Enhancement of Curie Temperature under Built-in Electric Field in Multi-Functional Janus Vanadium Dichalcogenides^*

Since the first one-atom-thick material graphene was discovered in 2004,^[1] two-dimensional (2D) materials have attracted a great deal of attention. Among these 2D materials, transition-metal dichalcogenides (TMDs) have been widely studied for their excellent properties, including a wide range of energy bandgap, tunable electronic and optical properties, etc.^[2] TMDs have the potential to be applied in the field of spintronics and high-performance sensors. In addition, functionalized 2D TMDs can generate new characteristics, for example, Li-exfoliated MoS₂ possesses enhanced catalytic activity.^[3–10]

Multiferroic materials are considered as a special class of functional compounds with two or more ferroic properties, including ferromagnetism, ferroelasticity and ferroelectricity.^[11,12] Coupling between different ferroic orderings can create new phenomena, which has great potential for applications in electronic devices.^[11–15] For example, in BiFeO₃, the multiferroics can be induced by defect engineering or the magnetic proximity effect.^[16] However, the coupling effect between different magnetic orderings is so weak, since the origins of the multiferroics are diverse from each other. Ferromagnetism originates mainly from unpaired electrons on partially filled d/f orbitals on transition metals, ferroelectricity is derived from empty d/f orbitals, while ferroelasticity originates from the lattice distortion. This means that ferroelectricity and ferromagnetism are derived from the space- and time-symmetry breakings, respectively, while ferroelasticity demands neither. Therefore, to date, there have been very few multiferroic materials that can be maintained at room temperature.

Recently in 2017, the first intrinsic ferromagnetic two-dimensional CrI₃ and CrGeTe₃ were successfully fabricated for the first time,^[17,18] bringing the study of ferromagnetism from 3D down to 2D. The curie temperature (T_c) of the monolayered CrI₃ and CrGeTe₃ are 45 and 30 K, respectively. The virtual exchange between a half-occupied and an empty orbital in a super-exchange system results in the weak FM coupling, which makes a low Curie temperature in monolayered CrI₃ and Cr₂Ge₂Te₆.^[19] After that, the Curie temperature is enhanced to 100K found in three-layered Fe₃GeTe₂,^[20] and further be modulated up to the room temperature. Very recently, monolayered VSe₂ is reported to have a high Curie temperature,^[21–25] in which the T_c of electrochemical exfoliation 1T-VSe₂ monolayer even reaches up to 470 K (1 or 2 stands for how many layers in the unit cell, T stands for trigonal).^[21] However, the absence of ferromagnetism in 1T-VSe₂ is still controversial. Molecular beam epitaxial VSe₂ monolayer is found no long-range ferromagnetic order on highly oriented pyrolytic graphite substrates.^[26,27] There could be two reasons for this being controversial: on the one hand, synthesis methods of VSe₂ sheets may affect the geometric structure, and thus further affect the ferromagnetic order; on the other hand, the charge-density-wave phase transition found in 1T-VSe₂ at low temperatures opens an energy band gap at the Fermi surface, resulting in the disappearance of the ferromagnetic ordering.^[23,27,28] Recently, Wang et al. predicted the Curie temperature of GdI₂ monolayer to survive near room temperature, arousing intense discussion.^[29]

In this letter, we predict a new family of multiferroic 2D materials of Janus VXX' (X/X' = S, Se, Te) monolayers. Firstly, contributed by the V 3d orbitals, Janus VXX' monolayers have intrinsic ferromagnetism. In particular, the computed T_c of 2H-VSeTe (H, hexagonal) is up to 150 K, which is higher than the other 2D ferromagnetic materials.^[17,18,20] The enhancement of the Curie temperature is due to the built-in electric field effect, resulting from the distinct electronegativity of chalcogen atoms on both sides of the V layer. Secondly, piezoelectricity is found in Janus VXX' monolayers. Due to the different electronegativity values of chalcogen atoms on both sides of the V layer, net intrinsic electric field is formed in the vertical direction, thus V d, Se p and Te p orbitals significantly affect the energy levels of bands. This result is also confirmed by the analysis of partial density of states (PDOS). Thirdly, the reversible ferroelastic strains of 1T- and 2H-VSSe monolayer reach up to 73% and the overall ferroelastic switching energy barriers are 0.26 eV/atom, which is 4 times lower than black phosphorus (73%, 0.99 eV/atom).^[30] In consequence, Janus vanadium dichalcogenides VXX' are multi-functional with intrinsic ferromagnetism, effective ferroelasticity and high piezoelectricity.

All first-principle calculations were performed within the frame of density functional theory (DFT) using the Vienna ab initio simulation package (VASP) based on the projected augmented wave (PAW) method^[31–33] The electron configurations of V, S, Se and Te are 3p⁶3d⁴4s¹, 3s²3p⁴, 4s²4p⁴ and 5s²5p⁴, respectively. The plane-wave cut-off energy was set to 450 eV and a k-point grid of 10 × 10 × 1 was used. The force on each atom was smaller than 1 meV/Å. The Heyd–Scuseria–Ernzerhof hybrid functional (HSE06 method) was carried out on the calculation of band structure, partial density of states and piezoelectric polarization.^[34] The 'Berry phase' theory was used in the piezoelectric polarization calculations (LCALCPOL = TRUE).^[35,36] The phonon-dispersion relations was calculated by employing the density-functional perturbation theory (DFPT) and PHONOPY,^[37] and a 2 × 2 × 1 supercell was applied in the DFPT calculations. The nudged-elastic-band (NEB) method was used in the ferroelastic calculation. The Mcsolver software was used in the Monte Carlo simulations.^[38] A 16 × 16 lattice was used for all the simulations. In the piezoelectric polarization calculations, a uniaxial strain was applied along the armchair direction of the orthorhombic supercell (Fig. 1), the constant of zigzag direction and the atom positions were totally relaxed.

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As is shown in Fig. 1, Janus VXX' monolayers consist of a layer of V atoms sandwiched between two different chalcogen atoms (S, Se and Te). Janus VXX' monolayers have 1T and 2H phases. The monolayered 2H-Janus VXX' is an ABA stack, while the monolayered 1T-Janus VXX' is an ABC stack. Table S1 and Fig. S1 in the supplementary material shows the lattice constant, bond length, bond angle of the Janus VXX' monolayers. The constants of 2H-VS₂, 2H-VSe₂ and 2H-VSSe are consistent with the previous theoretical reports.^[39,40] The lattice constant of Janus monolayered VXX' is between those of monolayered VX₂ and VX'₂. In addition, we find that as the number of electrons increases, the bond length and lattice constant also change from small to large in turn. Because of the different chalcogen atoms on both sides, the Janus monolayered VXX' has the space-inversion symmetry in vertical direction. The phonon spectrum calculations of all VXX' monolayers shown in Fig. S2 indicate that they are all dynamically stable and can exist as freestanding 2D monolayers.

Figure 2 shows the band structures of 2H and 1T Janus monolayered VXX'. In order to accurately calculate the electronic structures of the Janus vanadium dichalcogenides, we decide to use the Heyd–Scuseria–Ernzerhof hybrid functional (HSE06) method. All of the three 2H-VXX' monolayers are indirect semiconductors, while the 1T-VXX' is metal. In Fig. 2(a), the 2H-VSSe monolayer has a band gap of 1.01 eV. The spin-down (red) and spin-up (black) band gaps are 1.47 eV and 1.01 eV, respectively. The valence band maximum (VBM) and the conduction band minimum (CBM) are both spin-up bands. Based on the PDOS in Fig. 3(a), we can conclude that V d_z² and S p_x orbitals contribute to the VBM in VSSe, and V d_z² orbital also contributes to the CBM in VSSe.

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In Fig. 2(b), 2H-VSeTe has a band gap of 0.47 eV, which is smaller than that of 2H-VSSe. The spin-up and spin-down gaps are 0.87 eV and 0.88 eV. In contrast to the 2H-VSSe, 2H-VSeTe has a staggered band gap: the VBM is the spin-down band located at the Γ point, while the CBM is the spin-up band located at the K point. Meanwhile, the VBM and CBM are separately contributed by spin-up and spin-down electrons. The valence bands and conductive bands shift down simultaneously. Based on the PDOS in Fig. 3(b), we conclude that the VBM is contributed by V d_xy, d_yz, d_yz, d_x²−y² and Te p_x, p_y orbitals, while the CBM is contributed by V d_z²orbital. In contrast to the 2H-VSTe [Fig. 3(c)], the spin-up band near the Fermi level in 2H-VSeTe is contributed by V d_z² and Se p_z orbitals, which opens the spin-up band gap.

Figure 2(c) shows the band structure of 2H-VSTe without and with SOC. The band gap of 2H-VSTe is 0.36 eV, which is smaller than 2H-VSSe and 2H-VSeTe. The spin-up and spin-down gaps are 0.36 eV and 0.82 eV. Here 2H-VSTe has a flat band-gap: the CBM, located at the K point, is the spin-up band contributed by V d_z² orbital, while the VBM, located at Γ point, is composed of spin-up and spin-down bands, which is contributed by V d_z² orbital and V d_xy, d_yz, d_xz, d_x² − y², Te p_y, p_x orbitals, respectively. Due to the contribution of V d_xy, d_yz, d_yz, d_x² − y² and Te p_x, p_y orbitals, 2H-VSeTe and 2H-VSTe have the similar energy level in spin-down bands.

Based on the band structure and PDOS analysis of the Janus monolayered VXX', different electronegativity values of chalcogen atoms greatly affect the both spin-up and spin-down bands. Obviously, the 2H-VSSe is the nested band-gap contributed by V d and S p orbitals, while 2H-VSeTe and 2H-VSTe are staggered band gap and flat band gap, respectively, contributed by V d and Te p orbitals.

It can be noticed from the band structure and PDOS that the spin-up and spin-down bands of VXX' are separate, which represents the ferromagnetism order in VXX' monolayers. Then we turned to confirm the ferromagnetism order at higher temperature. The Curie temperature T_c can determine how high temperature the ferromagnetic material can remain in ferromagnetism order. Based on the magnetic anisotropy energy (MAE) calculation (Table S3), the MAE of monolayered 2H-VSeTe reaches −1.094 meV/V atom. We decide to use the Monte Carlo simulations with the Ising model to calculate the Curie temperatures. The Hamiltonian of the Ising model is

$$\begin{eqnarray*}&&H=-\displaystyle \sum _{i,j}J\cdot {\boldsymbol s}_{i}\cdot {\boldsymbol s}_{j},\end{eqnarray*}$$

where J is the exchange coupling constant, s is the spin of the V atom. The Curie temperature of CrI₃ calculated by Monte Carlo simulations with the Ising model is 60 K, which is close to the experimental result (Fig. S3).^[41] As shown in Fig. 4, based on the PBE method, the Curie temperature of 2H-VSeTe monolayer is ∼150 K, which is far higher than the temperature of liquid nitrogen (77 K), indicating that 2H-VSeTe monolayer can maintain ferromagnetic order in liquid nitrogen environment. Meanwhile, the Curie temperature of 1T-VSeTe is much higher than those of 1T-VSe₂ and 1T-VTe₂. The electronegativity of chalcogen atoms greatly affects the exchange coupling constants (Table S2), resulting in the enhancement of Curie temperature in Janus VXX' monolayers. Although the Curie temperatures of VXX' monolayers are lower than room temperature, T_c's of 2H-VSSe (∼100 K), VSeTe (∼150 K), VSTe (∼100 K) and 1T-VSeTe (∼100 K), VSTe (∼80 K) are still higher than those of CrI₃ and CrGeTe₃ monolayers,^[17,18] indicating that monolayered VXX' materials are promising ultrathin nanomaterials for potential applications in spintronics.

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Since the 2D-VXX' unit cells are all rhombic, we have chosen a rectangular supercell [Fig. 5(a)]. Compared with the rhombic unit cell, the orthorhombic supercell contains six atoms, which form a hexagon (Fig. 1). State I define the directions along armchair and zigzag as a and b. The lattice constant a along the armchair direction is larger than b along the zigzag direction. In Fig. 5(a), state II is the intermediate state of square unit cell. In order to obtain an accurate intermediate structure, we first use the method of molecular thermodynamics to generate a square unit cell with a lattice constant of a' = b' at a constant temperature of 1000 K for 2 ps, and then perform structural optimization. State III can be regarded as the structure in which state I is rotated by 90°, the reversible ferroelastic switching strain is defined as $\left(\frac{b}{a}-1\right)\times 100 \%$.

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In order to explore the evolution of configuration under uniaxial strain, the energy as a function of the uniaxial strain was calculated by the NEB method,^[42] as shown in Figs. 5(a)–5(f). Outstandingly, an impressive high reversible stress 73% for the ferroelastic switching is observed for 2H-VSSe monolayer. In the starting state I and the end state III, three S/Se/Te atoms are bonded with the interlayer V atoms, respectively; while in the transition state II, opposite trends are found for the S–V and Se–V bonds: the bond length of S–V is increased from 2.35 to 2.40 Å, while that of Se–V is decreased from 2.51 to 2.48 Å. As a whole, the configuration energy first increases by 0.26 eV/atom, then drops by 0.07 eV/atom. Similarly, under further stress, the configuration changes from state II to III. Overall, a significant ferroelastic switching effect is found in the VSSe monolayer with a moderate energy barrier of 0.26 eV/atom and a very large reversible ferroelastic strain of 73%, which is 4 times smaller than black phosphorus (73%, 0.99 eV/atom),^[30] and much higher than those in other classical ferroelastic materials (0.5%–3%).^[43,44] For other Janus VXX' monolayers, the energy barrier and ferroelastic strain are similar to the 2H-VSSe monolayer. Thus, Janus VXX' monolayers are suggested to produce stronger ferroelastic switching signals than phosphorene, which is crucial for designing sensors and data-storage devices.^[44]

The third-rank piezoelectric tensor is e_ijk = ∂ P_i/∂ ε_jk, where P is electrical polarization and ε is strain. In order to accurately calculate the electrical polarization, we decide to use the Heyd–Scuseria–Ernzerhof hybrid functional (HSE06 method) to apply a uniaxial strain along the armchair direction of the orthorhombic supercells. As shown in Fig. 6, like other TMDs, Janus 2H-VXX' monolayers have the in-plane piezoelectric coefficients. The in-plane piezoelectric coefficients e₁₁ of the Janus monolayered 2H-VSSe, VSeTe and VSTe are 3.59 × 10⁻¹⁰, 2.9 × 10⁻¹⁰ and 2.0 × 10⁻¹⁰ C/m, respectively. The in-plane piezoelectric coefficient e₁₁ of the Janus 2H-VSeTe monolayer is equal to MoS₂ monolayer (2.9 × 10⁻¹⁰ C/m).^[45,46]

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In contrast to monolayered MoS₂, 2H-VXX' monolayers have the intrinsic out-of-plane piezoelectricity in the out-of-plane direction since the different chalcogen atoms on both sides of the V atom on Janus 2H-VXX' monolayers, forming the built-in intrinsic electric field, break the space-inversion symmetry. The out-of-plane piezoelectric coefficients e₁₃ of the VSSe, VSeTe and VSTe monolayers are calculated to be 1.31 × 10⁻¹⁰ C/m, 0.11 × 10⁻¹⁰ C/m and 0.36 × 10⁻¹⁰ C/m, respectively. Although the out-of-plane piezoelectric coefficients e₁₃ of VSSe, VSeTe and VSTe are slightly smaller than the in-plane piezoelectric coefficients, 2D materials which have the out-of-plane piezoelectric coefficient e₁₃ are scare. Furthermore, as a piezoelectric material, it is easier to measure the out-of-plane piezoelectric polarization by applying in-plane strain in the experiment.

In summary, we have predicted a class of multiferroic Janus VXX' (X/X' = S, Se, Te) monolayers with ferromagnetism, ferroelasticity and piezoelectricity. The high Curie temperature of 2H-VSeTe reaches 150 K, which is higher than 2D ferromagnetic materials in the experiments. The built-in electric field effect, resulting from the distinct electronegativity of chalcogen atoms on both the sides of the V layer, enhances the curie temperature of Janus 1T-VXX' monolayers. Compared to other classical ferroelastic materials (0.5–3%), the reversible ferroelastic strain of 73% makes VXX' monolayers have potential applications in sensors and data-storage devices. In addition, due to the space-inversion symmetry, 2H-VXX' monolayers have the out-of-plane and in-plane piezoelectricity, indicating that 2H-VXX' monolayers can be piezoelectric sensors in the future. Furthermore, due to different electronegativity values of S, Se, Te atoms, the Janus 2H-VXX' monolayers have three types of band gaps. The Janus VXX' monolayers calculated by first-principles have many attractive properties, we hope that this study will facilitate further experimental work in this field.