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Weiyang Yu^{abc},
Chun-Yao Niu^{c},
Zhili Zhu^{c},
Xiaolin Cai^{a},
Liwei Zhang^{a},
Shouyan Bai^{c},
Ruiqi Zhao*^{d} and
Yu Jia*^{bc}
^{a}School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, 454000, China
^{b}Key Laboratory for Special Functional Materials of Ministry of Education, Henan University, Kaifeng, 475001, China. E-mail: jiayu@henu.edu.cn
^{c}International Laboratory for Quantum Functional Materials of Henan, Zhengzhou University, Zhengzhou, 450001, China
^{d}School of Materials Science and Engineering, Henan Polytechnic University, Jiaozuo, 454000, China. E-mail: zhaoruiqi@hpu.edu.cn

Received
12th April 2017
, Accepted 16th May 2017

First published on 25th May 2017

Topological insulator (TI) is a peculiar phase of matter exhibiting excellent quantum transport properties with potential applications in lower-power-consuming electronic devices. Searching for inversion-asymmetric quantum spin Hall (QSH) insulators persists as an effect for realizing new topological phenomena. Using first-principles density functional theory calculations, we investigate the geometry, dynamic stability, and electronic structures of monolayer β-BiSb. We find that it presents QSH state under biaxial tensile strain of 14%. The nontrivial topological situation in the strained system is confirmed by the identified band inversion, Z_{2} topological invariant (Z_{2} = 1), and an explicit presence of the topological edge states. Owning to the asymmetric structure, remarkable Rashba spin splitting is produced in both the valence and conduction bands of the strained system. These results provide an intriguing platform for applications of monolayer β-BiSb in future alternative quantum Hall spintronic devices.

To design 2D TIs for practical utilization, there must possess two essential points of the materials: one is the sizable bulk band gap, and the other, layered structure with van der Waals' (VDW) force between layers. The reason is that a large energy gap in 2D TIs is required to stabilize the boundary current against the influence of thermally activated bulk carriers, and layered structure, easy to get the chemical stability of 2D system. In recent years, the search for 2D TIs has been extended to Bi and Sb single-element materials^{26–30} in view of their strong SOC, a precondition which is indispensable for realizing robust topological insulators at high temperature. Interestingly, there exists a bulk BiSb compound (β-BiSb, space group Rm, no. 166), which is a natural form of antimonide and bismuth.^{31} Recently, Singh et al. predicted the lowest ground state structure of bulk BiSb with space group of R3m, no. 160.^{32} Actually, the natural form of β-BiSb crystal has the same layered structure as arsenic and antimony (β-SbAs)^{33} with space group of Rm, no. 166. S. Zhang et al. have made progress in group VA materials, such as the theoretical prediction of arsenene and antimonene and experimental preparation of antimonene.^{34,35} So the theoretical study of the 2D monolayer β-BiSb, which has not been synthesized so far, can not only enhance our understanding of their intrinsic characteristics but also facilitate the applications of the family of 2D VA–VA compound semiconductors.

In the present work, we have provided the electronic and topological properties of monolayer β-BiSb. Firstly, a basic geometry of monolayer β-BiSb was established with honeycomb structure. By means of first-principles density functional theory (DFT) computations, we calculated the cohesive energy and phonon vibrational spectra of monolayer β-BiSb, which confirm the thermodynamic and kinetic stabilities. Then, we investigate the electronic structure of monolayer β-BiSb with PBE, as well as PBE+SOC calculations. Interestingly, a robust SOC in monolayer β-BiSb results in a band-gap reduction of 60 meV. Under biaxial tensile strain, the gap of monolayer β-BiSb can be closed and reopened with a concomitant change of band shapes, which is reminiscent of band inversion known in many topological insulators. Accordingly, the QSH effect in monolayer β-BiSb is robust under biaxial tensile strain of 14%, which is confirmed by the direct calculation of the Z_{2} topological invariant and the nontrivial topological edge states. The giant band gap of monolayer β-BiSb and the robust topological properties under tensile strain are attractive features for potential applications of monolayer β-BiSb in quantum Hall spintronic devices.

Fig. 1 (a) Top and side views of monolayer β-BiSb. (b) Vibrational phonon spectra of monolayer β-BiSb. No soft modes present in the structure. |

To learn the thermodynamic stability of β-BiSb, we calculated the cohesive energy (E_{coh}), which is defined as the following formula:

E_{coh} = [E_{BiSb} − (E_{Bi} + E_{Sb})]/2 |

Now we turn to the electronic structure of monolayer β-BiSb. The calculated band structures with PBE method, along with PBE+SOC method are presented in Fig. 2(a). From Fig. 2(a) we can find that the fundamental band gap (E_{g}) are both direct band gap with 1.01 and 0.95 eV by PBE functional and PBE+SOC, respectively. However, bulk BiSb is known to exhibit with the band gap of an indirect band gap semiconducting nature.^{43} In monolayer β-BiSb, the conduction band minimum (CBM) shows a parabolic nature near Γ point, indicating the presence of highly mobile light electrons (nearly free-electrons), while the valence band maximum (VBM) indicates the presence of relatively heavy holes near Γ point. Intriguingly, the band gap with PBE+SOC is slightly smaller than that of PBE with 60 meV because of the split of valance band with SOC. The calculated strength of Rashba spin splitting, including the Rashba energy (E_{R}), the Rashba momentum (k_{o}), and the Rashba constant (α_{R}) are E_{R} = 2.12 meV, k_{o} = 0.0101 Å^{−1}, α_{R} = 2E_{R}/k_{o} = 0.42 eV Å, which are a little smaller than that of Singh's recent results.^{44}

To further shed light on the underlying bonding mechanism of Bi and Sb atoms in monolayer β-BiSb, we show in Fig. 2(b) the total and partial density of states (PDOS) of monolayer β-BiSb using PBE functional with and without SOC, respectively. As shown in Fig. 2(b), the partial density of states (PDOS) projected onto s and p orbitals of Bi and Sb atoms shows similar pattern and peak positions whether in valence band or conduction band, indicating a strong hybridization of s and p orbitals between Bi and Sb. When SOC is included in the calculations, all electronic states broaden a little. The Bi-p and Sb-p states still dominate the valence band. While the conduction states move towards the low energy region and narrow the energy gap. Meanwhile, CB states also split into two parts distinctly due to degenerated states and destroyed symmetry induced by SOC.

As we know, the character of Frontier states is not only of interest for a microscopic understanding of the conduction channels but also of great concern for the design of optimal contacts.^{45} The charge density corresponds to VBM and CBM with and without SOC are presented in Fig. 2(c), respectively. The VBM and CBM are similar and a typical lone pair electron state are found in these Frontier states, which is similar to those of phosphorene.^{46} For the semiconducting nanosheet, strain engineering is a favorable strategy to induce a switch between a trivial and a nontrivial topological phase in the system. We demonstrate that the Rashba effect in β-BiSb monolayer can be efficiently tuned under biaxial tensile strains, as shown in Fig. 3. We find that there exists direct band gap at Γ point as the tensile strain varying from 2% to 12%, and the band gap decreased gradually with maximum valence band transformed from “Λ shape” to “M shape”. Such a trend eventually leads to the smallest band gap near Γ point when the strain is 14%. On account of the asymmetric structure of β-BiSb, Rashba spin splitting is produced in both the valence and conduction bands in the strained system. Excitingly, the band gap opens again when the strain is larger than 14%. From Fig. 3 we can see that the Rashba energy (E_{R}) along with the Rashba momentum (k_{o}) become bigger and bigger with the tensile strains increasing.

The characteristic of band gaps closing and reopening associated with the change of band shapes is reminiscent of band inversion, which characterizes many known topological insulators (TIs).^{7,14,20} In order to ascertain the topological phase transition in the strained monolayer β-BiSb, we calculated the Z_{2} topological invariants. In the presence of time-reversal symmetry, Kramer's theorem dictates that the energy eigenstates must come in pairs. This allows us to enforce the so-called time-reversal constraint on the Bloch functions:

|u_{n}(−k)〉 = Θ|u_{n}(k)〉 |

where is the Berry connection and F(k) = ∇

In order to give a general idea about the energetic stability of the strained systems, the calculated formation energies of the strained systems are −2.22, −2.20, −2.19, −2.19, −2.18, −2.18, −2.16, −2.15, −2.15, −2.14 eV per atom, with the strain from 2% to 20% by interval of 2%, respectively. Meanwhile, we check the dynamical stability of the system under different external strains, we performed the phonon spectra calculations. As shown in Fig. 4, there are no negative frequencies, suggesting dynamical stabilities of monolayer β-BiSb under strains.

Fig. 4 Vibrational phonon spectra of monolayer β-BiSb under in-layer biaxial tensile strain of n% (n = 10, 12, 14, 16, 18, 20). No soft modes present in the structure. |

The 2D nontrivial insulating state is often characterized by topologically protected conducting edge states with in the bulk gap.^{50–52} Thus, the β-BiSb monolayer under the tensile strain of 14% should hold an odd number of topologically protected Dirac-like edge states connecting the conduction and valence-band edges at Γ high-symmetry points. To further confirm the nontrivial features of β-BiSb monolayer under the tensile strain of 14%, we constructed a zigzag β-BiSb nanoribbon structure, and the edge unsaturated atoms are terminated by hydrogen atoms to eliminate all dangling bonds, as seen in Fig. 5(a). The width of the zigzag β-BiSb nanoribbon adopted here is 100 Å, which is enough to avoid interactions between edge states of the two sides. The band structure of the nanoribbon is shown in Fig. 5(b). The gapless edge states appear and cross linearly at the Γ point, which further confirms the nontrivial topological phase in the β-BiSb monolayer under the tensile strain of 14%. Thus, our results provide a promising strategy for designing 2D VA–VA QSH insulators.

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## Footnote |

† PACS numbers: 73.43.-f, 71.70.Ej, 71.70.Fk. |

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