High Curie temperature and intrinsic ferromagnetic half-metallicity in two-dimensional Cr₃X₄ (X = S, Se, Te) nanosheets

Abstract

Two-dimensional (2D) intrinsic magnetic materials with room temperature ferromagnetism and complete spin-polarization are highly desirable for realizing advanced spintronic devices. However owing to the weak d–p–d super-exchange interaction, many recently realized 2D ferromagnetic (FM) materials present a very low Curie temperature (T_C). Here, based on first principles calculations, we report a series of Cr₃X₄ (X = S, Se, Te) monolayers, in which the coexistence of two oxidation states of Cr atoms results in double-exchange interaction and thereby enhances FM coupling greatly and improves T_C up to 370 K for Cr₃Se₄ and 460 K for Cr₃Te₄, respectively. Spin-polarized calculations further demonstrate that Cr₃Se₄ and Cr₃Te₄ monolayers are FM half-metals, which ideally could achieve 100% spin-polarized currents. Meanwhile, excellent dynamical and thermal stabilities are identified, and a possible synthetic strategy is proposed. Our work not only provides two competitive ferromagnetic candidate materials for nanoscale spintronic applications, but also shows that double-exchange interaction may be a way to realize 2D room temperature FM half-metals.

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Conceptual insights

Recently realized two-dimensional (2D) intrinsic ferromagnetic (FM) materials open the door for nanoscale spintronic applications. However, the development is highly limited by the incomplete spin-polarization and low Curie temperature (T_C) originating from the weak d–p–d super-exchange interaction. In this article, we present a series of 2D magnetic materials, namely, Cr₃X₄ (X = S, Se, Te) monolayers with two different oxidation-state Cr ions. A delocalized unpaired electron tends to hop from one Cr ion to another neighboring different oxidation-state Cr ion to effectively save energy, i.e. a double-exchange mechanism, which leads to enhanced FM coupling and high T_C (370 K for Cr₃Se₄ and 460 K for Cr₃Te₄). The new insights into strong FM coupling being attainable in monolayers via double-exchange interaction provide a feasible way for experiments to realize 2D room temperature ferromagnetism. More than that, Cr₃Se₄ and Cr₃Te₄ monolayers are found to be FM half-metals with relatively large spin gaps and 100% spin-polarization at the Fermi level. The thermal and mechanical stabilities have been further confirmed and a possible synthesis route is also proposed. The coexistence of these unique properties in one 2D material is rarely reported but crucial for advanced electronic and spintronic applications.

Introduction

Spintronics or spin-based electronics, using the spin of electrons, offers great opportunities for next generation information technology.^1,2 Two-dimensional (2D) ferromagnetic (FM) materials, as one of the most promising candidates for nanoscale spintronic devices, have been highly sought out.^3–5 While much effort has been made for inducing magnetism in 2D non-magnetic systems, success is still limited due to the induced magnetism lacking long-range FM coupling.^6–10 Very recently, ferromagnetic Cr₂Ge₂Te₆ bilayers,¹¹ CrI₃ monolayers,¹² and Fe₃GeTe₂ monolayers¹³ have been successfully synthesized, and open the door for achieving long-range FM order in 2D intrinsic ferromagnetic materials. However, the T_C of these intrinsic 2D FM materials is far below room temperature because of the weak ferromagnetic super-exchange interaction, which cannot meet most technology applications. In fact, to date, experimental observation of 2D intrinsic room temperature FM materials has never been reported.

Half-metals exhibit a metallic density of states at the Fermi level for one spin channel and a band gap for the other spin channel, showing enormous potential for an ideal electrode in spintronic devices. Half-metal electrodes not only can provide fully spin-polarized currents and large magnetoresistance (MR) in giant MR and tunneling MR devices, but also can realize efficient spin injection into semiconductors, opening the way for more applications, including spin-based transistors and spin Seebeck devices. Although numerous half-metallic bulk compounds have been identified,^14–19 and spintronic devices based on these materials have been realized,^20,21 experimental evidence for 2D half-metallic behavior is still rather scarce. Several recent studies showed that an applied electric field²² and edge modification^23–25 can make graphene nanoribbons create 100% spin-polarized currents. Nevertheless, device operation without an electric field is a fundamental objective of spintronics, and edge modifications will add additional challenges to the synthesis process. Therefore, 2D FM materials with high T_C and intrinsic half-metallicity remain an important frontier for discovery.

In this paper, for the first time, we propose a family of Cr₃X₄ (X = S, Se, Te) monolayers; in particular, the Cr₃Se₄ and Cr₃Te₄ monolayers exhibit large perpendicular magnetic moments, high-temperature ferromagnetism, 100% spin-polarization ratio, and sizable magnetocrystalline anisotropy energy (MAE), suitable for nanoscale spintronics applications. Notably, the Cr₃Se₄ and Cr₃Te₄ monolayers possess two different oxidation-state Cr atoms, which is different from many other 2D Cr-based FM materials, such as CrY₃ (Y = Cl, Br, I)^26,27 and CrMTe₃ (M = Si, Sn, Ge).^28,29 The coexistence of two oxidation states of Cr atoms results in a dominant interlayer double-exchange interaction and hence a strong interlayer FM coupling. Meanwhile, the intralayer nearest-neighboring Cr atoms also exhibit FM coupling behaviors because of Cr–X–Cr super-exchange interaction. As a result, the T_C values of the Cr₃Se₄ and Cr₃Te₄ monolayers estimated with Monte Carlo (MC) simulation based on the 2D Heisenberg Hamiltonian model are as high as 370 K and 460 K, respectively. In addition, our calculation results reveal that Cr₃Se₄ and Cr₃Te₄ monolayers exhibit intrinsic FM half-metallicity with relatively large spin gaps and have an out-of-plane easy axis. Their MAEs can be comparable to those of the Fe and Co monolayers deposited on Rh (111) and Rt (111) substrates.³⁰ Phonon dispersion curves and ab initio molecular dynamics (AIMD) simulation further demonstrate their dynamical and thermal stabilities. Experimentally, Cr₃Se₄ and Cr₃Te₄ monolayers are possibly obtained by reducing the thickness of CrSe and CrTe thin films recently synthesized^31–34 at the nanoscale.

Results and discussion

Magnetic coupling and electronic properties

Cr₃X₄ (X = S, Se, Te) monolayers are isostructural with each Fe₃S₄ sheet from the layered Fe₃S₄ bulk³⁵ (Fig. S1, ESI†). The top and side views of the Cr₃X₄ (X = S, Se, Te) monolayers are given in Fig. 1a, belonging to the P3̄m1 space group. Similar to double-transition-metal MXene monolayers (e.g. Hf₂MnC₂O₂, Hf₂VC₂O₂ and Hf₂VC₂F₂),^36,37 the Cr₃X₄ (X = S, Se, Te) monolayers are also composed of seven hexagonal layers, with X₁–Cr₁–X₂–Cr₂–X₂–Cr₁–X₁ atomic layers stacking along the z-direction. To study the magnetic properties of the Cr₃X₄ monolayers, we first study the magnetic coupling between the Cr atoms by considering six different spin orderings in a 2 × 2 × 1 supercell (Fig. S2, ESI†). The total energies of the ferromagnetic (FM), antiferromagnetic (AFM), and ferrimagnetic (FIM) configurations are calculated with the HSE06 functional and the relative energies between the FM and AFM/FIM configurations are listed in Table S1, ESI.† The results indicate that the Cr₃S₄ monolayer exhibits an FIM ground state, while the ground-states of the Cr₃Se₄ and Cr₃Te₄ monolayers are FM. The net ground-state magnetic moments are 2.00, 10.00 and 10.00 μ_B f.u.⁻¹ for Cr₃S₄, Cr₃Se₄, and Cr₃Te₄ respectively, indicating large spin polarization in these 2D Cr₃X₄ sheets. The major magnetisms in 2D Cr₃X₄ crystals are mainly located on the Cr atoms as illustrated by the spin charge density distribution in Fig. S2, ESI,† and the magnetic moments carried on the Cr₁ and Cr₂ atoms are listed in Table 1. From Table 1, we can find that the Cr atoms in Cr₃Te₄ have a slightly larger magnetic moment than those in Cr₃S₄ and Cr₃Se₄, which is likely due to the relatively weak Cr–Te bonds driving localization of the electrons (Fig. 1b).

Fig. 1 (a) Top and side views of Cr₃X₄ (X = S, Se, Te) monolayers with the unit cell highlighted. (b) The calculated electron localization function (ELF) of Cr₃X₄ (X = S, Se, Te) monolayers for the (110) plane. Each unit cell contains two inequivalent Cr atoms (Cr₁, Cr₂) and two inequivalent X atoms (X₁, X₂), respectively. Electronic band structures and projected density of states for Cr₃S₄ (c and d), Cr₃Se₄ (e and f), and Cr₃Te₄ (g and h) monolayers. “Up” and “down” arrows represent the spin-up and spin-down polarization, respectively. Fermi levels are set to zero.

Table 1 Magnetic moment on Cr₁ and Cr₂ atoms (M_Cr1/M_Cr2), total magnetic moment (M_tot), total energy difference between the ferromagnetic (FM) and antiferromagnetic (AFM) states (E_FM–AFM), exchange integral (J₁, J₂, J₃), and Curie temperature (T_C) for Cr₃X₄ (X = S, Se, Te) monolayers

Materials	M Cr1 (μB)	M Cr2 (μB)	M tot (μB f.u.^−1)	E FM–AFM (meV f.u.^−1)	J 1 (meV)	J 2 (meV)	J 3 (meV)	T C (K)
Cr3S4	2.94	−3.53	2.00	146.6	—	—	—	—
Cr3Se4	3.24	3.66	10.00	−58.6	12.12	10.89	−0.88	370
Cr3Te4	3.45	3.80	10.00	−218.6	30.60	9.24	2.18	460

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The fundamental character of the FM ordering in Cr₃X₄ monolayers at low temperature can be further understood by the MAE, which is directly related to the thermal stability of magnetic data storage. Here, spin orbit coupling (SOC) calculations at the HSE06 level are performed to accurately obtain the relative magnetization stability in-plane and out-of-plane. As summarized in Table S2, ESI,† the easy magnetization axes of the Cr₃X₄ monolayers are along the out-of-plane (001) direction. The calculated MAE of the Cr₃Se₄ and Cr₃Te₄ monolayers are 0.10 and 0.15 meV per Cr, respectively, which are comparable to the values of the Fe and Co monolayers deposited on Rh (111) and Rt (111) substrates.³⁰ These sizable MAEs make Cr₃Se₄ and Cr₃Te₄ monolayers good candidates for magnetic devices. Having studied the magnetic properties of the Cr₃X₄ monolayers, we further turn to their ground-state electronic structures. The calculated ground-state spin-dependent electronic structures and corresponding ground-state spin configurations of 2D Cr₃X₄ unit cells are displayed in Fig. 1 and Fig. S3, ESI.† Obviously, the Cr₃S₄ monolayer is an FIM semiconductor with a band gap of 0.25 eV, while the Cr₃Se₄ and Cr₃Te₄ monolayers exhibit FM half-metallicity, i.e., metallic character for the majority-spin electrons while semiconducting nature for the minority-spin electrons. This means that the majority-spin electrons make a dominant contribution to the charge transport, and the electric current is 100% spin-polarized. As shown in Fig. 1, the spin gaps between the bottommost unoccupied and topmost occupied minority-spin bands are 2.27 eV for the Cr₃Se₄ monolayer and 1.59 eV for the Cr₃Te₄ monolayer, respectively, which are relatively large to effectively block the thermally induced spin–flip transition. The total and partial density of states shows a predominant contribution of Cr-d and X-p orbitals to the majority-spin channel and a strong hybridization between the Cr-d and X-p states. This confirms that the important indirect exchange interactions between the two Cr 3d orbitals are mediated by the neighboring X np (n = 3/4/5) orbitals. In addition, the Cr atom in 2D Cr₃X₄ crystals is under D_3d symmetry, and therefore its five Cr 3d orbitals split into a single a (d_z²) orbital and two 2-fold degenerate e₁ (d_xz + d_yz) and e₂ (d_xy + d_x²−y²) orbitals, which is in good agreement with the calculated results displayed in Fig. 1d, f, and h and Fig. S4, ESI.† Besides, we find that the PDOS of Cr₂ atoms shows a much larger peak near the Fermi level than that of Cr₁ atoms, indicating that the Cr₂ atoms make a main contribution to the half-metallicity in Cr₃X₄ monolayers.

As studied above, the ground-state of the Cr₃Se₄ and Cr₃Te₄ monolayers both exhibit FM coupling between the two nearest-neighboring intralayer Cr atoms and interlayer Cr atoms, while that of the Cr₃S₄ monolayer shows interlayer AFM coupling (Fig. S3, ESI†). Actually, these ground-state intralayer and interlayer coupling behaviors in Cr₃X₄ can be understood by the competition between the direct-exchange and super-exchange/double-exchange interaction. The distances between the two intralayer nearest-neighboring Cr₁ atoms (Table S3, ESI†) are relatively large, therefore, the direct-exchange interaction (usually favoring an AFM arrangement, Fig. 2a) between these intralayer Cr₁ atoms in 2D Cr₃X₄ crystals is weak. Meanwhile, we notice from Table S3, ESI,† that the θ₁ (Cr₁–X₁–Cr₁) of 2D Cr₃X₄ crystals are all close to 90°. As explained by the well-known Goodenough–Kanamori–Anderson (GKA) rule, the super-exchange interaction (Fig. 2a) usually favors FM ordering for the systems with a Cr–X–Cr bond angle of 90°. As a result, the ground-state FM coupling behavior between the two nearest-neighboring intralayer Cr₁ atoms can be easily understood. Furthermore, the two nearest-neighboring Cr atoms along the z-direction, unlike the intralayer Cr atoms, are not equivalent. The deformation charge density (Fig. 2b) shows that the Cr₁ atom loses more electrons than the Cr₂ atom, which leads to the two Cr atoms presenting different valence states. Under an ionic model, each Cr₁ (or Cr₂) atom will donate 3e (or 2e) to neighboring X. X anions belong to relatively weak ligands, resulting in a high-spin configuration, and therefore the arrangement of electrons are localized on the Cr₁³⁺ and Cr₂²⁺ ions as displayed in Fig. 2a.

Fig. 2 (a) Illustrations of the Cr–Cr direct-exchange, Cr–X–Cr super-exchange, and Cr–X–Cr double-exchange interactions in Cr₃X₄ (X = S, Se, Te) monolayers. (b) The deformation charge density distribution of Cr₃X₄ (X = S, Se, Te) monolayers. The isosurface values are set to 0.009, 0.005, and 0.003 e Bohr⁻³ for Cr₃S₄, Cr₃Se₄, and Cr₃Te₄ monolayers, respectively.

The double-exchange model is superficially similar to the super-exchange model. However, super-exchange interaction generally occurs between two metal atoms with the same valence, while double-exchange interaction occurs only when one metal atom has an extra electron compared to the other.^38,39 The interlayer coupling in the Cr₃X₄ monolayers belongs to the latter case. The X atom connecting the nearest-neighboring Cr₁³⁺ and Cr₂²⁺ ions, gives up its spin-up or spin-down electron to Cr₁³⁺, and then its vacant orbital can be filled by an electron from Cr₂²⁺. At the end of the double-exchange process, the electron moves between the neighboring Cr ions retaining its spin. This electron movement from one to another will be facilitated more easily if the electron does not have to change spin direction. The ability to hop (to delocalize) reduces the kinetic energy, and thus leads to FM alignment of the nearest-neighboring Cr ions along the z-direction. Moreover, we notice that the magnetic moments on Cr₁ and Cr₂ atoms, shown in Table 1, slightly deviate from the ionic picture (Cr₁³⁺, 3 μ_B; Cr₂²⁺, 4 μ_B). It is also likely due to the delocalized hopping electron shared by the two nearest-neighboring Cr ions along the z-direction during the double-exchange process. In addition, the distances of the interlayer nearest-neighboring Cr ions in 2D Cr₃S₄ are only 3.02 Å and much smaller than those in 2D Cr₃Se₄ (3.19 Å) and 2D Cr₃Te₄ (3.26 Å). This suggests that the interlayer direct-exchange interaction in 2D Cr₃S₄ is much stronger than that in 2D Cr₃Se₄ and 2D Cr₃Te₄. Meanwhile, Cr–S bonds are stronger than Cr–Se/Te bonds (see Fig. 1 and Fig. 4), which makes the electron hopping process (interlayer double-exchange process) in a Cr₃S₄ monolayer more easily hindered than that in Cr₃Se₄ and Cr₃Te₄ monolayers. As a result, direct-exchange interaction plays a major role in the interlayer magnetic coupling behaviors in a Cr₃S₄ monolayer, and thus the interlayer nearest-neighboring Cr ions exhibit AFM ordering. By contrast, a stronger interlayer double-exchange interaction leads to the interlayer FM coupling in Cr₃Se₄ and Cr₃Te₄ monolayers.

Curie temperatures of the single-layer FM Cr₃Se₄ and Cr₃Te₄

Next, we calculate exchange interaction parameters and then use these exchange parameters to predict the T_C of single-layer Cr₃Se₄ and Cr₃Te₄. Based on the Heisenberg model, the spin Hamiltonian of Cr₃X₄ monolayers can be considered as , where J₁ and J₂/J₃ are defined as interlayer and intralayer exchange coupling parameters (Fig. 3a), A is anisotropy energy parameter, and S^z_l,i is the spin vector of each atom (more details see Fig. S5, ESI†). The interlayer (J₁) and intralayer (J₂, J₃) exchange coupling parameters for Cr₃X₄ monolayers are estimated by computing the energy differences between FM configurations and different AFM/FIM configurations at the HSE06 functional level. The large and positive interlayer exchange coupling parameter J₁ and intralayer exchange coupling parameter J₂ fail to make the AFM coupling occur, and thus lead to a robust FM order in Cr₃Se₄ and Cr₃Te₄ monolayers (Table 1). Furthermore, it is important to note that their J₁ originated from double-exchange interaction is larger than the J₂ derived from super-exchange interaction. This indicates that double-exchange interaction makes a dominant contribution to the robust FM arrangement of 2D Cr₃Se₄ and Cr₃Te₄ crystals.

Fig. 3 (a) Magnetic structure for estimating the exchange-interaction constants of Cr₃Se₄ and Cr₃Te₄ monolayers. J₁ is an interlayer exchange parameter and J₂ and J₃ are intralayer exchange parameters. Dark blue and gray balls represent Cr and Se/Te atoms, respectively. (b) The magnetic moment (black data), magnetic susceptibility χ (red data), and specific heat C_V (magenta) as functions of temperature for Cr₃Se₄ and Cr₃Te₄ monolayers by means of Monte Carlo simulations on the basis of a 2D Heisenberg Hamiltonian model. The peak positions of χ and C_V are marked with blue dashed lines.

The T_C of FM Cr₃Se₄ and Cr₃Te₄ monolayers are computed by carrying out MC simulations with a 50 × 50 × 1 supercell. In the MC simulations, 2 × 10⁵ MC steps for each temperature are employed and spins on all magnetic sites flip randomly. The calculated magnetic moments, magnetic susceptibility (χ), and specific heat capacity (C_v) as functions of temperature for Cr₃Se₄ and Cr₃Te₄ monolayers are illustrated in Fig. 3b. We notice that the FM orderings with large magnetic moments (∼5 μ_B f.u.⁻¹ for Cr₃Se₄ monolayer; ∼7 μ_B f.u.⁻¹ for Cr₃Te₄ monolayer) could be preserved at room temperature (300 K). Furthermore, under the exact solution to the spin Hamiltonians, the T_C can be accurately obtained from the peak positions of χ and C_v, and is 370 K for the Cr₃Se₄ monolayer and 460 K for the Cr₃Te₄ monolayer. These values are significantly higher than those of bilayer Cr₂Ge₂Te₆ (20 K),¹¹ monolayer CrI₃ (45 K),¹² and even monolayer VSe₂/van der Waals heterostructures.⁴⁰ Meanwhile, we have estimated the T_C (∼43 K) of monolayer CrI₃ using the same approach (Fig. S6, ESI†), which agrees well with the experimental measurement one.¹² We further calculate the relative energies of the FM/AFM/FIM states as a function of strain and carrier concentration. The FM states are always lower than the AFM/FIM states and the relative energies change little under different strain conditions and carrier concentrations (see Fig. S7, ESI†), indicating that the FM states are quite robust under external perturbations. This indicates that the Cr₃Se₄ and Cr₃Te₄ monolayers are potential candidates for spintronic applications.

Stability and practical synthesis approaches

To assess the structural stability of Cr₃X₄ (X = S, Se, Te) monolayers, we first calculate their cohesive energies by the definition of E_coh = (E_Cr3X4 − 3 × E_Cr − 4 × E_X)/7, where E_Cr3X4 is the total energy of E_Cr3X4 monolayers, E_Cr is the energy of an isolated Cr atom, and E_X is the energy of the isolated X atom. The calculated cohesive energies are −7.78, −6.65, and −5.96 eV per atom for Cr₃S₄, Cr₃Se₄, and Cr₃Te₄ monolayers, respectively. These values are comparable to that of graphene (−7.85 eV per atom), which suggests their possible stability. Phonon spectra computed by density functional perturbation theory (DFPT) implemented in the Phonopy program⁴¹ indicate that the Cr₃X₄ monolayers are dynamically stable since no imaginary phonon mode appears in the whole Brillouin zone (Fig. 4a). Meanwhile, the acoustic vibration modes become softer from Cr₃S₄ to Cr₃Te₄ due to the gradually weakening bond strength from Cr–S to Cr–Te (Fig. 4a and Fig. S2b, ESI†). We further perform ab initio molecular dynamics (AIMD) simulations to assess the thermal stability by employing a 4 × 4 × 1 supercell containing 112 atoms to reduce lattice translational constraints. Fig. 4b displays the structural snapshots of Cr₃X₄ monolayers at 5 ps at 800 K. Neither geometric reconstruction nor broken bonds have been discovered during the entire process, suggesting that MXenes-like Cr₃X₄ monolayers have high thermal stability.

Fig. 4 (a) Phonon spectra and (b) ab initio molecular dynamics (AIMD) snapshots (top and side views) at the temperature of 800 K after 5 ps evolution for Cr₃X₄ (X = S, Se and Te) monolayers. (c) Comparison of crystal frames between CrX thin films and Cr₃X₄ monolayers. Grey balls represent X (X = S, Se, Te).

Considering that Cr₃Se₄ and Cr₃Te₄ monolayers exhibit a promising spintronic application and excellent stability, practical synthetic methods are expected to be attractive to experiments. Note that the crystal structures of Cr₃X₄ monolayers are similar to the layers produced by periodically interrupting the basic CrX framework (Fig. 4c). Moreover, CrSe and CrTe thin films have been successfully produced via chemical bath deposition on substrates (commercial glass) and molecular beam epitaxy on SrTiO₃ (111) substrates, respectively.^31–34 Therefore, it is possible to obtain 2D Cr₃Se₄ and Cr₃Te₄ sheets by reducing the thickness of CrSe and CrTe thin films to the nanoscale (Fig. 4c), which has recently been used to obtain a MnSe₂ monolayer.⁴² Alternatively, it is also possible to obtain 2D Cr₃Se₄ and Cr₃Te₄ sheets by the method of chemical vapor deposition, which has been extensively used for obtaining 2D materials on selected substrates.⁴³

In summary, we have predicted a series of stable 2D Cr₃X₄ (X = S, Se, Te) crystals with novel electronic and magnetic properties utilizing first-principles calculations. A Cr₃S₄ monolayer is found to be a ferrimagnetic semiconductor while Cr₃Se₄ and Cr₃Te₄ monolayers are ferromagnetic half-metals with relatively large spin gaps. Importantly, 2D Cr₃Se₄ and Cr₃Te₄ crystals also exhibit large magnetic moments at room temperature and high T_C (370 K for Cr₃Se₄ and 460 K for Cr₃Te₄). The further analysis shows that the Cr atoms in Cr₃Se₄ and Cr₃Te₄ monolayers present two different oxidation states and a delocalized unpaired electron tends to hop between the two neighboring different oxidation-state Cr ions to retain its spin. This magnetic coupling mechanism (double-exchange process) can effectively save energy, and thus facilitates the realization of high-temperature ferromagnetism. Moreover, phonon spectrum calculations and AIMD simulations confirm that the Cr₃Se₄ and Cr₃Te₄ monolayers have excellent dynamic and thermal stabilities. Thus, 2D Cr₃Se₄ and Cr₃Te₄ crystals are ideal candidates for nanoscale spintronic devices. Additionally, our study suggests that strong FM coupling is attainable via a double-exchange process and perhaps experiments should pay more attention to the double-exchange systems for achieving 2D intrinsic room temperature FM half-metals.

Computational methods

All calculations were performed on the basis of spin-polarized density functional theory (DFT) as implemented in the Vienna ab initio simulation (VASP) package⁴⁴ with projector-augmented wave (PAW) potentials.⁴⁵ To exactly deal with the strongly correlated effect on 3d electrons of Cr atoms and give accurate electronic structures, the HSE06 hybrid functional⁴⁶ was employed. The plane-wave cutoff energy of 400 eV for basis and a vacuum of 15 Å for eliminating adjacent layer interactions were used. The total-energy convergence criterion of 1 × 10⁻⁶ eV and the force convergence criterion of −10⁻² eV Å⁻¹ were chosen in the complete relaxations including the lattice constants and atomic positions.

Phonon dispersion was calculated on the basis of DFPT as embedded in the Phonopy program.⁴¹ AIMD simulations were carried out to evaluate the thermal stability of Cr₃X₄ (X = S, Se, Te) monolayers at the temperature of 800 K. AIMD simulation in the NVT ensemble lasts for 5 ps with a time step of 1 fs. The temperature is controlled by using the Nose–Hoover method.⁴⁷

Conflicts of interest

The authors declare no competing financial interests.

Acknowledgements

This work was supported by the National Key Research and Development Program of China (2017YFA0204800), the Natural Science Funds of China (21525311, 21773027, and 51435003), the Jiangsu 333 project (BRA2016353), the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1773), and the Fundamental Research Funds for the Central Universities of China. The authors are thankful for the computational resources at the SEU and National Supercomputing Center in Tianjin.

Notes and references

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9nh00038k

‡ These authors contributed equally to this work.

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