Prediction of dual quantum spin Hall insulator in NbIrTe₄ monolayer

Two-dimensional (2D) materials with VHS near the Fermi level are ideal platforms for studying the interplay between topological electronic structure and correlation effects[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. As a typical layered material, TaIrTe₄ monolayer was predicted to be a 2D topological insulator with a band gap around 32 meV[12, 13, 14]. Very recently, a new quantum topological state of duel QSHI was discovered in the TaIrTe₄ monolayer by the observation of quantized longitudinal conductance, which is rare in other QSHIs[15]. It was found that, in addition to the QSHI state at the charge neutrality point, two new non-trivial Z₂ gaps are generated when the Fermi level shifts to the VHS points by weak electron and hole doping effects, respectively.

Owing to the divergence of density of states (DOSs), a CDW phase transition is induced when the Fermi level shifts to the VHSs, resulting in the correlated $Z_{2}$ band gap. Different from traditional $Z_{2}$ band gaps, the VHSs generated topological band gap is expected to naturally relate to the novel correlated electronic topological states of the fractional quantum spin Hall effect and helical quantum spin liquid[16, 17, 18, 19, 20, 21, 22]. Though the VHSs generated $Z_{2}$ band gap and integer quantized longitudinal conductance can exist in TaIrTe₄, the corresponding fractional topology and other strongly correlated topological states have, so far, not been observed. Therefore, it is necessary to find out some more materials candidates that host both topology and correlation[23, 24, 25, 26, 27, 28, 29, 30, 31]. One possible reason for the absence of strongly correlated topological phenomena in TaIrTe₄ is the weak strength of electron correlation. With this inspiration, we analyzed the dual $Z_{2}$ topological state in NbIrTe₄, an isoelectronic counterpart of TaIrTe₄. Due to the difference between Ta-5d and Nb-4d orbitals near VHSs, the electronic correlation effect is expected to be more pronounced in NbIrTe₄ than in TaIrTe₄. This makes the NbIrTe₄ to be a good candidate material for observing the correlated topological phenomena that are absent in TaIrTe₄.

The crystal structure of a NbIrTe₄ monolayer shares the same structural prototype as 1T^′-MoTe₂ monolayer[32] and features a sandwich-like structure, as depicted in Fig. 1(a). In this arrangement, the Nb and Ir atoms are positioned in the middle layer, forming parallel zigzag chains along the $b$-axis. Each Nb or Ir atom is surrounded by six Te atoms, forming a distorted octahedron. Fig. 1(c) shows the energy dispersion along the high-symmetry lines (Fig. 1(b)) calculated by employing Vienna Ab initio Simulation Package (VASP)[33] with generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) aproximation[34]. One can observe that a band inversion exists near the $X$ point with a band gap of around 35 meV, in agreement with previous reports[13, 35]. In addition, two VHSs are located around $\Gamma$ and $X$, respectively, with the former one above the Fermi level and the latter below the Fermi level. Correspondingly, two narrow peaks of DOSs generated from the VHSs can be found above and below the Fermi level with energy of $\sim$E_F + 60 meV and $\sim$E_F - 60 meV, respectively. From the orbital projected band structure and DOS, the two VHSs below and above the Fermi level are dominated by the $4d$ orbitals of Nb and hence a strong correlation is expected here. With the above understanding of the electronic band structure near the Fermi level, we projected the Bloch wavefunctions into maximally localized Wannier functions (MLWFs) and constructed effective tight binding model Hamiltonians based on the overlaps of these Wannier orbitals[36].

To verify the topological state, we calculated the Wannier center evolution and edge state with chemical potential locating at the charge neutrality point[37]. The wanneir center evolution in Fig. 1(e) presents as a zigzag form with changing partner at the edges of k_x=0 and $\pi$/a, which is a typical feature in $Z_{2}$ topological insulators[38]. Consistent with the bulk state analysis, the edge state with open boundary condition along the $b$-axis direction shows linear band crossing at the $\Gamma$ point and connects the occupied and unoccupied bulk bands. Hence, the $Z_{2}$ band gap at the charge neutrality point can be confirmed from both bulk band order and edge Dirac point.

In order to identify the detailed positions of the VHSs near the charge neutrality point, we plot the energy dispersion of band-$n_{occ}+2$ and band-$n_{occ}-2$ in the whole 2D Brillouin zone (BZ), where $n_{occ}$ refers to the index of the highest occupied band in a primitive cell considering the spin. As presented in Fig. 2(a), the four VHS points on band-$n_{occ}+2$ are located at (0.211$\frac{2\pi}{a}$, 0.024$\frac{2\pi}{b}$), and its inversion and mirror partners. Considering the one-dimensional Nb-chains are similar to the Ta-chains in TaIrTe₄, the wavevectors along the $b$ direction roughly corresponds to 1$\times$ 20 $\times$ 1 and 1$\times$ 21 $\times$ 1 supercells for the nearby VHSs on band-$n_{occ}+2$ and band-$n_{occ}-2$, respectively.

We further calculated the charge susceptibility by shifting the chemical potential at $\sim$E_F + 60 meV and $\sim$E_F - 60 meV, corresponding to the position of VHSs on band-$n_{occ}+2$ and band-$n_{occ}-2$, respectively. The charge susceptibility $\chi(\textbf{q})$ at a given energy is computed by following the formula of the Lindhard function $\chi(\textbf{q})$ = $\sum\frac{f_{\textbf{k}}(1-f_{\textbf{k}+\textbf{q}{)}}}{\varepsilon_{\textbf{k}+\textbf{q}}-\varepsilon_{\textbf{k}}+i\delta}$, where $f$ refers to the Fermi-Dirac distribution, $\varepsilon$ is a momentum-dependent energy and $\delta$ refers to the broadening in data processing. Guided by the fact that the direction of Q is parallel to the Ta-chains in the TaIrTe₄ monolayer[15], we specifically chose a vector aligned with the Nb-chains in the NbIrTe₄ monolayer prior to further analysis. From Fig. 2(b) and (c), one can see that a local maximum of the charge susceptibility is located at the wavevectors of $\textbf{Q}^{*1}=(0,0.046\frac{2\pi}{a})$ and $\textbf{Q}^{*2}=(0,0.048\frac{2\pi}{b})$ for the valence bands and conduction bands, respectively. Therefore, a CDW phase transition may occur when the chemical potential is shifted to near $\sim$E_F + 60 meV and $\sim$E_F - 60 meV via weak doping or gating effects.

The wavevectors of $\textbf{Q}^{*1}=(0,0.046\frac{2\pi}{a})$ and $\textbf{Q}^{*2}=(0,0.048\frac{2\pi}{b})$ in Fig. 2 can be described by the supercell of 1$\times$ 21 $\times$ 1 and 1$\times$ 20 $\times$ 1, respectively. To study the possible CDW phase transition, we constructed an effective Hamiltonian model for a supercell of 1$\times$ 20 $\times$ 1 based on the overlaps of Wannier orbitals. Following the Fr$\ddot{o}$hlich-Peierls Hamiltonian approach with supercell along the $b$-axis and only the intra-orbital interaction considered, the CDW modulated electronic band structure can be described by $H=H_{r}+Vcos(Qy)\psi_{r}^{\dagger}\psi_{r}$[39]. Here, $V$ represents the potential induced by the superlattice. The periodicity of this potential is determined by the parameter $Q$, which can be identified by examining Q^∗ in Fig. 2.

By increasing the potential $V$, a new band gap emerges near E_F + 80 meV, originating from the original VHSs, as illustrated by the pink range in Fig. 3(a). However, when the potential V was varied from 0.01 eV to 0.15 eV, no indirect band gap was observed between band-N_occ and band-N_occ - 10 for the VHSs below the charge neutrality point. Here $N_{occ}$ refers to the index of the highest occupied band in a supercell, taking into account the spin. This lack of an indirect band gap may be attributed to the competition between the detailed shape of the electronic band structure and the strength of the modulation potential. From Fig. 3(a), one can see that there are two global band gaps in the energy window from $\sim$E_F - 150 meV to $\sim$E_F + 150 meV. One is located at the charge neutrality point and the other one is generated between the band-$N_{occ}+4$ and band-$N_{occ}+6$. The band gaps at and above the charge neutrality point are around 15 and 10 meV, respectively, when the modulation potential V is set to 0.1 eV.

To check the topological charge of these two gaps, we calculated the bulk Wannier center evolutions and energy dispersions of edge states. Fig. 3(b) shows the Wannier center evolutions by modulating the chemical potential at the charge neutrality point. One can observe that each pair of evolution lines exhibits a fixed partner and becomes degenerate at two time-reversal-invariant points $k_{x}$=0 and $\pi/a$. Therefore, the Wannier center evolutions do not intersect with the reference lines, indicating that the gap at the charge neutrality point is topologically trivial. However, the Wannier center evolutions switch partners at k_x = 0 and $\pi$/a while keeping the occupied bands fixed at band-N_occ + 4, as depicted in Fig. 3(c). The zigzag pattern of the Wannier center evolutions ensures an odd number of intersections with the reference lines, resulting in a non-trivial Z₂ band gap.

The topological phases were further checked by calculating edge states with open boundary conditions along the $b$-axis. From the energy dispersion illustrated in Fig. 3(d), we are mainly interested in the three zones that are indexed by the energy windows of $\sim$E_F - 0.04 eV to $\sim$E_F + 0.01 eV, $\sim$E_F + 0.01 eV to $\sim$E_F + 0.06 eV, and $\sim$E_F + 0.06 eV to $\sim$E_F + 0.1 eV. In the first region, although edge bands are present within the bulk band gap, they intersect the Fermi level twice. Consequently, these edge states lack topological protection and can be eliminated through external perturbations, consistent with the behavior exhibited by the Wannier center evolutions depicted in Fig. 3(b). In the second region, we observe a Dirac-point-like edge band structure connecting the bulk states originating from band-N_occ + 2 and band-N_occ + 4, as depicted in Fig. 3(a). Notably, there exists a noticeable band anti-crossing between band-N_occ + 2 and band-N_occ + 4 around the X point, although there is no indirect band gap present. Therefore, the band structure resulted from band-N_occ + 2 and band-N_occ + 4 forms a non-trivial Z₂ topological semimetal state when considering the CDW modulation potential. This differs from the case in monolayer TaIrTe₄, where a direct non-trivial Z₂ band gap of approximately 5 meV is formed by band-N_occ + 2 and band-N_occ + 4.

While there is not a global band gap between band-$N_{occ}+2$ and band-$N_{occ}+4$, we do observe a direct gap formed by band-$N_{occ}+4$ and band-$N_{occ}+6$ in the range of $\sim$E_F + 60 meV to $\sim$E_F + 100 meV. From the edge energy dispersion in this energy window, one can observe that a Dirac-point state connecting the bulk bands originating from original band-$N_{occ}+4$ and band-$N_{occ}+6$. It is a typical feature of a 2D $Z_{2}$ topological insulator and fully agreement with the analysis from Wannier charge evolutions in Fig. 3(c). Therefore, NbIrTe₄ is expected to undergo a CDW phase transition along the $b$-axis direction and results in two non-trivial $Z_{2}$ band gap. One is a direct band gap with a globally semimetallic nature, and the other one is an indirect band gap of around 10 meV. Since both gaps are not far away from the charge neutrality point, they can be achieved via weak electron doping or gating.

In summary, we have predicted a dual QSHI state in monolayer NbIrTe₄ via first-principles calculations. In addition to the original non-trivial $Z_{2}$ band gap at the charge neutrality point, a new global $Z_{2}$ band gap of around 10 meV can be obtained when the chemical potential is shifted to the VHS point in the electron doping range. Besides, an indirect band inversion in the CDW phase was found in the energy window around E₀ + 20 meV to E₀ + 30 meV near the X point. Since both of these two nontrivial band gaps are close to the charge neutrality point, they are expected to be experimentally detected after weak electron doping or gating. Moreover, the VHS point above the Fermi level is mainly dominated by the $4d$ orbitals of Nb. In comparison with Ta-$5d$ orbitals in TaIrTe₄, the dual QSHI state in NbIrTe₄ should manifest a strong correlation effect and some more correlated topological states are expected.