Prediction of a large-gap quantum spin Hall insulator: Diamond-like GaBi bilayer

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1 Nano Research DOI /s z Nano Res 1 Prediction of a large-gap quantum spin Hall insulator: Diamond-like GaBi bilayer Aizhu Wang 1,2, Aijun Du 2 ( ), and Mingwen Zhao 1 ( ) Nano Res., Just Accepted Manuscript DOI /s z on August 19, 2015 Tsinghua University Press 2015 Just Accepted This is a Just Accepted manuscript, which has been examined by the peer-review process and has been accepted for publication. A Just Accepted manuscript is published online shortly after its acceptance, which is prior to technical editing and formatting and author proofing. Tsinghua University Press (TUP) provides Just Accepted as an optional and free service which allows authors to make their results available to the research community as soon as possible after acceptance. After a manuscript has been technically edited and formatted, it will be removed from the Just Accepted Web site and published as an ASAP article. Please note that technical editing may introduce minor changes to the manuscript text and/or graphics which may affect the content, and all legal disclaimers that apply to the journal pertain. In no event shall TUP be held responsible for errors or consequences arising from the use of any information contained in these Just Accepted manuscripts. To cite this manuscript please use its Digital Object Identifier (DOI ), which is identical for all formats of publication.

2 Revised Manuscript for Nano Research TABLE OF CONTENTS (TOC) Prediction of a large-gap Quantum Spin Hall Insulator: Diamond-like GaBi Bilayer Aizhu Wang 1,2, Aijun Du 2 *, Mingwen Zhao 1 * 1 Shandong University, China 2 Queensland University of Technology, Australia The 2DCD HGaBi lattice, based on cubic diamond-like symmetry, was predicted to be a stable topological material accompanying by the inversion of the s-p bands. Aijun Du, Mingwen Zhao,

3 Nano Research DOI (automatically inserted by the publisher) 2 Nano Res. Review Article/Research Article Research Article Prediction of a large-gap Quantum Spin Hall Insulator: Diamond-like GaBi Bilayer Aizhu Wang 1,2, Aijun Du 2 ( ), Mingwen Zhao 1 ( ) Received: day month year Revised: day month year Accepted: day month year (automatically inserted by the publisher) Tsinghua University Press and Springer-Verlag Berlin Heidelberg 2014 KEYWORDS Topological insulators, First-principles calculations, 2D cubic-diamond-like lattice, Rashba spin splitting, Band inversion ABSTRACT Quantum spin Hall (QSH) state has been achieved experimentally, but the critical temperature is quite low due to the small bulk band gaps of the materials. Searching for two-dimensional (2D) topological insulators is of critical importance for realizing novel topological applications. Using density functional theory (DFT), we demonstrated that the hydrogenated GaBi bilayers (HGaBi) is a stable topological insulator with a large nontrivial band gap of ev based on the state-of-the-art hybrid functional method, which is implementable for achieving QSH states at room temperature. The topological nontrivial property of the HGaBi lattice can be also confirmed by the appearance of gapless edge states in the nanoribbon structure. Our results offer a versatile platform for hosting nontrivial topological states with important application in nanoelectronic devices. 1 Introduction Topological insulators (TIs), typically insulating or gapped in bulk but possessing conducting states at their edges or on the surfaces as a result of the topology, are distinguished from ordinary insulators [1, 2]. So far, numbers of TIs have been theoretically predicted and experimentally confirmed. HgTe-based quantum well [3, 4] is the first quantum spin Hall (QSH) system. Subsequently, binary (Bi2Te3, Bi2Se3 and Sb2Te3) [5-7], ternary (TlBiTe2, TlBiSe2 and PbBi2Te4) [8, 9] and Heusler compounds [10-12] have been identified as 3D TIs. Most recently, the bismuth bilayer [13] and actinide-based binary compounds [14] have been predicted as new families of 2D and 3D TIs. And the edge states of the single bilayer Address correspondence to aijun.du@qut.edu.au; zmw@sdu.edu.cn

4 Nano Res. 3 bismuth or bismuth (111) islands on substrates with a sizable gap (~100 mev) [15, 16] have been confirmed experimentally. 2D TIs possess unique advantages over 3D as the electrons in the surface states of 3D TIs are only free from exact 180 -backscattering and suffer from scattering of other angles, whereas the electrons at the edges of 2D TIs can only move along two directions with opposite spins and thus free from backscattering caused by nonmagnetic defects [17]. This leads to dissipationless charge or spin current carried by edge states, and the charge carriers can be also easily controlled by gating [18]. Actually, the search for 2D realistic TIs materials including graphene (~10-3 mev) [19], silicene (1.55 mev) [20, 21], graphyne (0.59 mev) [22], organometallic frameworks (2.4~13.6 mev) [23-25] has benefited from the fruitful interplay between topological band theory and band structure. However, the potential applications of TIs are substantially hindered by their extremely smaller bulk band gaps due to weak spin-orbit coupling (SOC). Therefore the explorations of new and stable 2D TIs with large band gaps are highly desired toward the realistic application. Structural symmetry and SOC are two key essential ingredients in TIs. For the lattice symmetry, we give priority to the hexagonal graphene-like honeycomb lattice and cubic diamond-like lattice. It has been reported that static compression of hexagonal graphite (HG) can form metastable hexagonal diamond (HD) at temperatures of K and cubic diamond (CD) at higher temperatures [26]. Similar 2D CD lattice is expected to be easily formed in a wide range of bilayer hexagonal-like inorganic materials and has attracted great research efforts on the structural transformations via layered hexagonal-based materials [27, 28]. Most of the 2D TIs are based on a hexagonal (specifically, graphene-like honeycomb) lattice. But TIs with the cubic diamond-like symmetry have not been reported yet. SOC is significant in heavy elemental materials such as bismuth with an effectively stable isotope [29], and will constitute another important ingredient in the design of new TIs [13-15, 30]. The purpose of this work is not only to predict a stable topological material based on heavy-element (Bismuth)-based materials for realistic room temperature application but also provides a new lattice with cubic diamond-like symmetry for engineering topological states. Here, using first-principles calculations, we demonstrated that the 2D cubic-diamond-like (2DCD) GaBi bilayer with the surface atoms being passivated by hydrogen atoms (HGaBi), as shown in Figure 1a, is a stable 2D TI with strong SOC. Accompanied by the inversion of the pxy bands, SOC opens a topological nontrivial band gap at the Fermi level. The bulk band gap can be as large as ev based on the state-of-the-art hybrid functional method, which is quite efficient for achieving QSH states at room temperature. More importantly, such a large topologically nontrivial gap is mainly related to the pxy orbitals of the sp 3 -hybridized atoms of the 2DCD lattice, which are sandwiched by two hydrogenated HGa and BiH layers. Such unique structure is beneficial for the robustness of topological nontrivially against the interaction between a substrate and the 2DCD lattice. The gapless edge states of the HGaBi nanoribbons also confirm the topological nontrivially of the HGaBi bilayer. Our new structure opens an avenue for searching for 2D TIs with strong SOC and large bulk band gap, which are quite crucial for achieving QSH states at high temperatures. 2 Computational Details Our first-principles calculations were performed within the framework of density functional theory (DFT) as implemented in the Vienna Ab-initio Simulation Package (VASP) [31]. The electron-electron interactions were treated within generalized gradient approximation using Perdew-Burke-Ernzerhof (PBE) [32] exchange-correlation functional. The energy cut-off employed for plane-wave expansion of electron wavefunctions was set to be 520 ev and the electron-ion interaction was described by projector-augmented-wave (PAW) [33] potentials. The supercell are repeated periodically on the x-y Nano Research

5 4 Nano Res. plane while a vacuum region of about 30 Å was applied along the z-direction to avoid mirror interaction between neighboring images. The Brillouin zone (BZ) integration was sampled on a grid of k-points and structural optimizations were carried out using a conjugate gradient (CG) method until the remaining force on each atom is less than ev/å. It is well known that the general gradient approximation (PBE exchange correlation functional) always significantly underestimate the band gap of semiconductors and overestimate the ferromagnetic coupling strength due to self-interaction error [34, 35]. Here, a hybrid functional in the form of HSE06 [36] was adopted to correct the well-known band gap problem by the generalized gradient approximation. Our phonon spectrum is calculated by Phononpy code [37] interfacing with VASP. quite promising for synthesizing 2D GaBi lattice. Secondly, previous works showed that the hydrogenation facilitates the formation of diamond-like structures [40, 41]. Finally, the dynamic stability of the HGaBi lattice was verified by calculating the phonon spectrum as shown in Figure 1c. And all the phonon dispersion curves have positive frequencies, confirming the 2DCD HGaBi lattice is dynamically stable. These results suggest that the 2DCD HGaBi lattice will be likely fabricated in future experiments. 3 Results and discussions The optimized structure of the 2DCD HGaBi lattice is shown in Figure 1a. The combination of two half-hydrogenated single bilayers, H-GaBi and GaBi-H, turns out to be diamond-like structure without any activation barrier (more details can be found in Figure S1 in the Electronic Supplementary Material (ESM)). The 2D HGaBi lattice has six atoms in one unit cell with an angle of 60 between the two basis vectors. Each Ga (or Bi) atom is bonded to one H atom and three Bi (or Ga) atoms on the two surfaces, while the bulk atoms exhibit a diamond-like-structure. The lattice constant of 2DCD HGaBi is calculated to be Å, which is slightly shorter than that of single layer H2-GaBi lattice (4.588 Å ) [38]. The Ga-H and Bi-H distances are Å and Å, respectively and the Ga-Bi distance can be Å, Å and Å depending on the positions of Ga and Bi, Bi-H and Ga-H bond. We then focus on the possibility of synthesizing the 2DCD HGaBi lattice. Firstly, the alloying of bismuth with III-V semiconductors has been achieved by using molecular beam epitaxy (MBE) technique. For example, crystalline InBi has been grown on (100) GaAs substrate [39]. This technique is Figure 1 (Color online) (a) Crystal structure for decorated GaBi with a cubic diamond lattice (2D HGaBi from the top (side) view [left (right)]). (b) Band structure without SOC and (c) the phonon spectrum along the high symmetric points in Brillouin zone of HGaBi lattice. The inset figure in (b) shows a zoomed in energy dispersion near the Γ point. The Fermi level is indicated by the dashed line. Figure 1b presented the electronic structures of the 2DCD HGaBi lattice obtained from DFT calculations in the absence of SOC. The ground state is spin-unpolarized and exhibits gapless state at the Γ point. The valences band maximum (VBM) and the conduction band minimum (CBM) are mostly contributed from the p-orbital of Bi atom as shown in Figure S2 in the ESM. Interestingly, both the

6 Nano Res. 5 conduction and valence bands display a clear camelback shape near the Γ point in the 2D Brillouin zone (BZ), suggestive of band inversion at the Γ point. To understand the nature of the inverted band near the Γ point, we carefully examined their orbital characters, and found there is an obvious band Figure 2 (Color online) Band structure of the 2DCD HGaBi lattice without and with SOC based on PBE (a) and HSE (b) functional. The direct inversional band gap at the Γ point is labelled as Eg (Γ). inversion between s- and p-orbital as shown in Figure S3 in the ESM. Then we turned on the SOC effect in the band structure calculations. A direct band gap (Eg (Γ)=0.293 ev) can be opened up at the Γ point with the lifted spin degeneracy in the BZ as shown in Figure 2 and Figure S3. Such spin splitting has also been found in recent work on the single layer GaBiCl2 lattice, where the lifting of spin degeneracy due to SOC leads to linear terms in electron wave vector k in the effective Hamiltonian [39, 40]. The origin of the linear terms in low dimensional systems is due to the structurally inversion asymmetry, leading to a Rashba spin-orbit term in the Hamiltonian [41]. It should be noted that the spin splitting is crucial for the field of spintronic because it can allow the electric field to control spin polarization, determines the spin relaxation rate, and can be utilized for all-electric spin injection. This is the advantage of HGaBi comparing to bismuth bilayers with structural-inversion symmetry, although the SOC gap in the bismuth bilayers is larger. Here, without considering the SOC effect, the band structure based on HSE06 calculation showed a semiconductor nature with a direct band gap (0.084 ev) located at the Γ point, which significantly corrects the gapless state in the PBE calculation as shown in Figure 2. By projecting the bands onto different atomic orbitals, we found that the two valence bands closest to the Fermi level that are energetically degenerated at the Γ point arise mainly from the px and py orbitals of Ga and Bi atoms (denoted as pxy band), whereas the lowest conduction band mainly consists of s orbital of Ga and Bi (denoted as s band), as shown in Figure 3. The states of the pxy band at the Γ point have the E symmetry, whereas that of the s band has the A1 symmetry. The A1 state (s band) is higher in energy than the E states (pxy bands), suggesting a normal band order in the absence of SOC, similar to the cases of GaBi lattice [39] and H2-GaBi lattice [38]. In analogy to the PBE results, the strong SOC effect of Bi atom in the 2DCD HGaBi lattice lifts the energy-degeneracy of the valence band maximum at the Γ point (see Figure 2). The degenerated valence bands are downshifted with different amplitude and are separated from each other with a huge gap of ev. The CBM and VBM move away from the Γ point, leading to an indirect band gap of ev. By considering the orbital contribution of the band, one can find that, with SOC, two bands with different parities are inverted around the Fermi level. To further understand the origin of the inverted band structure near Γ point, we have analyzed the orbital character of the bands scientifically. Figure 3 shows the orbital-resolved band structures near the Fermi level for different atoms with and without the inclusion of SOC using the HSE06 functional. It can be seen clearly that the band inversion strongly points to the inversion asymmetric nontrivial topological phase, which is attributed to the s-orbital of Ga1 and the pxy-orbital of Bi3, indicating that atoms belong to diamond layer are mainly responsible for Nano Research

7 6 Nano Res. the inversion of bands. However, the band contributions from Ga2 and Bi4 also display interesting topological properties and thus cannot be ignored absolutely. Such an inversion mechanism was shown in Figure 4a. The inverted band gap by the HSE06 functional is calculated to be as high as ev, suggesting that the QSH effect can be achieved at room temperature. Figure 3 (Color online) Orbital-resolved band structures without (left) and with (right) SOC around the Γ point obtained from HSE functional. The red, green and yellow dots represent the contributions from the s, pxy and pz atomic orbital of Ga and Bi atoms. Hydrogenation has been proven to be an efficient way to modulate the electronic properties of 2D materials. For example, semi-hydrogenated graphene (graphone) exhibits stable ferromagnetism [45]. Full-hydrogenation kills the topological nontriviality of stanene, making stanane a normal insulator [46]. Recently, some reports demonstrated that novel valley-polarized quantum anomalous Hall (VP-QAH) states can be found in either full-hydrogenated or half-hydrogenated Bismuth film [42, 43]. So it is worth to discuss the role of hydrogen in the 2D Bismuth-based (2DCD HGaBi) system. The bared 2DCD GaBi lattice is a normal insulator in its equilibrium structure. But there are imaginary frequencies in the phonon spectrum as shown in Figure S4b, suggesting that it is dynamically unstable. Imaginary frequencies have also been found in the semi-hydrogenated 2DCD GaBi and thus is unlikely realized in experiments. More details can be found in the ESM. Due to the loss of inversion symmetry in 2DCD HGaBi thin film, the method proposed by Fu and Kane [44] cannot be used to calculate the Z2 invariant. Normally, the 2D TIs is expected to display an odd number of Dirac-like edge states connecting the conduction and valence bands. In particular, the 2DCD HGaBi lattice contains one-dimensional infinite molecular chain as its building block; therefore, it could be naturally torn to nanoribbons with clean and atomically sharp edges, which are much desired for the observation of topological edge states. If so, then the 2DCD HGaBi lattice could serve as ideal one-dimensional (1D) dissipationless conducting wires. In order to explicitly demonstrate the edge states for the 2DCD HGaBi lattice, a zigzag nanoribbon (ZNR) with mirror symmetry was constructed with the edges passivated by hydrogen atoms [see Figure 4b]. Following the convention of graphene ZNR [45], we classify the 2DCD HGaBi-ZNR by the number of zigzag chains across the ribbon width. To avoid the interaction between the edge states, a wide 2DCD HGaBi-ZNR was selected and the calculated band structure is presented in Figure 4b. Clearly, the helical edge states emerge from the bulk conduction band, cross at the Γ point, and enter the bulk valence band, exhibiting the topologically nontrivial property. Since the mirror symmetry is broken, the topologically nontrivial edge states are splitted and mainly distributed at the two opposite edges (Ga-edges and Bi-edges), respectively. Collectively, the above results consistently indicate that 2DCD HGaBi lattice is an ideal 2D topological insulator.

8 Figure 4 (Color online) (a) Schematic diagram the band evolution at the Γ point driven by SOC for the orbitals around the Fermi level. (b) Schematic representations (top and side views) of the HGaBi nanoribbon with zigzag edges. The width of the nanoribbon is nm. The edge atoms are passivated by H atoms. The edge state of the 1D HGaBi ribbon was also shown in this Figure. Red and green dots mark the Dirac points of the edge states localized at different terminals. The Fermi level is set to zero. 4 Conclusions By using first-principles modelling approach, we have systematically studied crystal and electronic structures of a bilayer 2D HGaBi in diamond-like lattice. The nontrivial topological characteristics of inverted bands are observed with a band gap as large as ev base on the HSE calculation, suggesting potential spintronics applications for the 2DCD HGaBi film. This is mainly attributed to sp 3 hybridization of the cubic diamond lattice, leading to the band inversion of the pxy orbitals of Ga and Bi. In addition, the topological nontrivial property of the 2DCD HGaBi lattice has been further confirmed by the appearance of gapless edge states in the HGaBi nanoribbon. The room-temperature band gap and the special cubic diamond-like lattice in the HGaBi film might allow these states to reach the long-sought topological spin-transport regime, suggesting that the 2DCD HGaBi lattice could be a versatile platform for hosting nontrivial topological states with potential application in 2D spintronics and computer

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