Abstract
Using firstprinciples electronic structure calculations, we predict halffluorinated GaBi honeycomb under tensile strain to harbor a quantum anomalous Hall (QAH) insulator phase. We show that this QAH phase is driven by a single inversion in the band structure at the Γ point. Moreover, we have computed the electronic spectrum of a halffluorinated GaBi nanoribbon with zigzag edges, which shows that only one edge band crosses the Fermi level within the band gap. Our results suggest that halffluorination of the GaBi honeycomb under tensile strain could provide a new platform for developing novel spintronics devices based on the QAH effect.
Introduction
Topological materials are attracting intense current interest in search of new materials platforms for energyefficient device applications^{1,2,3,4,5}. These include twodimensional (2D) topological insulators (TIs), also called quantum spin Hall (QSH) insulators^{3,6,7}, topological superconductors^{2}, topological crystalline insulators (TCIs)^{8,9}, and quantum anomalous Hall (QAH)^{5,10} insulators, among others. At the heart of these developments is the discovery of the quantum Hall effect, which led to the realization of the topological nature of the underlying electronic structures and the associated dissipationless spintransport around the edges of a 2D insulator under a strong magnetic field^{11,12}.
QAH effect is similar to the quantum Hall effect except that in the case of the QAH the driving magnetic field is generated internally in the material through spinorbit coupling (SOC), while in the more conventional quantum Hall case the magnetic field is applied externally^{13}. The existence of a QAH insulator phase was first inferred via an analysis of a tightbinding model on a honeycomb lattice^{14,15}, where the Hall conductance was found to be quantized even in the absence of an external magnetic field^{16}. The natural place to search for materials realization of the QAH phase is a ferromagnetic insulator with a topologically nontrivial band structure^{17,18,19}. A strong enough magnetization in the QSH phase could then perhaps drive the material into the QAH phase^{17}.
QAH effect was first realized in magnetically doped TI thin films^{13,18,20,21,22,23,24}, and has been the subject of extensive theoretical studies^{19,25,26,27}. In principle, we might expect the QAH effect to be more robust compared to the QSH effect where one needs to be in the submicron regime in order to prevent spinflip scattering^{17}. Notably, quantization can be achieved even at low mobility^{17,21}. In any event, the fact remains that the presently available QAH materials are constrained to very low temperatures and are based on magnetic doping of thin TI films. There is great need thus to find new QAH materials based on nonmagnetic thin films, which would be viable for room temperature applications.
Several thin films of elements of groups IV^{28,29} and V^{30,31,32}, which have been predicted to be QSH insulators, have also been predicted to harbor the QAH phase^{33,34,35,36}. In contrast, even though a number of alloys of elements of groups III and V have been predicted to support QSH phases^{37,38,39,40,41} in freestanding^{37} and functionalized forms^{38,40,41}, the possible existence of QAH phases in these alloys has, to our knowledge, not been explored.
With this motivation, here we examine the possible presence of QAH phases in thin films of GaBi IIIV honeycombs using firstprinciples calculations, including effects of halffluorination of the films. The Chern numbers in a number of cases are computed to show that the films support the QAH phase over a reasonable range of lattice constant values. We further confirm our results by computing edge states for fluorinated GaBi nanoribbons. Our results imply that on a suitable substrate, which will induce the appropriate tensile strain, the GaBi IIIV films could provide a new materials platform for applications based on the QAH effect.
Results
Figure 1(a,b) show the top views of the 2D crystal structure of our GaBi honeycomb with fluorine atoms adsorbed on Bi and Ga layer, respectively. Figure 1(c) shows the side view of the fullyfluorinated GaBi. For halffluorinations, F atoms can be adsorbed either on Bi (denoted by GaBiF) or on Ga (denoted by BiGaF). The sideviews of different fluorinations of GaBi are shown in Fig. 1(d–i). There are three types of possible configurations: planar (PL), buckled (BK), and inversely buckled (IBK) honeycomb. Here, BK [(e) & (h)] refers to cases where the adsorbed F atom retains the buckling of the original unpassivated GaBi structure^{37,38}, while IBK [(f) & (i)] refers to the case in which the buckled honeycomb is inverted after halfpassivation.
The computed total energies in Fig. 2 show that halffluorination of Ga atoms is energetically more favorable than that of Bi atoms. For halffluorination of Ga, the inversely buckled honeycomb is seen in Fig. 2 to transform into the buckled honeycomb as the lattice constant is increased. In sharp contrast, halffluorination of Bi, transforms the buckled honeycomb into the inversely buckled honeycomb. By tracking evolution of the band structure and computing the corresponding Chern numbers as a function of strain, we can identify the emergence of topological phases in these films. The resulting phase diagram for halffluorination on Ga (labeled as BiGaF) is shown in Fig. 2(a). For the planar case, we obtain a QAH semimetal for lattice constant (a) values below 4.70 Å. PL then yields a QAH insulator between a = 4.70 to 5.325 Å and, eventually turns into a trivial insulator above a = 5.325 Å. As for the buckled structure, it is a trivial insulator over the range a = 4.80–5.26 Å, but it becomes a QAH insulator for a = 5.26 to 5.50 Å. The inversely buckled honeycomb (red squares in Fig. 2(a)) is very unstable and transforms into the buckled honeycomb beyond a = 4.5 Å. On the other hand, for F adsorbed on Bi atoms, shown in Fig. 2(b), (labeled as GaBiF), we obtain a metallic state for all configurations. For this reason, for the remainder of this study, we will focus on films with halffluorination of Ga. Notably, we also find QAH phases for halfhydrogenation and halfhalogenation with Cl or Br on Ga (see Figs S1 and S6 in the Supplementary Information).
In order to understand the role of buckling in driving the topological phase transition, we consider in Fig. 2 how the band structure of the film with fixed a = 4.80 Å evolves as the buckling distance, δ, is decreased from 0.72 Å until the structure becomes planar. The band gap is seen to close at δ = 0.15 Å as, see Fig. 2(e). By plotting the p_{x}orbital contributions (green circles), we can differentiate between the trivial band gap in Fig. 2(c,d) and the inverted band gap in Fig. 2(f). We will return below to show that the Berry curvature near the inverted band gap region assumes large values, yielding a Chern number of −1. For the three aforementioned crystal structures, the topological phase transitions due to strains in the PL, BK and IBK structures are further discussed in Figs S2–S4 in Supplementary Information. In particular, Fig. S3 demonstrates the transition from an insulator to a QAH insulator in the buckled BiGaF with a critical point at a = 5.26 Å. In Fig. S4, the planar honeycomb is seen to go from a QAH semimetal to a QAH insulator, and to then transition from a QAH insulator to a trivial insulator with increasing lattice constant. In short, for the halffluorinated and planar GaBi films, we obtain a stable QAH phase over a reasonable range of lattice constants from 4.7 Å to 5.325 Å. Notably, the BiGaF film at a = 4.80 Å exhibits a band gap of 56 meV, but this gap value increases to 105 meV at a = 5.50 Å, which is large enough for realizing the QAH phase above roomtemperature.
Recent firstprinciples studies have shown that freestanding^{37} and functionalized GaBi^{38,40} nonmagnetic films can support a robust QSH insulator phase. Halffluorination or halfhydrogenation of GaBi on Ga (BiGaF) leads to an intrinsic magnetization strong enough to drive the film into the QAH phase. In this connection, band structures of fluorinated GaBi films in a planar honeycomb are presented in Fig. 3. Fullyfluorinated GaBi (a = 4.80 Å) without and with SOC are considered in Fig. 3(a,d), respectively. Fullyfluorinated GaBi is gapless without SOC, but adding SOC in the calculations is seen to open a bulk band gap of ~0.77 eV; we have verified that the film exhibits a QSH phase with Z_{2} = 1. At the equilibrium lattice constant of halffluorinated planar GaBi (4.80 Å), without SOC, we found the magnetic ground state in the spinpolarized calculations [Fig. 3(c)] to have a lower energy than the nonmagnetic state [Fig. 3(b)]. However, when the SOC was turned on in spinpolarized calculations [Fig. 3(e)], we obtained the QAH phase. Note that in Fig. 3(c,e), the spinup and spindown polarized states are marked with red and blue lines/circles, respectively.
Our nonspinpolarized band calculations without the SOC [Fig. 3(b)] show that the material is a zerogap metal with degenerate states at the Fermi energy at Γ. When spinpolarization is included in the computations [Fig. 3(c)], the exchange field causes the electronic spectrum to split into two sets of bands with different spin polarizations with an intrinsic magnetization of approximately 0.7 μB per unit cell with coexisting gapped spinup and gapless spindown dispersions; the spindown bands continue to be degenerate at Γ. Note that the p_{x}orbitals (green circles) around Γ of the spindown bands have become inverted compared to the corresponding bands in Fig. 3(a). Furthermore, the inclusion of SOC leads to an insulating state with a gap (~56 meV), see Fig. 3(e). We thus realize the QAH phase via a p_{x}orbital inversion in the spinup band in the presence of an SOC induced band gap. Further plots of p_{x,y}orbital contributions are provided in Fig. S4 of the Supplementary Information.
We have considered effects of halfhalogenations of the planar GaBi honeycomb using other elements of the halogen group (Cl, Br, and I). Like F and H atoms, we found more generally that halfhalogenations all prefer to adsorb on Ga atoms. At their equilibrium lattice constants both halfchlorinated and halfbrominated films are in the QAH phase, whereas halfiodinated films are trivial, although the QAH phase can be induced in halfiodinated films under a compressive strain. Detailed band structures of the halogenated films are given in Fig. S6 of the Supplementary Information.
We turn next to discuss the nature of edge states of GaBi nanoribbons. For this purpose, we use a tightbinding Hamiltonian, which is parameterized via Wannier functions, and as Fig. 4(a) shows, our tightbinding model reproduces the firstprinciples band structure quite well. The Berry curvature of the halffluorinated GaBi film at a lattice constant of 4.80 Å along a few symmetry directions in the Brillouin zone (BZ) is shown in Fig. 4(b). The value of the Berry curvature is seen to be large around the band inversion region near the Γ point, and when this curvature is integrated over the first BZ, it yields a Chern number of −1 as expected. Chern number is an important quantity for monitoring the QAH effect because the Hall conductivity is proportional to the Chern number. In order to further insight into the nature of the QAH edge state in the halffluorinated GaBi film as compared to the QSH edge states in a fullyfluorinated GaBi film, we constructed a nanoribbon with zigzag edge as shown in Fig. 4(e). Using our tightbinding model Hamiltonian to calculate the edge states, results for the fullyfluorinated and halffluorinated nanoribbons are given in Fig. 4. The sizes of blue and red circles are proportional to the contributions of left and right hand side zigzag edges, respectively. Figure 4(c) shows the electronic structure of the fullyfluorinated QSH nanoribbon. The plot exhibits an odd number of band crossings with the Fermi level between π/a and Γ^{38} and the related timereversed partners −π/a and Γ, and thus establishes clearly the existence of helical edge states. In Fig. 4(d), on the other hand, for each side of the halffluorinated nanoribbon edge, we can see that there is only one chiral edge state connecting the conduction and valence bands, which lies in the middle of the band gap. Note that the number of chiral edge state gives the the absolute value of the Chern number.
In order to assess the robustness of our calculations to the underestimation of the band gap within the GGA, we have computed electronic structures and Berry curvatures of our fullyfluorinated as well as halffluorinated GaBi films using the hybrid functional HSE06^{42}. We find that our GGAbased results in Fig. 2 correctly capture the evolution of the topological phases with strain, some differences in the exact values of the strain at which various phase transitions occur for different exchangecorrelation functionals notwithstanding.
Regarding experimental realization, our predicted halfpassivated IIIV honeycombs could be realized by growing these honeycombs on a suitable substrate. A recent study^{33} has shown that halfiodinated stanene could be realized in stanene grown on CdTe. In addition, one study^{43} SiCH utilized ultrahigh purity hydrogen etching to modify SiC(0001) into Hpassivated SiC which is equivalent to halfhydrogenation, and might be appropriate for halfhalogenations more generally. Studies toward realizing GaBi films via Bidoped GaAs^{44}, and related works attempting honeycomblike InBi on Si(111)^{45}, has been also reported. Most recently, another study^{46} explored the growth of TlBi film by depositing Bi on Tlcovered Si(111) surface for which a variety of different surface reconstructions such as, and 4 × 4 for Tl_{0.75}Bi_{0.25} and Tl_{0.632}Bi_{0.368}, respectively, were observed. Our study demonstrates the tunability of the halfhalogenated/halfhydrogenated IIIV films, which implies a suitable substrate could induce the QAH phase in these IIIV films.
Summary and Conclusions
We have presented an ab initio study of the electronic and topological properties of GaBi films functionalized through hydrogenation/halogenation. The presence of a QAH insulator phase is predicted in halffluorinated GaBi honeycomb under tensile strain. This QAH phase is driven by a single inversion in the band structure at the Γ point. Our computations on a halffluorinated GaBi nanoribbon with zigzag edges reveal that the edge state spectrum consists of a single chiral band crossing the Fermi level within the band gap. Our study suggests that halffluorinated GaBi honeycombs could provide a new materials platform for exploiting the QAH effect in spintronics applications.
Methods
Our firstprinciples calculations were performed within the framework of the density functional theory (DFT) utilizing the generalized gradient approximation (GGA)^{47,48,49,50,51}. Projectoraugmentedwave (PAW)^{52} wave functions with energy cutoffs of 300 and 400 eV for hydrogenation and fluorination, respectively, were used in the Vienna AbInitio Simulation Package (VASP)^{53,54}. Atomic positions were optimized for each lattice constant value considered until the residual forces were no greater than 10^{−3} eV/Å. Convergence criterion for selfconsistency in electronic structure computations was set at 10^{−6} eV. A vacuum layer of at least 30 Å along the z direction was used to simulate thin films. A Γcentered MonkhorstPack^{55} grid of 12 × 12 × 1 was used for 2D integrations in the Brillouin zone. Berry curvatures and edge states were calculated based on an effective tightbinding Hamiltonian obtained by using maximallylocalized Wannier functions via the WANNIER90 package^{56}. In connection with topological properties, Chern number C were computed for various cases using^{15,25,57}:
where Ω_{n}(k) is the Berry curvature for the nth band^{15,58,59}.
Additional Information
How to cite this article: Chen, S.P. et al. Prediction of Quantum Anomalous Hall Insulator in halffluorinated GaBi Honeycomb. Sci. Rep. 6, 31317; doi: 10.1038/srep31317 (2016).
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Acknowledgements
F.C.C. acknowledges support from the National Center for Theoretical Sciences and the Ministry of Science and Technology of Taiwan under Grants Nos MOST1042112M110002MY3 and MOST1032112M110008MY3. He is also grateful to the National Center for Highperformance Computing for computer time and facilities. The work at Northeastern University was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences grant number DEFG0207ER46352 (core research), and benefited from Northeastern University’s Advanced Scientific Computation Center (ASCC), the NERSC supercomputing center through DOE grant number DEAC0205CH11231, and support (applications to layered materials) from the DOE EFRC: Center for the Computational Design of Functional Layered Materials (CCDM) under DESC0012575. H.L. acknowledge the Singapore National Research Foundation for support under NRF Award No. NRFNRFF201303.
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Author notes
 SungPing Chen
 & ZhiQuan Huang
These authors contributed equally to this work.
Affiliations
Department of Physics, National Sun YatSen University, Kaohsiung 804, Taiwan
 SungPing Chen
 , ZhiQuan Huang
 , Christian P. Crisostomo
 , ChiaHsiu Hsu
 & FengChuan Chuang
Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, Singapore 117546
 Hsin Lin
Department of Physics, National University of Singapore, Singapore 117542
 Hsin Lin
Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA
 Arun Bansil
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Contributions
F.C.C. and H.L. conceived and initiated the study. S.P.C., C.H.H. and Z.Q.H. performed first principles calculations. Z.Q.H. performed the Chern number calculation. C.H.H., Z.Q.H., C.P.C., F.C.C., H.L. and A.B. performed the detailed analysis and contributed to the discussions. C.H.H., Z.Q.H., C.P.C., F.C.C., H.L. and A.B. wrote the manuscript. All authors reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to FengChuan Chuang or Hsin Lin.
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