Two-dimensional dual topological insulator in hexagonal IrO

Double topological insulators (DTIs) are systems with two kinds of nontrivial topological indexes. Three-dimensional materials with dual-topological character (DTC) have received extensive research, whereas two-dimensional (2D) DTIs have received less attention. In this paper, we propose a new 2D DTI IrO material by using first-principles calculations. Further calculation reveal it is energetically, dynamically, and mechanically stable. There is a direct band gap in the electronic band structures with and without spin–orbit coupling (SOC), and a SOC-induced band inversion occurs at the M point. Based on the symmetry analysis and the k · p model, the topological invariant Z2 and mirror Chern number C m are determined to be nontrivial. The DTC is supported further by the different behavior of the edge states with and without the mirror symmetry breaking. Our theoretical prediction broadens the domain of candidate 2D DTIs in terms of topological manifestations as well as material realization.


Introduction
As a hot research topic in condensed matter physics, topological insulators (TIs) differ from ordinary insulators in that they have topologically protected surface or edge states [1][2][3][4]. Topology protection is usually derived from some particular symmetries. For example, in quantum spin Hall insulators (QSHIs), the edge states are protected by the time-reversal (TR) symmetry [5][6][7][8], whereas in topological crystalline insulators (TCIs), the edge states are protected by either point or mirror symmetry [9][10][11]. Topological invariants are used to classify insulators for various symmetric protections [12]. QSHIs and TCIs, for example, are characterized by Z 2 [6,13,14] and mirror Chern number C m [15,16], respectively. When both the topological invariants Z 2 and C m are nontrivial, the material exhibits a dual-topological character (DTC) and is referred to as a dual topological insulator (DTI).
For a system with DTC, when one of the topology protections is broken, it can still behave as a non-trivial topology state. For instance, strain (magnetic field) can break the mirror (TR) symmetry, but the edge states could still be protected by the TR (mirror) symmetry. Such advantages will make the preparation and application of topological materials more appealing in the future. Three-dimensional systems with DTC have been predicted and observed [17], e.g. Bi 1 Te 1 synthesized by molecular beam epitaxy has been theoretically determined as DTI and experimentally observed using spin and angular resolution optical emission spectroscopy [18]. Furthermore, DTC has also been predicted theoretically for Bi 2 Te 3 [19], Bi 4 Se 3 [20], and Bi 1−x Sb x [21]. Only odd number energy band inversion in the reciprocal space is required for a DTI. Given that TR invariant points always appear in pairs in a 2D hexagonal lattice, the Γ and M points can meet the conditions. However, the band inversion of all reported 2D hexagonal DTIs, including Na 3 Bi monolayer [22] and HF-honeycomb materials [23], occurs at the Γ point.
In this paper, we predict a 2D hexagonal DTI IrO with the band inversion occurring at the M point. Using the first-principles method, we systematically studied the geometric structure, stability, electronic performance, and DTC of this system. Band inversion at the M point is determined by examining the irreducible representations (irreps) of the valence band maximum (VBM) and conduction band minimum (CBM) with and without spin-orbit coupling (SOC). The topological invariant Z 2 of the system is calculated from the parities of the Bloch states at TR invariant points, and the effective Hamiltonian is used to explore the mirror Chern number C m . The edge states with and without mirror symmetry breaking are also investigated to further illustrate the DTC of the system.

Computational details
First-principles calculations have been carried out using the projector augmented wave [24] potentials, as implemented in the Vienna ab initio simulation package [25,26]. A plane wave basis with a 500 eV cut-off energy is used to expand the wave functions. The electron exchange-correlation potential is treated in the form of Perdew, Burke, and Ernzerhof (PBE) [27] in the generalized gradient approximation [27]. A 20 Å vacuum layer is used to avoid interaction between adjacent layers. We use a 15 × 15 × 1 Γ-centered Monkhorst-Pack k-point grid to perform self-consistent calculations. For all the calculations, the energy convergence parameter was set to 10 −6 eV, and the Hellman-Feynman forces on each atom in structural optimization were less than 0.01 eV Å −1 . Phonon dispersion is obtained from the density-functional perturbation theory and the PHONOPY code [28]. Irreducible representations of electronic bands were determined using the IRREP subprogram in WIEN2k code [29].

Structure and stability
The geometric structure of 2D IrO with the space group p3m1 (No. 164) is depicted in figure 1 represents the total energy of IrO per unit cell and E s (Ir) and E s (O) represent the energies of single Ir and O atoms, respectively. The cohesive energy of 5.47 eV per atom, which is less than that of 2D graphene (7.86 eV per atom) [31] but greater than that of silicene (3.71 eV per atom) [32], indicates that the 2D IrO is energetically stable. The dynamic stability of 2D IrO is evaluated using its phonon spectrum, and the results are shown in figure 1(c). It is dynamically stable because there is no imaginary frequency. The partial density of states (DOS) shows that the low (high) frequency peaks correspond to Ir (O) atom states, indicating that the large band gap in phonon dispersion is caused by the significant mass difference between Ir and O atoms.
Elastic constant is an important physical quantity for the mechanical properties of 2D materials. With the method of linear response, we obtain the linear elastic constants of 2D IrO: C 11 = C 22 = 280.8, C 12 = 154.5, and C 44 = 63.2 N m −1 . Obviously, they satisfy the Born-Huang criterion [33,34] (C 11 C 22 -C 2 12 > 0, and C 44 > 0), which suggests that it is mechanically stable. We also evaluate the mechanical properties of in-plane Young's modulus, which can be calculated using the following formula [35]: where c = cos θ, s = sin θ, and θ is the angle with respect to the lattice vector ⃗ a. As shown in figure 1(d), the values of Young's modulus are independent of the angle θ, indicating that the 2D IrO has isotropic mechanical properties. It has a lower Young's modulus (195.9 N m −1 ) than graphene (384 N m −1 ) [36], indicating that it is softer. This value, however, is greater than that of germanene (42 N m −1 ) and stanene (24 N m −1 ) [37], indicating stronger covalent bonds. Figures 2(a) and (b) show the band structure of 2D IrO without and with SOC, respectively. In the absence of SOC, there is a direct band gap at the M point, with a value of 143 meV. We anticipate that SOC will have a  significant impact on energy band structure due to the large atomic radius of Ir atoms. When the SOC is taken into account, the band gap shrinks to 13 meV. According to the partial projected energy band and DOS with SOC, we find the Ir-d z 2 contribute most at the VBM and the CBM, as shown in figure 2(c). In order to better understand the effect of SOC, we calculate the irreps of the bands near the Fermi level at M point (see figures 2(a) and (b)). In the absence of SOC, the symmetry properties of Bloch states are described by single-valued irreps of little group. Moreover, for symmorphic space group p3m1, the little group at any wave vector k can be written into the direct product of the corresponding point group and translation group. Therefore, the point group irreps is enough to describe the symmetry properties of Bloch states. For IrO, the point group of M point is C 2h , and table 1 is the corresponding character table. In the absence of SOC, the irreps at the VBM and the CBM are M + 1 and M − 2 , respectively. After considering SOC, the Bloch states must be described by the double-valued irreps (I d ), which can be obtained from the corresponding single-valued irresp (I s ) with I d = I s ⊗ D (1/2) [38], where D (1/2) is the irreps for half-integer spinors. According to the characters in table 1, we find the irreps of M + 1 and M − 2 will transform into M 3 + M 4 and M 5 + M 6 , respectively. However, as shown in figure 2(b), the irreps at VBM and CBM, respectively, are M 5 + M 6 and M 3 + M 4 , indicating that band inversion occurs at M point. It is well-known that the PBE generally underestimate the band gap, we also employ the local modified Becke-Johnson exchange-correlation functional (LMBJ) + local density approximation (LDA) [39] to calculate the energy bands, as illustrated in figure S1(a). Despite having a wider band gap than the PBE result, the energy band inversion can be induced by external strains and the details can be found in the supplemental material.  [40].

Electronic properties
To further investigate its topological properties, we compute the Z 2 topological invariant of 2D IrO. For systems with inversion symmetry, Z 2 can be determined by the parities of the Bloch states at TR-invariant momenta (Γ i ) [41]: where ξ 2m (Γ i ) = ±1 is the parity eigenvalue of the mth occupied energy band at the Γ i point. The Kramers pairs appear only once in this product (ξ 2m = ξ 2m−1 ). The ν invariant of Z 2 is then as follows: The Z 2 invariant ν = 0, 1 corresponds to the trivial states and the topological states, respectively. A 2D hexagonal lattice's TR-invariant momentum are Γ and three equivalent M points, with the calculated δ i are +1, −1, −1, and −1. As a result, the Z 2 invariant ν = 1 proves that the 2D IrO is a TI.
In order to determine the TCI characteristics of 2D IrO, we study the mirror Chern number C m along the high symmmetry line Γ-M. In the first BZ, Γ and M points are exactly located on the intersection line between the mirror σ h and the 2D plane. The wave-vector point group of the M is C 2h and it includes space inversion symmetry I, the rotation symmetry C 2y , mirror symmetry σ h , and their symmetry matrices can be expressed as follows: The bands of the VBM and CBM at the M points are double-degenerate because of the simultaneous presence of TR symmetry and I. By analyzing the symmetry properties of their wavefunction, we obtain the irreps of them are M 5 + M 6 and M 3 + M 4 , respectively. With the help of their characters in table 1, we can construct the 4 × 4 k · p Hamiltonian [42] with the following form, and where , σ x , σ y and σ z are Pauli matrices, and σ 0 is a 2 × 2 identity matrix. In the 2D BZ, both of k y = 0 and k y = π are mirror-invariant, on which M x k = k is satisfied, and then [M x , H(k)] = 0. However, for the 2D IrO, the energy band inversion occurs at the M point and can cause a topological phase transition at k y = 0 but not at k y = π. Therefore, we only consider the case of k y = 0. By using the representation matrix M x = iσ z ⊗ σ 0 , the Hamiltonian can be converted into a block diagonal matrix:  where , The diagonal matrix associated with the mirror eigenvalues of ±i is then given as h ±i = h 0 ± (h 1 + h 2 ). Since the mirror plane at M point is perpendicular to the plane of the 2D structure, the C m can be expressed by winding numbers N ±i [43][44][45], which is related to the Zak phase of the 1D effective Hamiltonian h ±i [44,45]. To better understand the mirror symmetry protection, we compute the winding numbers by substituting the continuum model with k x → sin k x and k 2 x → 2(1 − cos k x ). Then, h ±i = σ zε ± (σ yγ + σ xᾱ ), whereε = −ε 0 + 2A 1 (1 − cos k x ),γ = A 3 sin k x , andᾱ = A 2 sin k x . Therefore, the winding numbers here is considered to be directly related to the band indexes δ ±i k = sgn(ε)sgn(±γ), where k represents the high symmetry point in the mirror-invariant plane. In the case of IrO, k = Γ, M [46]. Then the winding numbers is N ±i = − 1 2 (δ ±i k=Γ + δ ±i k=M ). By fitting the first-principles energy bands, we obtain the fitted parameters ε For 0 < ε 0 /A 1 < 4, the mirror Chern number is C m = 1 2 (C i + C −i ) = 1, which indicates that the 2D IrO material is a TCI. Combined with the nontrivial Z 2 invariant, 2D IrO is classified as a DTC protected by both of the mirror and TR symmetries.
In order to further confirm the DTC of 2D IrO, we study its edge DOS, as the presence of gapless edge states is an important signature of both the 2D TIs and the 2D TCIs. To better understand the effect of mirror symmetry on the edge states, the mirror symmetry is broken by applying 3% uniaxial compressive strain along the a-axis (shown in figure 3(a)). Since the VBM and the CBM are mainly contributed by Ir-d z 2 and O-p orbits, as shown in figure 2(c), we construct a Wannier-based tight-binding model with these basis to calculate the edge states. The edge states of a 1D IrO nanoribbon along the direction of [010] are shown in figures 2(c) and (d). When mirror symmetry is preserved, the edge states with opposite mirror eigenvalues intersect between Γ − M, as shown in figure 3(c). When mirror symmetry is broken, the edge states between Γ − M repulse with each other because they are no longer protected by the mirror symmetry. However, because the system still retains TR symmetry, the edge states of different spin textures intersect at M point, as illustrated in figure 3(d). Simultaneously, we find the Z 2 still equal to 1 when the mirror symmetry is broken, confirming that it is topologically nontrivial.
Next, we investigate the system's topological properties when TR symmetry is broken. Here, by adding a Zeeman splitting term to the k · p model [47,48], we introduce a magnetic field that naturally breaks the TR symmetry. The splitting term is given by [49,50] where c i represents the intensity of the Zeeman splitting induced by the magnetic field in the direction of i.
Here, we assume the magnetic field is along the z-axis. Combined with equation (8), the Hamiltonian is given by where According to the calculation method of the mirror Chern number above, we find that the magnetic field introduced in the z-axis direction only changesᾱ in the one-dimensional effective Hamiltonian h ±i = σ zε ± (σ yγ + σ xᾱ ), that is,ᾱ ′ =ᾱ + c z . Apparently, the calculation outcome of the mirror Chern number is unaffected by theᾱ item. The value of the mirror Chern number remains 1, proving that the system is still topologically nontrivial. The above results demonstrate, once again, that the 2D IrO material has DTC.

Conclusion
In summary, we propose a hexagonal DTI IrO material with band inversion at the M point. The phonon spectrum, mechanical constants, and cohesive energy results show that it is thermodynamically, mechanically, and energetically stable. The band gaps without and with SOC are 143 and 13 meV, respectively. The symmetry analysis reveals that the SOC induced band inversion occurs at the M point. The system has DTC, which is supported by the mirror Chern number C m = 1 and the topological invariant Z 2 = 1. Further supporting the DTC are the distinct topological behaviors of the one-dimensional edge local state density for 2D IrO with and without breaking mirror symmetry, and the fact that the mirror Chern number remains 1 when TR symmetry is broken. The gapless edge states of IrO are resistant to various external perturbations because they coexist with the TCI phase and Z 2 topological phase. Different from the reported hexagonal DTI materials where the energy band inversion happen at the Γ, the case of energy band inversion at the M point expands our understanding of the formation of DTI phase in hexagonal materials, which is fascinating to find more DTI in hexagonal materials. Furthermore, the topological property that can be changed by strain makes it suitable for use in next electronic devices.

Data availability statement
No new data were created or analysed in this study.