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Quantum spin Hall effect in two-dimensional transition-metal chalcogenides

Based on first-principles calculations, we have found a family of 2D transition-metal (TM) chalcogenides MX5 (M = Zr, Hf and X = S, Se and Te) can host quantum spin Hall (QSH) effect. The molecular dynamics simulation indicate that they are all thermal-dynamically stable at room temperature, the largest band gap is 0.19 eV. We have investigated MX5's electronic properties and found their properties are very similar. The single-layer ZrX5 are all gapless semimetals without consideration of spin-orbit coupling (SOC). The consideration of SOC will result in insulating phases with band gaps of 0.05 eV (direct), 0.18 eV (direct) and 0.13 eV (indirect) for ZrS5, ZrSe5 to ZrTe5, respectively. The evolution of Wannier charge centers and edge states confirm they are all QSH insulators. The mechanisms for QSH effect in ZrX5 originate from the special nonsymmorphic space group features. In addition, the QSH state of ZrS5 survives at a large range of strain as long as the interchain coupling is not strong enough to reverse the band ordering. The single-layer ZrS5 will occur a topological insulator (TI)-to-semimetal (metal) or metal-to-semimetal transition under certain strain. Monolayer MX5 expand the TI materials based on TM chalcogenides and may open up a new way to fabricate novel low power spintronic devices at room temperature.


I. INTRODUCTION
Under the influence of graphene stripping from graphite, more and more two-dimensional (2D) materials are obtained by stripping block layered materials [1][2][3] . These successfully prepared 2D materials have aroused great interest of scientists because of their excellent properties and great application prospects [4][5][6][7][8] . Among them, 2D topological insulators (TIs) are of great significance to the study of condensed matter and materials science because of their fresh topological phases.
The 2D TIs, also called quantum spin Hall (QSH) insulators, have fully spin-polarized gapless edge states into an insulating bulk. The QSH effect was first proposed by Kane and Mele in graphene [9][10][11] , but its small energy gap of 10 −3 meV makes it hard to be observed experimentally.
The HgTe/CdTe quantum wells 12,13 have been first realized in experiment, but it needs precisely controlled molecular beam epitaxy (MBE) growth at ultralow temperature. In the passed years, extensive efforts have been devoted to search new QSH insulators, a growing number of compounds have been predicted to be 2D TIs [14][15][16][17][18][19][20][21][22][23] . Due to the lack of suitable materials that are easy to prepare, stable in structure and large in band gap, the research of 2D TIs has been seriously hindered. Therefore search larger gap 2D TIs from the common used materials is indispensable for their ultimate realization.
At present, intensive research on transition metal (TM) based QSH insulators has greatly enriched the family of QSH insulators [24][25][26] , showing band inversion is caused by strong electron interaction rather than spin-orbit coupling (SOC). TM atom-based TIs are rare and only several examples are reported. Among them, the MTe 5 (M = Zr, Hf) have attracted broad attention because of their topogical properties [27][28][29][30][31][32][33][34][35] . Three-dimensional (3D) MTe 5 is a layered crystal with weak interlayer coupling, which is comparable to graphite. The MTe 5 monolayer may be obtained via the mechanical exfoliation from the 3D bulk phase as like producing graphene from graphite 36,37 . The 2D crystals are predicted to be QSH insulators with the band gap about 0.1 eV 27 . In addition, TM halide MX (M = Zr, Hf; X = Cl, Br and I) monolayer has to be found the 2D QSH insulators 19 .
Material design or atomic substitution may lead to more and better 2D TIs. Here we study TM chalcogenides MX 5 (M = Zr, Hf; X = S, Se and Te) and different TM chalcogenides may have different topological properties. What's more, we try to use the strain to increase the band gap of TM chalcogenides.
In this work, we have investigated the structural stability and electronic structure of single-layer MX 5 by using first-principles calculations. The single-layer ZrX 5 and HfX 5 have very similar properties, they are all QSH insulators with the largest band gap of 0.19 eV. Then we take ZrX 5 as an example, they are all gapless semimetals without consideration of SOC. The consideration of SOC will result in QSH insulators with band gaps of 0.05 eV (direct), 0.18 eV (direct) and 0.13 eV (indirect) for ZrS 5 , ZrSe 5 to ZrTe 5 , respectively. The topological invariant Z 2 =1 and edge states confirm the nontrivial topological nature of these materials. The mechanism of QSH effect in ZrX 5 originates from the special nonsymmorphic space group features. In addition, the QSH state of ZrS 5 survives over a very large strain range until the interchain coupling is strong enough to reverse the band ordering.

II. COMPUTATIONAL DETAILS
The Vienna ab initio Simulation Package (VASP) 38,39 is used to study the structural and electronic properties of MX 5 . The exchange correlation interaction is treated within the generalized gradient approximation (GGA) 40 of Perdew, Burke and Ernzerhof (PBE) 41 . The Brillouin zone (BZ) is integrated with 13 × 4 × 1 Γ-centered Monkhorst-Pack grid 42 , the plane-wave cutoff energy is set to 500 eV with the energy precision of 10 −5 eV. A more than 20Å vacuum slab along z-direction is set to avoid interactions between neighboring slabs. The atoms in the unit cell are relaxed until the force on each atom is less than 0.01 eV/Å. Since PBE methods often underestimate the band gap, the hybrid functional HSE06 43 is used to check the results and the band gaps are very similar. Further, the local orbital basis suite towards electronic-structure reconstruction (LOBSTER) [44][45][46] has been used to extract the chemical-bonding information for ZrX 5 . The tight binding matrix elements are calculated by projecting the Bloch states onto maximally localized Wannier functions (MLWFs) 47,48 using the VASPWANNIER90 interface. The MLWFs are derived from M's d and X's p orbitals by using the Wannier90 code 47,48 . The WannierTools 49 is used to analyse topological properties after successful constructions of the MLWFs.

III. RESULTS AND DISCUSSION
The MX 5 monolayer has the orthorhombic structure with Pmmn (D 13 2h ) nonsymmorphic space group shown in Fig. 1 (a). The trigonal prismatic chains X1-X3-X1 oriented along the x axis, and these prismatic chains are linked via parallel zigzag chains of X2 atoms along the y axis to form a 2D sheet of MX 5 in the x-y plane. We choose the origin of the coordinate system located  Table I. The relaxed lattice constants (a= 4.047Å, b= 13.860Å) of ZrTe 5 are nearly same to the reference value ( a= 4.036Å, b= 13.843Å) 27 . The lattice constants have increase trend with increasing atomic radius of X atom in ZrX 5 and HfX 5 monolayer. While fixed chalcogenide atoms, the atomic radius of Zr and Hf are affected by lanthanide contraction effect 50 , so the lattice constants show opposite variation trends, similar to the ZrCl and HfCl 19 . The single-layer ZrX 5 and HfX 5 have very similar properties, hereafter, we take ZrX 5 as an example.
First, we use the crystal orbital Hamilton population (COHP) method to extract the chemicalbonding information, shown in Table II. The bond lengths increase from ZrS 5 , ZrSe 5 to ZrTe 5 , and it makes sense because the atomic radius increases from S, Se to Te. The Fermi levels in   The calculated band structures for a single-layer ZrX 5 are displayed in Fig. 4 and Fig. S6.
They are all gapless semimetals without consideration of SOC, the conduction band minimum (CBM) and valence band maximum (VBM) touch each other at the Γ-X direction. In Fig. 5 (a), we plot the ZrS 5 's band dispersion around this crossing point, which demonstrates that the Dirac point is isolated and has linear dispersion. The consideration of SOC will result in insulator phase and the large band gaps at Γ point decrease from ZrS 5 , ZrSe 5 to ZrTe 5 , shown in Table I. The global band gaps of 0.05 eV (direct), 0.18 eV (direct) and 0.13 eV (indirect) for ZrS 5 , ZrSe 5 to ZrTe 5 , respectively. The ZrS 5 and ZrSe 5 are more suitable than ZrTe 5 as materials for making photoelectric devices. We can clearly see a small gap will be opened at the original Dirac point and the low-energy electrons become 2D massive Dirac fermions in Fig. 5 (b). The evolution lines of Wannier centers in Fig. 6 (a) and Fig. S7 show they are all nontrivial QSH insulators with Z 2 =1 and the important character of helical edge states also appear, as shown in Fig. 6 (b, c) and It can be expected that the ZrSe 5 and ZrTe 5 are more suitable than ZrS 5 as candidate materials for high-speed devices. For the edge along the y axis, the symmetric edge structure leads to two Dirac cones located at opposite Y points. The topological nature of monolayer ZrX 5 is further confirmed by the nontrivial metallic edge states. To get a physical understanding of the topological nature, we present the orbitals-resolved band structures in Fig. 4 (c, d) and SM Fig. S6, it can be seen the bands around the Fermi level are mainly derived from X1-p and X2-p orbitals for ZrX 5 , so they have the same band inversion mechanism as ZrTe 5 27 . The mechanism for QSH effect in ZrX 5 originates from the special nonsymmorphic space group features, X1 state has odd parity lead to the total parity of the occupied states is negative, which leads to the QSH state. the E Γ increases first then decreases from -10 % to 10% and reaches a maximum value of 0.97 eV at 4% tensile strain. The E g varies with strain and reaches a maximum value of 66.0 meV at 2% compressive strain. When the compressive (or tensile) strain is more than 6%, the band structure produces a TI-to-metal transition. When the compressive (or tensile) strain is more than 8%, the band structure produces a metal-to-semimetal transition.

Me t a l T I S e mi me t a l S e mi me t a l S e mi me t a l T I Me t a l
For the [010] uniaxial strain, the E Γ also increases first then decreases from -12% to 10%, reaches a maximum value of 0.84 eV at 4% compressive strain. However, the E g value changes very little with strain, reaches a maximum value of 74.2 meV at 4% compressive strain. When the compressive strain is more than 10%, the band structure produces a TI-to-semimetal transition.
When the tensile strain is more than 8%, the band structure produces a TI-to-metal transition. In a word, the nontrivial topological phases survive over a wide strain from -6% to 6% along [100] direction or -10% to 8% along [010] direction, as long as the interchain coupling is not strong enough to reverse the band ordering near the Fermi energy, the QSH state should be stable, such robust topology gainst lattice deformation may make it easier for experimental realization and characterization on different substrate.
Finally, we have analyzed the electronic and topological properties for HfX 5 monolayer and draw the same conclusions as ZrX 5 . HfX 5 mnolayer materials also possess QSH states with Z 2 = 1, remain thermal dynamically stable at room temperature, and their band structures and band-gap sizes are very similar to ZrX 5 's, see Table II, Fig. S9 and Fig. S10.

IV. CONCLUSION
In summary, we have found a family of 2D MX 5 can host QSH effect. The MD simulation indicates they are all thermal-dynamically stable at room temperature, they are all QSH insulators with the largest band gap is 0.19 eV. The single-layer ZrX 5 and HfX 5 have very similar properties.
The evolution of WCC and edge states verifies the nontrivial topological nature of these materials.
The Dirac points of these materials have high velocities, so they can be used as candidates for high-speed electronic devices materials. The mechanism for QSH effect in ZrX 5 originates from the special nonsymmorphic space group features, X1 state has odd parity lead to the total parity of the occupied states is negative, which leads to the QSH state. In addition, the QSH state of ZrS 5 survives at a large range of strain as long as the interchain coupling is not strong enough to reverse the band ordering. The band structure of single-layer ZrS 5 can produce transition among TI, semimetal and metal under certain strain. These simple monolayer ZrX 5 without chemical decoration, which are very favorable for the future experimental implementation via simple exfoliation from its 3D system, making them highly adaptable to various environments. These significant results may promote the further study of QSH insulators based on TM chalcogenide.