is 17/232 µeV per Mn. We found that the distribution of MAE satisfies the equation: E_MAE = E₀ + K₁ sin² θ + K₂ sin⁴ θ − K_u cos θ in the xy plane, as evidenced by the corresponding fitted curve presented in Fig. 5. Here, K₁ and K₂ are system-dependent anisotropy constants for cubic or hexagonal lattices, which were estimated to be −8.53, 3.93 µeV, respectively, and K_u is the unidirectional anisotropic exchange parameter, which is 108.2 µeV. Clearly, the MAE exhibits a strong dependence on the unidirectional anisotropic exchange parameter. Exchange anisotropy exerts a unidirectional “bias” on uniaxial magnetic anisotropy, breaking its energy symmetry and cooperating with it to greatly enhance the effective anisotropy, coercivity and stability of the magnetization state of the system.

Fig. 5 Angular dependence of the MAE of the Mn₂OS monolayer with the direction of magnetization lying on the xy planes and the distribution of MAE over the whole space.

To estimate the Néel temperature T_N for the predicted Mn₂OS monolayers, the exchange coupling parameters were extracted according to the classical spin model Hamiltonian expressed as

where J₁ and J₂ are the first and second neighboring exchange parameters, respectively, and is the spin vector at the i/j site and is treated as 1 in our calculations. The energy of each unit cell of these four magnetic configurations (Fig. 1) can be written as follows:
where J^O₂ and J^S₂ correspond to the exchange parameters that measure the strength of the interaction established by the next-nearest Mn atoms via O and S, respectively. The estimated exchange coupling parameters are presented in Table 1, obtained by GGA+U and HSE06 schemes, revealing antiferromagnetic characteristics. According to the results of the Monte Carlo simulations, the specific heat C_v as a function of temperature is shown in Fig. 6(a). The calculated T_N is 320 and 410 K at the GGA+U and HSE06 levels, exceeding room temperature and comparable to that of the altermagnet Mn₂PSe (408 K).²¹

Fig. 6 (a) Heat capacity as a function of temperature. (b) The average electrostatic potential along the z axis. The inset shows the difference charge density with electron depletion shown in yellow and electron accumulation in cyan. (c) Energy variations based on the switching step number in the NEB calculations for the Mn₂OS system. (d) The computed absorption coefficient of the Janus Mn₂OS monolayer using the G₀W₀ + BSE scheme.

3.3. Piezoelectricity

Breaking the horizontal mirror symmetry and the distinct electronegativity of the O and S atoms induces an out-of-plane spontaneous polarization. Fig. 6(b) illustrates the electrostatic potential difference (ΔΦ) of 1.64 eV between the O- and S-terminated surfaces. The difference in charge density, which is defined as the difference in charge density between the Mn₂OS monolayer and the superposition of the charge densities of the neutral constituent atoms (Mn, O and S), indicates the presence of a built-in electric field pointing from the Mn sub-layer to the S sub-layer and an electric field pointing from the Mn sub-layer to the O sub-layer. The resultant electric field is directed from the O layer to the S layer. Bader charge-population analysis shows that the charge transfers from the Mn sublayer to the O and S layers are 1.330 and 1.075 electrons per unit cell, respectively. Because the distance between the Mn and the S layer (1.355 Å) is larger than that between the Mn and the S layer (0.632 Å), the calculated net dipole moment is 4.19 Debye (0.0872 eÅ) with the direction from S pointing to the O layer. The estimated spontaneous polarization P_s, i.e., the electric dipole polarization per unit area, is 3.38 pC m⁻¹. At the same time, the P_s is estimated to be 4.14 pC m⁻¹ based on Berry-phase calculation. The spontaneous polarization suggests that the Mn₂OS Janus monolayer might exhibit out-of-plane ferroelectricity. To determine the feasibility of ferroelectric switching, we calculate the energy barrier height based on the nudged elastic band (NEB)⁴⁶ approach. Fig. 6(c) shows the ferroelectric polarization switching pathway and energy evolution. The calculated energy barrier is 368 meV per atom, indicating that the ferroelectric switching is generally impracticable in practice.

The relaxed ion piezoelectric stress tensor e_ijk can be obtained as the sum of ionic and electronic contributions: , where P_i and ε_ijk are respectively the polarization vector and the strain tensor, and superscripts elec and ion respectively denote the electronic and ionic contributions to the total e_ijk. We calculate the e_ijk using the density functional perturbation theory (DFPT), implemented in VASP. For 2D materials, the third-rank tensor e_ijk is reduced to e_ij. The piezoelectric strain tensor d_ij and stress tensor e_ij are related by the Voigt notation e_ij = d_ikC_kj, where C_jk is the elastic stiffness tensor. The relationship between them can be expressed in matrix form as follows:

The Janus Mn₂OS monolayer has only one independent piezoelectric component e₃₁, meaning that in-plane strain in a uniaxial direction induces out-of-plane piezoelectric polarization. The obtained e₃₁ is −4.1 pC m⁻¹, and the d₃₁ is −0.074 pm V⁻¹. The e₃₁/d₃₁ values are larger than those in Janus transition metal dichalcogenides (0.010–0.038 pC m⁻¹)/(0.007–0.030 pm V⁻¹).⁴⁷

3.4 Photocatalytic water splitting

For photocatalytic applications in water splitting, the high adsorption coefficient in the visible light range is an important factor, as solar energy is primarily concentrated in this region. The calculated absorption coefficient of the Mn₂OS monolayer is shown in Fig. 6(d). For the random-phase approximation (RPA) scheme, the threshold energy occurs at approximately 2.6 eV, which corresponds to the electronic band gap of 2.66 eV obtained using the G₀W₀ method. Therefore, the RPA scheme describes the optical absorption spectrum of the Mn₂OS monolayer well. It is seen that the absorption edge obtained from the BSE scheme shows a substantial red-shift of 0.8 eV compared to the RPA scheme. This is due to the exciton effect included through solving the BSE, resulting in a red shift of the absorption spectrum. The redshift corresponds to an exciton binding energy of 0.8 eV. The exciton effect in Mn₂OS results in an absorption spectrum that almost covers the entire visible and ultraviolet regions, greatly enhancing the optical absorption performance. Moreover, the absorption coefficient reaches up to 1.8 × 10⁵ and 3.5 × 10⁵cm⁻¹ in the visible and ultraviolet regions, respectively. Therefore, the Mn₂OS monolayer is an excellent material for absorbing solar energy and has potential applications in photocatalytic water splitting to produce hydrogen.

From an application perspective, materials with low effective carrier mass and high mobility are expected. The mobility can be deduced from the expression⁴⁸

where e is the electron charge, ħ is the reduced Planck constant, C is the in-plane elastic modulus, k_B is the Boltzmann constant, and T is the temperature and is set to 300 K in this work. If the band structures near the VBM and CBM are parabolic, the carrier effective mass m* will be deduced from the equation m* = ħ²/(∂²E(k)/∂k²), where E(k) is the energy, and k is the momentum. By quadratic fitting of the energy band curvature around the VBM and CBM, the effective mass of hole and electron can be obtained, respectively. The details of the deformation potential for electron (E_1e) or hole (E_1h) states were described in our previous work.⁴⁹ The mobilities of the electron and hole are only 5.95 and 1.66 cm² V⁻¹ S⁻¹, respectively.

In addition to their excellent ability to absorb sunlight, the reasonable band structure and suitable electron transfer mechanisms are also key factors determining the photocatalytic efficiency of materials for water splitting. As previously stated, the intrinsic built-in electric field in Mn₂OS, with direction from the O to the S side, enhances the separation and transfer efficiency of the photogenerated electron–hole pairs. As an ideal photocatalyst, the position of the conduction band minimum (CBM) should be higher than the water reduction potential of H⁺/H₂ (−4.44 eV vs. vacuum level), while its valence band maximum (VBM) should be lower than the water oxidation potential of H₂O/O₂ (−5.67 eV). Fig. 7(a) illustrates the band edge positions of the VBM and CBM with respect to the vacuum level at pH = 0. It is observed that the VBM is below the oxidation potential by 1.21 eV, enabling the oxygen evolution reaction to proceed. As illustrated in Fig. 7(b), the photo-induced carriers are separated by the intrinsic built-in electric field, and the holes and electrons are transferred to the S and O surfaces respectively. At the S surface, the oxygen evolution reaction occurs (2H₂O + 4h⁺ → O₂ +4H⁺). On the other hand, the intrinsic built-in electric field results in a realignment of energy levels through the bending of energy bands.^50,51 Consequently, the CBM of Janus Mn₂OS is below the reduction potential by 2.26 eV, which can provide considerable driving force for the hydrogen evolution reaction (4H₂O + 4e⁻ → 2H₂ + 4OH⁻). It is evident that Mn₂OS demonstrates excellent visible light absorption, suitable band edge position and band gap, the ability to effectively separate photogenerated electron–hole pairs, and a large reaction surface area. This makes it a potential catalyst for visible light photocatalytic applications.

Fig. 7 Schematic plots of (a) energy levels for the photocatalyst system of Mn₂OS monolayer and (b) the photocatalytic process.

4. Summary

In summary, we investigated the fundamental properties of the Mn₂OS monolayer by using DFT calculations. Analysis of the binding energy, phonon spectrum, elastic constants, and AIMD simulations demonstrated that the structure is dynamically, mechanically, and thermally stable. The monolayer is predicted to be an altermagnetic semiconductor, exhibiting zero net magnetization, but with momentum-dependent spin-splitting, a Néel temperature above room temperature and unique uniaxial magnetic anisotropy with an MAE of 215 µeV per Mn. The monolayer exhibits strong optical absorption across the entire visible and ultraviolet spectrum. The band edge positions of the monolayer straddle the redox potentials of water. Additionally, horizontal mirror symmetry breaking results in an inherent electric field that effectively improves the separation and migration of photoinduced carriers. These outstanding characteristics make the monolayer a promising material for potential applications in spintronic and optoelectronic nanodevices and as a photocatalyst for water splitting.

Author contributions

Wen-Zhi Xiao: supervision, project administration, software, conceptualization, data curation, investigation, formal analysis, writing – original draft, methodology, funding acquisition, validation, and writing – review & editing.

Conflicts of interest

There are no conflicts to declare.

Data availability

All data generated or analyzed during this study are included in this published article.

Acknowledgements

This work was supported by the Scientific Research Fund of Hunan Provincial Education Department No. 20A106 and the Hunan Provincial Natural Science Foundation under Grant No. 2021JJ30179.

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