Quantum spin Hall insulators and quantum valley Hall insulators of BiX/SbX (X = H, F, Cl, and Br) monolayers with a record bulk band gap

Large bulk band gap is critical for application of the quantum spin Hall (QSH) insulator or two dimensional (2D) topological insulator (TI) in spintronic device operating at room temperature (RT). Based on the first-principles calculations, here we predict a group of 2D topological insulators BiX/SbX (X = H, F, Cl, and Br) monolayers with extraordinarily large bulk gaps from 0.32 to a record value of 1.08 eV. These giant-gaps are entirely due to the result of strong spin-orbit interaction related to px and py orbitals of Bi/Sb atoms around the two valley K and K' of honeycomb lattice, which is different significantly from the one consisted of pz orbital just like in graphene/silicene. The topological characteristic of BiX/SbX monolayers is confirmed by the calculated nontrivial Z2 index and an explicit construction of the low energy effective Hamiltonian in these systems. We show that the honeycomb structures of BiX monolayers remain stable even at a temperature of 600 K. These features make the giant-gap TIs BiX/SbX monolayers an ideal platform to realize many exotic phenomena and fabricate new quantum devices operating at RT. Furthermore, biased BiX/SbX monolayers become a quantum valley Hall insulator, showing valley-selective circular dichroism.


Introduction
The quantum spin Hall (QSH) insulators, also known as two-dimensional (2D) topological insulators (TIs), have generated great interest in the fields of the condensed matter physics and materials science due to their scientific importance as a novel quantum state and potential applications in ranging from spintronics to topological quantum computation [1][2][3] . The QSH insulators are characterized by an insulating bulk and fully spin-polarized gapless helical edge states without backscattering at the sample boundaries, which are protected by time-reversal symmetry. The prototypical concept of the QSH effect was first proposed by Kane and Mele in graphene in which the spin-orbit coupling (SOC) opens a band gap at the Dirac point 4,5 .
However, the rather weak second order effective SOC makes the QSH effect in graphene only appear at an unrealistically low temperature 6 .
Up to now only the HgTe/CdTe quantum well is verified to be a well-established QSH insulator experimentally 7,8 . Experimental evidence has also been presented recently for helical edge modes in inverted InAs/GaSb quantum wells 9 . The critical drawback of such reported QSH state is their small bulk gaps, which are too small to make the predicted QSH effect observable under experimentally easy accessible conditions. Thus, to observe QSH effect at room temperature (RT) in TIs, large bulk band gap is essential because they can stabilize the edge current against the interference of the thermally activated carriers in the bulk due to the fact that the carrier concentration in the bulk decreases exponentially with the band gap. Extensive effort has been devoted to search for new 2D TIs with a large bulk band gap [10][11][12][13][14] . Some layered materials such as silicene, germanene 15 and stanene 16 have been proposed, and the bulk band gap of 2D TI has been elevated to remarkable 0.3 eV in chemical modified tin film, SnX (X = F, Cl, Br, and I) 13 . Recently, ultrathin Bi films have drawn much attention as a promising candidate of the QSH insulator, and the 2D topological properties of the ultra-thin Bi(111) films have been reported 17 . To the best of our knowledge, no bulk band gap has exceeded 0.7 eV in both 2D and 3D TIs 18 .
Since Bi and Sb are well known for their strong SOC that can drive and stabilize the topological non-trivial electronic states, it is wise to search for large-band-gap QSH insulators based on the Bi/Sb related materials. Here, we predicted that the free-standing 2D honeycomb Bi/Sb halide and Bi/Sb hydride (We call them bismuthumane and stibiumane, respectively, by analogy with graphane, silicane, and stanane) systems are stable huge-band-gap QSH insulator based on the first-principles (FP) calculations of the structure optimization, phonon modes, and the finite temperature molecular dynamics as well as the electronic structures. The topological characteristic of these TIs is confirmed by the FP-calculated nontrivial Z 2 index.
The low-energy effective Hamiltonian (LEEH) is given to capture the low-energy long-wavelength properties of these systems. Significantly, among these new TIs, we found that the bulk band gap of about 1.0 eV related to the p x and p y orbitals of the Bi atoms in BiX (X = H, F, and Cl) monolayers. To our knowledge, these are the largest-band-gap TIs. Their gaps opened by SOC in QSH phase can be effectively tuned by the X atom. All of the above features make these compounds promising for the applications at RT. Moreover, when the inversion symmetry of the honeycomb lattice for BiX/SbX monolayers is broken, BiX/SbX monolayers become a quantum valley Hall insulator, and chiral optical selectivity of the valleys is obtained.

Methods
For these materials, we first carried out a geometry optimization including SOC interaction using the VASP package within the framework of the projector augmented wave (PAW) pseudopotential method using a plane-wave basis set. The Brillouin-zone integrations have been carried out on a 9 9 1  Γ-centered k mesh. Vacuum regions with thickness larger than 14 Å were placed to avoid interaction between the monolayers and its periodic images. Both the atomic positions and lattice constant were relaxed until the maximal force on each relaxed atom was smaller than 0.001 eV/Å. The cutoff energy for wave-function expansion was set as 1.3* E max of the X atoms. The stability of the optimized structure for BiH monolayer was confirmed by a vibrational analysis using the phonopy package 19 with a supercell of 55  unit cells. Fully relativistic band calculations were performed with the LAPW (linearized augmented plane wave) method implemented in the WIEN2K package, and the results are in good agreement with those generated by the VASP package. SOC was included as a second vibrational step using scalar-relativistic eigenfunctions as basis after the initial calculation being converged to self-consistency. The relativistic p 1/2 corrections were also considered for 6p orbital of Bi in order to improve the accuracy. A 20 20 3  k-points grid was utilized in the first Brillouin zone sampling and cutoff parameters max mt RK  were 4 for BiH/SbH monolayers and 6 for BiX/SbX (X = F , Cl, and Br) monolayers, respectively. The Fermi energy was calculated where each eigenvalue was temperature broadened using a Fermi function with a broadening parameter of 0.002 Ry. The exchange-correlation functional was treated using Perdew-Burke-Ernzerhof generalized gradient approximation throughout the paper. Figure 1(a) plots the typical optimized geometries for BiX monolayers, which have a three-fold rotation symmetry like that in graphene. The inversion symmetry holds for all tested compounds. The obtained equilibrium lattice constants, nearest neighbor Bi-X distances and buckling heights through structural optimization were listed in Table 1. A quasi-planar geometry is found to be more stable for BiH monolayer (bismuthumane), while a low-buckled configuration is more stable for BiF, BiCl, and BiBr monolayers. This is related to the bonding between Bi and the X atoms. Since F, Cl, and Br are more electronegative than H, the bond between Bi and F atoms is stronger than that between Bi and H atoms, leading to a low buckling in BiX (X = F, Cl and Br) monolayers. The lattice constants of BiX monolayers follows the sequence of a(F) < a(Cl) < a(Br), in accordance with the electronegativity. The bond distances of the Bi-X films slightly increases with the sequence of d(Br) > d(Cl) > d(F) determined by their covalent bond radii. The kinetic stability of these 2D TIs is further confirmed by the calculations of the phonon spectrum without SOC. Taking BiH monolayer for example ( Fig. 1(c)), there is no imaginary frequency along all momenta, which indicates that this structure is kinetically stable, corresponding to an energy minimum in the potential energy surface.

Results
Thermodynamical stability of BiX/SbX monolayers is then checked by calculating the per-atom Gibbs free energy of formation (δG), where -E represents the cohesive energy per atom of the BiX/SbX monolayers, n Bi and n X are the mole fractions of Bi and X atoms, respectively, for a given structure, and μ Bi and μ X are the per-atom chemical potentials of Bi and X, respectively, at a given state. We chose μ Bi and μ X as the binding energies per atom of bulk Bi, and X 2 molecule, respectively. We provide the formation energy data of all the checked BiX/SbX monolayers in Table 2. The calculated δG value for BiH monolayer is 0.30 eV. Bismuthine is a chemical compound with the formula BiH 3 . It is stable below −60 °C 20,21 . δG of bismuthine is 0.53 eV. δG of BiH monolayer is smaller than δG of Bismuthine, therefore it is possible to synthesize BiH monolayer.
Remarkably, bismuth/antimony-halide monolayers have negative δG, indicating a higher thermodynamical stability relative to their elemental reservoirs.
We carried out ab initio molecular dynamics (MD) simulations using a supercell of 33  unit cells at various temperatures (see Fig. 2 and Fig. S1) with a time step of 1.5 fs to check thermal stability of BiX monolayers. After running 1500 steps at 300 and 600 K, no bond is broken, suggesting that the structures of BiX (X = H, F, Cl, and Br) monolayers are thermally stable even at a temperature of 600 K. We also performed an ab initio MD simulation for a larger 44  supercell for BiH monolayer and found that the structure of BiH monolayer is stable after 2.25 ps at 600 K (See Fig. S2). In fact, it was found that the Bi-X bond energy is much higher than that of Bi-Bi bonds due to a large bond distance between Bi-Bi atoms. The snapshots of the MD simulations at higher temperature show that the Bi-Bi bonds are broken while Bi-X bonds remain at 700 K. SbX monolayers are also stable at 300-400 K (see Fig. S1). The thermal stability of these structures enables these films to be used at or even above RT, which is very important for the practical applications.
The typical band structures of the predicted systems BiH, BiF, and SbF are shown in which are comparable to those of the theoretically predicted chemically modified tin films 13 .
The band topology of BiX/SbX (X = F , Cl, and Br) monolayers can be characterized by the Z 2 invariant 22 . Z 2 = 1 characterizes a nontrivial band topology (corresponding to a QSH insulator) while Z 2 = 0 means a trivial band topology. The Z 2 invariants can be directly obtained from the FP lattice computation method 23 . Taking BiH monolayer for example, the n-field configuration is shown in Fig. 3 To the best of our knowledge, a bulk band gap of over Therefore, although the idea to realize a new TI by functionalization of a 2D material in this paper is similar to that in the previous work, our results predict a new generation of TIs and stand for an important breakthrough in TIs study.
Large lattice distortion really affects the energy band and the band gap of BiH monolayer.
Based on the MD calculation, we predicted that phase transition temperature of BiH monolayer is between 600 and 700 K. At a temperature of 600 K, the lattice distortion is very large, and the inversion symmetry is destroyed (See Fig 2). However, the band gap remains larger than 0.22 eV after 2.25 ps. The bands are split by SOC in absence of inversion symmetry, and the splitting is mainly a Rashba type. We calculated the Z 2 number of the structure of the MD simulation at 600 K after 2.25 ps and found that BiH monolayer remains a TI. Hence, the topology of BiH monolayer is very robust against a lattice distortion.
We also studied the effects of inversion symmetry breaking induced by an electric field on the band gap and topology of BiH monolayer. It was found that the band gap decreases with the increasing electric field (See Fig S4)  Taking into account that there are A and B two distinct sites in the unit cell ( Fig. 1(a)), the symmetry-adopted basis functions can be written as . The low-energy Hilbert subspace consisting of p x and p y orbitals differs significantly from the one consisting of p z orbital just like in graphene and silicene. Moreover, the SOC term is on-site rather than the next nearest neighbor as in Kane-Mele model 4,5,15,16 . This indicates that the SOC mechanism in BiX/SbX monolayers is totally different from that in the graphene or silicene.
To the first order of k, the symmetry-allowed four-bands LEEH involving SOC can be written as, where Pauli matrix  denotes 1  and 2  orbital degree of freedoms, and z  labels the valley degree of freedom K and K'. The energy spectrum of the total LEEH is   Fig. 4. It is obvious that in the vicinity of the Dirac K point there is a good agreement between the calculated results of these two methods.
There The low energy effective model with broken inversion symmetry in BiX/SbX monolayers is: where Δ 1 is the additional band gap induced by inversion symmetry breaking. In the low energy limit, the valley magnetic moments of BiX/SbX systems can be expressed as: The MD simulation indicates that BiX monolayers will deform to some degree in condition partial concentration of oxygen gas is high at RT, but their honeycomb structure can remain (See details in Supplementary Information (Ⅱ)). BiX monolayers should be protected under vacuum or using inert gases environment or an anti-oxidization layer such as two-dimensional graphene, BN, or MoS 2 . It should be pointed out that application of the strain can further modify the band gaps of the BiX/SbX monolayers. For example, a strain of 5% can increase the band gap of BiH monolayer by 0.06 eV (See Fig. S5).
In conclusion, we have identified a new family of huge-gap 2D TI phase BiX/SbX monolayers (X = H, F, Cl and Br) by FP calculations, especially BiH and BiF monolayers with known largest bulk band gaps (>1.0 eV) that far exceed the gaps of the current experimentally realized 2D TI materials. The topological characteristic of these TIs is confirmed by the calculated nontrivial Z 2 index and an explicit construction of the low energy effective model in the system. These giant-gaps are entirely due to the result of strong spin-orbit interaction being related to the p x and p y orbitals of the Bi/Sb atoms around the two valley K and K' of honeycomb lattice, which is sufficiently large for the practical application at RT. The newly discovered BiX monolayers structure survives even at a temperature of 600 K. These results represent a significant advance in TIs study, and they are expected to stimulate further work to synthesize, characterize, and utilize these new 2D TIs for fundamental exploration and practical applications at RT. Besides, the biased BiX/SbX monolayers become a quantum valley Hall insulator, and valley-selective circular dichroism is available. We find a strong coupling between the real spin and the valley pseudo-spin, which is induced by the large SOC and has modified the valley magnetic moments.