Quantum spin Hall states in MX₂ (M = Ru, Os; X = As, Sb) monolayers

Abstract

The quantum spin Hall (QSH) effect has attracted extensive research interest due to its great promise in topological quantum computing and novel low-energy electronic devices. Here, using first-principles calculations, we find that MX₂ (M = Ru and Os; X = As and Sb) monolayers are 2D topological insulators (TIs). The spin–orbit coupling (SOC) band gaps for RuAs₂, RuSb₂, OsAs₂, and OsSb₂ monolayers are predicted to be 80, 131, 118, and 221 meV, respectively. Additionally, the nontrivial topological states are further confirmed by calculating the topological invariant and the appearance of gapless edge states. More interestingly, for RuSb₂ and OsSb₂ monolayers, the position of node points in energy can be effectively tuned by applying in-plane strain. Our results consistently indicate that all MX₂ monolayers can serve as an effective platform for achieving the room-temperature QSH effect.

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1 Introduction

The band structure topology of insulating and semimetallic materials has garnered considerable attention in condensed matter physics and materials science.^1–4 The unique topological features, such as conductive channels on surfaces or edges protected by various symmetries, hold great promise for applications in topological quantum computing and innovative low-energy electronic devices.^5–7 By combining both an insulating bulk and a metallic surface, topological insulators (TIs) with quantum spin Hall (QSH) states have received special attention.^8–10 With the protection of time-reversal symmetry, these states are marked by an odd number of surface-state bands and non-zero topological invariants. Besides, the robust conductive surface states of TIs are closely linked to band inversions and remain resilient to scattering by nonmagnetic impurities.^11–13

Graphene was the first system to be proposed as a QSH insulator via the spin-orbital coupling (SOC) effect.¹⁴ Unfortunately, the SOC effect of graphene is too weak, resulting in a too small bulk band gap on the order of 10⁻³ meV.¹⁵ Two dimensional (2D) HgTe/CdTe quantum wells were predicted and subsequently experimentally verified to realize the QSH effect.^16,17 On the other hand, three-dimensional (3D) topological insulators, including Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃, were discovered, sparking great research enthusiasm due to their unique electronic properties and ease of experimental synthesis.¹⁸ Numerous novel physical phenomena, including the QSH effect and the quantum anomalous Hall (QAH) effect, have been observed in this material system. For instance, the QAH effect has been realized in Cr-doped (Bi; Sb)₂Te₃ and V-doped (Bi; Sb)₂Te₃ films.^19,20

2D TIs have demonstrated a fundamental interplay between the electronic structure and topology, offering a promising platform for applications in spintronics.^21–30 There is great interest in the discovery of new 2D TIs, as this provides more options for experiments and applications.^31,32 Consequently, intensive efforts have been devoted to the search for 2D materials with novel topological properties and a large amount of 2D TIs have been proposed theoretically.^33–37 However, these topological materials proposed in theoretical works have rarely been confirmed experimentally. The low temperature required for QSHs, which arises from the small bulk gap, limits the practical applications of 2D TIs. Therefore, searching and designing 2D TIs with large energy gaps is essential to overcome thermal fluctuations and thus advance the realistic applications of QSH insulators.

Transition-metal dichalcogenides (TMDCs) have been studied intensively in recent years due to their diverse electronic and magnetic properties.^38–40 For instance, the 1T′ structures of TMDC monolayers have been confirmed as quantum spin Hall (QSH) insulators.⁴¹ Following this discovery, other phases of TMDCs have also been predicted to exhibit the QSH state with large nontrivial gaps.^42,43 On the other hand, transition metal dipnictides (TMDPs) have also been found to exhibit nontrivial topological properties. For example, single crystals of NbAs₂ and TaAs₂ have been synthesized and identified as potential candidates for topological semimetals (TSMs).⁴⁴ Additionally, WP₂ and MoP₂ have been recognized as type-II Weyl semimetals with exceptionally high conductivity.⁴⁵ However, much less attention has been paid to 2D TMDPs, especially with respect to their topological properties.

In this study, based on first-principles calculations, we investigated the electronic and topological properties of MX₂ (M = Ru and Os; X = As and Sb) monolayers. These monolayers are confirmed to be dynamically and thermally stable by the phonon spectrum calculations and ab initio molecular dynamics (MD) simulations. The nontrivial band inversion occurs around the Γ point even when the SOC effect is neglected. These MX₂ monolayers become topological insulators because the band gap opens upon the inclusion of the SOC effect. The calculated topological invariant and the appearance of gapless edge states indicate that these MX₂ monolayers are promising candidates for realizing the QSH effect. More interestingly, for RuAs₂ and OsSb₂ monolayers, appropriate uniaxial tension strain can be applied to tune the position of node points in energy and further enlarge their nontrivial gaps.

2 Computational methods

Our calculations are carried out using the projector augmented wave (PAW) method within density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP).^46,47 Perdew–Burke–Ernzerhof (PBE) of generalized gradient approximation (GGA) is adopted for the exchange–correlation potential.⁴⁸ The cutoff energy of the plane-wave basis is set to 500 eV, and the convergence threshold of total energy is set to 10⁻⁶ eV. The atomic positions and lattice parameters are optimized until the forces are smaller than 0.001 eV Å⁻¹. A 20 Å thick vacuum layer is used to avoid interactions between the nearest slabs. The Brillouin zone is sampled with a 19 × 9 × 1 centered k point grid for geometry optimization and the computations on electronic properties,⁴⁹ which guarantees the convergency of topological properties. As the PBE functional underestimates the bulk band gaps of TIs, the HSE06 method is also adopted for attaining an accurate result.⁵⁰ The phonon spectra are calculated using the density functional perturbation theory implemented in the PHONOPY package.⁵¹ A 4 × 2 × 1 supercell is used for ab initio molecular dynamics (AIMD) simulations in the NVT ensemble at 300 K and 500 K for 5 ps.⁵² The tight-binding models are developed with the help of the WANNIER90 code on the basis of density functional theory (DFT) calculations.⁵³

3 Results and discussion

All possible 2D planar structures of MX₂ (M = Ru and Os; X = As and Sb) are searched by the particle swarm optimization (PSO) structure search method.^54,55 We focus our structure search on MX₂ monolayers with fixed atomic stoichiometry for M and X atoms (1 : 2). The population size and the number of generations are set to be 40 and 30, respectively.⁵⁶ Simulating cells containing two formula units are considered for MX₂ monolayers. Here, the most energetically stable monolayers are taken into account for further calculations, which belong to the monoclinic system with a space group of P2₁/M (1T′ phase). Without a loss of generality, we take the RuAs₂ monolayer as an example. The structure of the RuAs₂ monolayer can be viewed as an As bilayer intercalated by a Ru layer, forming the edge-shared RuAs₆ octahedron. The primitive unit cell contains two formula units of RuAs₂, as shown in Fig. 1a. The corresponding lattice parameters of optimized structures are listed in Table S1 (ESI†). It is indicated that the lattice parameters increase as the composition elements change from As to Sb and from Ru to Os. The equilibrium lattice constants are found to be a = 3.062 Å and b = 6.360 Å, a = 3.273 Å and b = 6.648 Å, a = 3.093 Å and b = 6.358 Å, and a = 3.285 Å and b = 6.635 Å for RuAs₂, RuSb₂, OsAs₂, and OsSb₂ monolayers, respectively.

Fig. 1 (a) Top and side views of the crystal structure and (b) 2D Brillouin zone with specific symmetry points for the RuAs₂ monolayer in the 1T′ phase. Top and side views of crystal structures for the RuSb₂ monolayer in the (c) 2H and (d) 1T phase.

Cohesive energy can be employed to access the energetic stability and experimental feasibility of predicted 2D materials.⁵⁷ Here, the cohesive energy of the MX₂ monolayer is calculated according to this expression:

E_C = E_MX2 − E_M − 2E_X

where E_MX2 represents the total energies of MX₂, while E_M and E_X are the energies of Ru(Os) and As(Sb) atoms, respectively. According to this definition, a more negative value of cohesive energy implies greater energetic stability of this monolayer.

With this conception, we then identify the cohesive energies of these four predicted MX₂ monolayers. As shown in Table S2 (ESI†), the resultant cohesive energies are −6.17, −5.70, −6.69, and −6.16 eV per atom for RuAs₂, RuSb₂, OsAs₂, and OsSb₂ monolayers, respectively, indicating high energetic stability. Thus, the high synthetic possibilities of the MX₂ monolayers are expected to enhance their practical applications. As a comparison, the crystal structures of 1T and 2H phases are shown in Fig. 1c and d. In addition, the cohesive energies of 1T and 2H MX₂ monolayers are also calculated and given in Table S2 (ESI†). We find that the cohesive energies of the 2H phases are slightly higher than those of 1T′ phases for all MX₂ monolayers. However, the 1T phases are significantly unstable compared to the 1T′ and 2H phases, indicating the low synthetic possibility.

Next, we examine the dynamic stability of MX₂ monolayers, and calculate the phonon spectra along the high-symmetry lines using the DFPT method, as plotted in Fig. 2a and Fig. S1 (ESI†). The absence of any appreciable imaginary frequency is found in the full Brillouin zone, confirming the dynamic stability of these MX₂ monolayers. To validate their thermal stability, we performed AIMD simulations at 300 K and 500 K that lasted for 5 ps with a time step of 1 fs. Taking the RuAs₂ monolayer as an example, the calculated results are shown in Fig. 2b and c. Although As atoms exhibit small deviations from their original sites due to thermal perturbations, neither bond breakage nor structure reconstruction is observed. Besides, it is also noted that the total energy fluctuation of the RuAs₂ monolayer is considerably small at this temperature, as indicated in Fig. 2d and e. As shown in Table S3 (ESI†), the energy oscillation intervals for the MX₂ monolayers are in the range of 0.47 to 0.57 eV at a heating temperature of 300 K. When the heating temperature reaches 500 K, the energy oscillation intervals significantly increase with values ranging from 0.70 to 0.91 eV. Next, we calculated the phonon spectrum of MX₂ monolayers after heating at 300 K and 500 K for 5 ps, as indicated in Fig. S2 (ESI†). For RuAs₂, RuSb₂, and OsAs₂ monolayers, a few imaginary frequencies appear near the Gamma point in acoustic branches when the heating temperature is 300 K. However, the OsSb₂ monolayer shows some imaginary frequencies in the optical branches, indicating its weaker thermal stability. At a heating temperature of 500 K, RuAs₂ and OsAs₂ monolayers still exhibit only a small number of imaginary frequencies in the acoustic branches, indicating their high thermal and dynamic stability.

Fig. 2 (a) Phonon dispersion of the RuAs₂ monolayer, indicating the dynamical stability of this monolayer. The structures of RuAs₂ after 5 ps of AIMD simulation at (b) 300 K and (c) 500 K. Total potential energy fluctuations of AIMD simulation at (d) 300 K and (e) 500 K.

To demonstrate the mechanical stability of MX₂ monolayers, the elastic properties are investigated, as shown in Table 1. Taking the RuAs₂ monolayer as an example, the calculated values of C₁₁, C₂₂, C₁₂, and C₄₄ are 79.77 N m⁻¹, 79.72 N m⁻¹, 21.90 N m⁻¹, and 25.90 N m⁻¹, respectively. Notably, the elastic constants for the RuAs₂ monolayer satisfy the Born criteria of mechanical stability: C₁₁C₂₂ > C₁₂² and C₁₁, C₂₂, C₄₄ > 0, confirming its mechanical stability. The 2D Young's moduli C^2D, shear modulus G^2D, and Poisson's ratios ν^2D for RuAs₂ are calculated to be 73.75 N m⁻¹, 28.93 N m⁻¹, and 0.27, respectively. It is worth noting that the in-plane Young's moduli of RuAs₂ are less than those of hexagonal h-BN (271 N m⁻¹) and graphene (340 N m⁻¹),^58,59 indicative of good mechanical flexibility.

Table 1 Calculated Young's modulus (N m⁻¹), shear modulus G (N m⁻¹), Poisson's ratio, surface density (kg m⁻²), and gravity deformation h/l for MX₂ (M = Ru, Os; X = As, Sb) monolayers

Systems	Y ^2Ds	G	ν	σ	h/l
RuAs2	73.69	28.93	0.27	1.849 × 10^−6	2.906 × 10^−4
RuSb2	65.81	22.49	0.26	2.656 × 10^−6	3.407 × 10^−4
OsAs2	91.29	35.37	0.26	2.896 × 10^−6	3.144 × 10^−4
OsSb2	68.93	29.53	0.23	3.335 × 10^−6	3.619 × 10^−4

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For these MX₂ monolayers, we estimate the typical plane deformation caused by gravity using the following formula:

where Y^2D_s is Young's modulus of a 2D material. σ, g, and l are the surface density, gravitational acceleration, and evaluated size, respectively.⁶⁰ The surface density of RuAs₂, RuSb₂, OsAs₂, and OsSb₂ is estimated to be 1.849 × 10⁻⁶, 2.656 × 10⁻⁶, 2.896 × 10⁻⁶, and 3.335 × 10⁻⁶ kg m⁻², respectively, as illustrated in Table 1. When the evaluated size l is set to 100 mm, the h/l values for RuAs₂, RuSb₂, OsAs₂, and OsSb₂ monolayers are 2.906 × 10⁻⁴, 3.407 × 10⁻⁴, 3.144 × 10⁻⁴, and 3.619 × 10⁻⁴, respectively. Notably, the calculated h/l values are comparable to those of graphene,⁶¹ indicating that these monolayers can maintain their planar structure without the need for substrate support.

A detailed symmetry analysis is needed before the investigation of electronic structures for MX₂ monolayers. It is known that the symmetry of MX₂ monolayers belongs to space group 11, which contains the inversion symmetry P: (x, y, z) → (−x, −y, −z), and rotation symmetries C_2x: (x, y, z) → (x + 1/2, −y, −z). Combining the two symmetry operations leads to the glide plane M_2x: (x, y, z) → (−x + 1/2, y, z). Notably, time reversal symmetry T is preserved because these systems are non-magnetic. Without including the SOC, we obtain T² = 1 and C_2x² = e^ikxa. The antunitary operator TC_2x with the product (TC_2x)² = −1 is considered along the X–M line (k_x, π, 0), leading to double degeneracy bands along this direction. With the presence of SOC, the bands are Kramers degenerate for all k due to the simultaneous satisfaction of P and T symmetries. Additionally, at the X (π, 0, 0) and M (π, π. 0) points, the C_2x eigenvalues are ±1. Simultaneously, P and C_2x satisfy the anticommutation relation {P, C_2x} = −1. Therefore, the fourfold degeneracy bands are located in the X and M points even in the presence of SOC.

Then, we move to study the electronic properties of MX₂ monolayers. The band structures are first calculated without considering SOC, as shown in Fig. 3a and Fig. S3 (ESI†). In addition, no spin polarization can be found for the band structures of MX₂ monolayers, indicating that these systems are nonmagnetic. Further test calculations have been conducted using the GGA+U method with U values ranging from 1 to 3 eV for the Ru and Os d electrons to correct for Coulomb interactions. The calculated results indicate that no magnetic moment is present in the Ru and Os atoms. Interestingly, the 1T′ phase MX₂ monolayers are gapless semiconductors, or alternatively semimetals. Obviously, the absence of a linear dispersion relation between the conduction and valence bands can be observed near the Fermi level. Thus, the band crossing points of these MX₂ monolayers are classified as type-II Dirac points due to the breakdown of Lorentz symmetry.^62–64 Notably, the conduction and valence bands of MX₂ monolayers are inverted around the Γ point and cross each other along the Γ–X line when SOC is ignored, suggesting the nontrivial band topology. In addition, an electron pocket can be observed around the Y point. By analyzing the orbital components of the states near the Γ point, we note that the bands near the Fermi level mainly originate from 4 d orbitals of Ru atoms and 4 p orbitals of As atoms, as shown in Fig. 3b. A careful analysis of these states near the Γ point is needed to further explore its topological properties. As indicated in Fig. 3c, the calculated fat-band structures show that these states above the Fermi level are mainly contributed by p_z orbitals of As atoms. While these states that are lower than the Fermi level are composed of d_z² orbitals of Ru atoms. Due to band inversion, the band crossing along the Γ–X line corresponds to accidental degeneracy. Notably, the band crossing in energy is about 0.05 eV above the Fermi level while the other bands are gapped. In addition, the band structures of RuSb₂, OsAs₂, and OsSb₂ monolayers share similar features to those of RuAs₂. After a careful symmetry analysis, we find that these crossing points are isolated rather than part of a nodal line, due to the absence of mirror symmetry protection. This accidental band touching near the Fermi energy can be attributed to the opposite parity of the two inverted bands. It is worth noting that the band inversion around the Γ point is not caused by SOC, as this interaction is absent. As shown in Fig. 3d, the 3D band structure confirms the presence of nodal points rather than a nodal line in the band structure near the Fermi level. To reveal the bonding characteristics of predicted 2D materials, the electron localization function (ELF) is calculated. The ELF is defined as:

Here, D(r) and D_h(r) represent the Pauli kinetic energy densities and homogeneous electron gas with the same density at a specific point in real space r, respectively.⁶⁵ As shown in Fig. 3e, the ELF values around As atoms are close to 1, indicating the presence of lone pair electrons. The ELF values of 0.5 are found in the region further outward around the As atoms, indicating that they gain electrons in systems. However, the ELF values around Ru atoms are found to be 0, indicating electron deficiency. Accordingly, electrons are transferred from Ru atoms to As atoms in the RuAs₂ monolayer. The ELF values of 0.4 between two neighboring As atoms indicate weak covalent bonding. While the ELF values in the intermediate region between Ru and As atoms are close to 0.25, implying the ionic bonding characteristics.

Fig. 3 (a) Band structures of the RuAs₂ monolayer without SOC and with SOC. The inset shows the gap opening in the presence of SOC. (b) and (c) Orbitally resolved band structure of the RuAs₂ monolayer without SOC. (d) 3D representation of the band structure around the crossing point. The Fermi level is set as the zero reference. (e) Electron localization function (ELF) map of the RuAs₂ monolayer. All results are calculated using the PBE functional.

It will be important and interesting to further investigate the potential topological properties of proposed MX₂ monolayers. Taking the SOC into account, the twofold degenerate bands can be observed at any k-vector of the Brillouin zone due to the presence of both P and T symmetries. Interestingly, the nodal points are gaped in the Γ–X line, as shown in Fig. 3a. In other words, a phase transition from gapless semimetal to an insulator can be realized. According to previous reports, the SOC-induced bandgap opening near the Fermi level indicates the nontrivial topological properties of these systems. It is also worth noting the lack of well-defined band gaps near the original crossing points when SOC is applied, as the indirect band is zero or negative. The hybrid functional calculation (HSE06) is employed to overcome the possible overestimation of band inversion in the PBE functional. Interestingly, we find that the band inversion remains, but the conduction band at the Y point is pushed above the Fermi level, resulting in the disappearance of the electron pocket observed in the PBE functional calculations, as illustrated in Fig. 4a. The calculated results indicate that the band inversions still occur using the HSE06 functional. The SOC band gap of the RuAs₂ monolayer is about 80 meV, which can be seen from the inset of Fig. 4b. As sketched in Fig. S4 (ESI†), the opened gaps for RuSb₂, OsAs₂, and OsSb₂ monolayers are as large as 131, 118, and 221 meV, respectively. Such large bandgaps imply that these monolayers are expected to realize the QSH effect at room temperature. In order to explore the nontrivial topological properties of RuAs₂, the Z₂ invariant is calculated by using the maximally localized Wannier functions. The evolution of the Wannier center of charges (WCCs) is adopted for the Z₂ calculations. We calculate the Z₂ number by counting the crossing number for any arbitrary horizontal reference line, as illustrated in Fig. 4c. As expected, an odd number of times in the k_z = 0 plane can be found for the crossing between any arbitrary horizontal reference line and evolution lines of Wannier centers, yielding Z₂ = 1. Similar to the RuAs₂ monolayer, a gap opening at the original gapless crossing points can be observed for RuSb₂, OsAs₂, and OsSb₂ monolayers, indicating the existence of the nontrivial topological phase, as shown in Fig. S4 (ESI†). The presence of topologically protected conducting edge states with a single Dirac cone at the Γ point can be observed, which connect the bulk valence and conduction bands, as displayed in Fig. 4d and Fig. S5 (ESI†).

Fig. 4 Band structures of RuAs₂ (a) without SOC and (b) with SOC, with the Fermi level as the zero reference. (c) Evolution of the Wannier charge center (WCC) for RuAs₂ along k. (d) Topological edge states of the semi-infinite RuAs₂ nanoribbon with SOC. All results are calculated by the hybrid functional (HSE06) method.

For the band structures of RuSb₂ and OsSb₂ monolayers, the crossing point in the X–Γ–X path remains above the Fermi level even in HSE06 calculations. In addition, some states of the conduction band cross the Fermi level, which may prevent the experimental observation of topological properties. As an effective way to tune the electronic properties of 2D materials, strain can be readily exerted by various strategies, such as lattice mismatch, bending, and application of stress on materials.⁶⁶ We believe that strain has a significant impact on the electronic structures of MX₂ monolayers. Therefore, the uniaxial strain is imposed on the planes of these systems by varying the planar lattice parameters. The strain magnitude is defined as ε = Δc/c, in which c and c + Δc represent the lattice constants of the unstrained and strained supercells, respectively. For RuSb₂ and OsSb₂ monolayers, the energy variations of the node points as a function of strain are calculated and given in Fig. 5, Fig. S6 and S7 (ESI†). Upon increasing the compressive strain from −1% to −5% in the x direction, the node point in energy shifts upward with its value varying from 0.24 to 0.40 eV for RuSb₂. Instead, the energy of the node point varies from 0.14 to 0.03 eV upon increasing the tension strain. We can also observe a similar trend in the node point shift when the strain is applied in the y direction. Interestingly, these states around the node point become occupied under 5% tension strain. For the OsSb₂ monolayer, the energy of the crossing point first increases and then decreases by increasing compressive strain in the x direction. For the OsSb₂ monolayer, the Os–Sb bond length decreases with increasing compressive strain. As a result, the hybridization between the Os 5d and Sb 5p orbitals is enhanced, leading to the separation of valence and conduction bands. As indicated by the topological phase diagram in Fig. 5c and d,³⁹ the node points disappear, and the topological phase transitions from nontrivial to trivial when the compressive strain in the x direction reaches −4% and in the y direction reaches −5%. The energy of the node point downshifts to 0.03 eV when the lattice constant of the OsSb₂ monolayer in the x direction is stretched by 5%. This value reaches 0.02 eV when the 5% tension strain is applied in the y direction.

Fig. 5 The energies of node points as a function of strain for (a) RuSb₂ and (b) OsSb₂ monolayers, with the Fermi level as the zero reference. Topological phase diagram of the OsSb₂ monolayer as a function of uniaxial strain along the (c) x and (d) y directions. All results are calculated using the PBE functional without including SOC.

As discussed above, applying tensile strain pushes up the conduction band around the Y point, bringing the nodal points closer to the Fermi level in both RuSb₂ and OsSb₂ monolayers. Taking the OsSb₂ monolayer as an example, we find that the energy gap at the Γ point becomes larger with a value of 236 eV at the HSE06 level, demonstrating the high possibility of observing QSH states at room temperature, as shown in Fig. 6a and b. In addition, the nontrivial band topologies can survive in a wide range of strain, which can be confirmed by topological invariants Z₂. Finally, the edge states of the OsSb₂ monolayer with 5% tension strain are studied. Fig. 6c summarizes the topological edge states of the semi-infinite OsSb₂ nanoribbon with SOC under 5% tension strain. It is noted that these edge states, protected by time reversal symmetry, locate at the Γ point, revealing a nontrivial topology of these states.

Fig. 6 The band structure of the OsSb₂ monolayer (a) with and (b) without SOC under 5% tension strain. (c) Topological edge states of the semi-infinite OsSb₂ nanoribbon with SOC under 5% tension strain.

4 Conclusions

In summary, based on first-principles calculation, we have proposed a new family of QSH insulators. Our results demonstrate that MX₂ (M = Ru and Os; X = As and Sb) monolayers exhibit appreciable energetic, thermal, and dynamic stabilities. These monolayers exhibit TI characteristics and large SOC band gaps with values of 80, 131, 118, and 221 meV for RuAs₂, RuSb₂, OsSb₂, and OsSb₂, respectively. In addition, the strain can effectively tailor the electronic structures and topological properties of these MX₂ monolayers. In detail, for the RuSb₂ and OsSb₂ monolayers, the node point in energies shifts downward toward the Fermi level under the tension strain. Our results consistently indicate that the proposed MX₂ monolayers can provide an ideal platform for realizing the QSH effect at room temperature.

Data availability

The data used to support the findings of this study are included within the article.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the open fund of Hubei Key Laboratory of Energy Storage and Power Battery (HUAT) (grant No. ZDK22023A04), Shiyan Key Laboratory of Quantum Information and Precision Optics (Grant No. SYZDK12024B02), Doctoral Scientific Research Foundation of Hubei University of Automotive Technology (Grants No. BK202431 and Grant No. BK202429), and Guizhou Provincial Major Scientific and Technological Program (QKHZDZXZ[2024]022).

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Footnote

† Electronic supplementary information (ESI) available: Details of the calculated lattice parameters, cohesive energies, phonon dispersion, and band structures for predicted MX₂ (M = Ru and Os; X = As and Sb) monolayers. See DOI: https://doi.org/10.1039/d4cp04025b

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