Robust large-gap topological insulator phase in transition-metal chalcogenide ZrTe$_4$Se

Based on density functional theory (DFT), we investigate the electronic properties of bulk and single-layer ZrTe$_4$Se. The band structure of bulk ZrTe$_4$Se can produce a semimetal-to-topological insulator (TI) phase transition under uniaxial strain. The maximum global band gap is 0.189 eV at the 7\% tensile strain. Meanwhile, the Z$_2$ invariants (0; 110) demonstrate conclusively it is a weak topological insulator (WTI). The two Dirac cones for the (001) surface further confirm the nontrivial topological nature. The single-layer ZrTe$_4$Se is a quantum spin Hall (QSH) insulator with a band gap 86.4 meV and Z$_2$=1, the nontrivial metallic edge states further confirm the nontrivial topological nature. The maximum global band gap is 0.211 eV at the tensile strain 8\%. When the compressive strain is more than 1\%, the band structure of single-layer ZrTe$_4$Se produces a TI-to-semimetal transition. These theoretical analysis may provide a method for searching large band gap TIs and platform for topological nanoelectronic device applications.


I. INTRODUCTION
The TIs are a new quantum state matter with gapped bulk band and gapless edge state, and the low-energy scattering of the edge states leads to dissipationless transport edge channels. The two dimension (2D) TI also called the QSH insulator was first theoretically predicted in 2006 and experimentally observed in HgTe/CdTe quantum wells 1,2 . The three dimension (3D) TI was first predicted and observed in the Bi 1−x Sb x alloy 3,4 . These pioneering works opened up the exciting field of TIs, expanding at a rapid pace. In the past decade, the more and more compounds have been predicted to be TIs [5][6][7][8][9][10][11][12][13][14][15] , which has undoubtedly a dramatic impact on the condensed matter physics. However, the extremely small bulk band gaps hinder their applications due to weak spin-orbit coupling (SOC). Therefore, the researchers have strong motivation for exploring new TIs or transforming materials into TIs with large band gaps.
The 3D crystal is located near the phase boundary between strong and weak TIs, the 2D is predicted to be a QSH insulator 16 . Later studies indicate that the topological nature in this bulk material is very sensitive to the crystal lattice constants and detailed composition 23,24 , so it is ideal platforms to investigate different intriguing physical properties. In addition, homologue substitution provides a useful guess for a novel material 25 . Here we modulate electronic structure by using selenium element substitution, which may change the topological properties of ZrTe 5 . In order to design a large-gap band topological nontrivial phase, one widely used approach is applying strain, which has been proved to be able to regulate the topological properties, such as SiGe 26 30,31 . We use the generalized gradient approximation (GGA) 32  Te's 5p orbitals by using the Wannier90 code 38,39 . After successful constructions of the MLWFs, the WannierTools 41 is used to evaluate Z 2 topological invariants, surface states and edge states. The ZrTe 4 Se has the orthorhombic layered structure with Cmcm (No. 63) space group symmetry, as shown in Fig. 1 Table II. They fulfill the Born criteria of stability 45 , C 11 > 0, C 11 C 22 > C 2 12 , C 11 C 22 C 33 + 2C 12 C 13 C 23 -C 11 C 2 23 -C 22 C 2 13 -C 33 C 2 12 > 0, C 44 > 0, C 55 > 0 and C 66 > 0, indicating bulk ZrTe 5 and ZrTe 4 Se are all mechanically stable. The calculated band structures for bulk ZrTe 4 Se are shown in Fig. 3 Fig. 4 (a). E g and E Γ represent the globe band gap and direct band gap at the Γ point, respectively. It can be seen that when the tensile strain is more than 1%, the as a function of strain is presented Fig. 4 (b). The E Γ increases monotonically under strain from 1% to 9%, the E g increases with strain and reaches a maximum value of 91.6 meV at 8%. When the strain is more than 2%, a semiconductor phase occurs. In addition, we find it is still a WTI with same Z 2 under 2% to 9% uniaxial strain. The band structure and (001) surface state for ZrTe 4 Se under 8% strain are presented in Fig. 5 (c) and (d), two Dirac cones located at the R and K points confirm it is WTI.

III. RESULTS AND DISCUSSION
The calculated band structures for a single-layer ZrTe 4 Se are displayed in Fig. 3  8 applications, is about 3.2 × 10 5 m/s, comparable to 10 6 m/s in graphene 47 . For the c axis edge, the symmetric edge structure leads to two Dirac cones located at opposite Y points. The nontrivial metallic edge states further confirm the nontrivial topological nature of the monolayer ZrTe 4 Se.
To get a physical understanding of the topological nature, we start from atomic orbitals and consider the effect of chemical bonding on the energy levels at the Γ point for monolayer ZrTe 4 Se.
For the convenience of discussion, we define coordinate system with x, y along a, c axes, respectively. The origin of the coordinate system located on Zr site, so the inversion center located at (0.25, 0.25). We note single-layer ZrTe 4 Se has space group Pmmn (D 13 2h ), which is nonsymmorphic, the Z 2 index of the material is fully determined by the energy order of the bands at the Γ point 16 . From the orbitals-resolved band structures, we find the band inversion happens between the Te 2 -p x and Se-p y , as shown in Fig. 8. For (I) process, there are four equivalent Te 2 atoms, they are fourfold degenerate. There are two equivalent Se atoms, they are double degenerate. For (II) process, the strong intrachain covalent bonding will split them into bonding and antibonding states. The Te 2 and Se have inversion symmetry and can be divided into two classes with p=+1 or p=-1. For (III) process, the weak interchain coupling will further change the Se's states and split Te 2 states to single non-degenerate states. As a result, only the Te 2 state has odd parity, the total parity of the occupied states is negative, which leads to the QSH state. The band gap is opened by SOC effect, but SOC effect has nothing to do with topological properties.  Fig. 4 (c). The E Γ increases monotonically under strain from -6% to 10%. The E g increases under tensile strain increases continuously and reaches a maximum value of 0.211 eV at 8% tensile strain. It can be seen that the nontrivial topological phases exists over a wide strain from -1% to 10%, such robust topology against lattice deformation makes it easier for experimental realization and characterization on different substrate. When the compressive strain is more than 1%, the band structure produces an TI-to-semimetal transition.

IV. CONCLUSION
In summary, the bulk and single-layer ZrTe 4 Se is mechanically and dynamically stable, so it is possibly to be prepared. survives at a large range of strain from -1% to 10%, indicating its robust stability against the strain.
The maximum global band gap is 0.211 eV at the tensile strain 8%. When the compressive strain is more than 1%, the band structure of single-layer ZrTe 4 Se produces a TI-to-semimetal transition.
These findings make the ZrTe 4 Se is an excellent candidate for large-gap TI and may provide a platform for realizing low-dissipation quantum spintronic devices.