Letter pubs.acs.org/NanoLett
Robust Two-Dimensional Topological Insulators in MethylFunctionalized Bismuth, Antimony, and Lead Bilayer Films Yandong Ma,*,†,‡ Ying Dai,‡ Liangzhi Kou,§ Thomas Frauenheim,§ and Thomas Heine*,† †
Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany School of Physics, State Key Laboratory of Crystal Materials, Shandong University, 250100 Jinan, People’s Republic of China § Bremen Center for Computational Materials Science, University of Bremen, Am Falturm 1, 28359 Bremen, Germany ‡
S Supporting Information *
ABSTRACT: One of the major obstacles to a wide application range of the quantum spin Hall (QSH) effect is the lack of suitable QSH insulators with a large bulk gap. By means of first-principles calculations including relativistic effects, we predict that methyl-functionalized bismuth, antimony, and lead bilayers (Me-Bi, Me-Sb, and Me-Pb) are 2D topological insulators (TIs) with protected Dirac type topological helical edge states, and thus suitable QSH systems. In addition to the explicitly obtained topological edge states, the nontrivial topological characteristic of these systems is confirmed by the calculated nontrivial Z2 topological invariant. The TI characteristics are intrinsic to the studied materials and are not subject to lateral quantum confinement at edges, as confirmed by explicit simulation of the corresponding nanoribbons. It is worthwhile to point out that the large nontrivial bulk gaps of 0.934 eV (Me-Bi), 0.386 eV (Me-Sb), and 0.964 eV (Me-Pb) are derived from the strong spin−orbit coupling within the px and py orbitals and would be large enough for room-temperature application. Moreover, we show that the topological properties in these three systems are robust against mechanical deformation. These novel 2D TIs with such giant topological energy gaps are promising platforms for topological phenomena and possible applications at high temperature. KEYWORDS: Topological insulators, Dirac states, Quantum spin Hall effect, Two-dimensional material, Large band gap, Strain engineering
T
silicene, germanene, ZeTe5 monolayers, and chemically modified stanene.10−16 However, up to now, only the HgTe/ CdTe and InAs/GaSb quantum wells are established to be 2D TIs by experiment.17−19 Nevertheless, the experimental activity on the QSH effect in HgTe/CdTe and InAs/GaSb quantum wells has remained limited by notorious practical difficulties. Besides, the small bulk band gap makes the realization of QSH effect in HgTe/CdTe and InAs/GaSb quantum wells only at ultralow temperature, which greatly obstructs further possible applications. Therefore, the search for new 2D TIs is of great interest and importance. In order to design 2D TIs for practical utilization, a large energy gap is essential. The reason is that a sizable bulk band gap in 2D TIs is required for stabilizing the boundary current against the influence of thermally activated bulk carriers. Recently, the search for 2D TIs has been extended to Bi and Sb single-element materials20−24 in view of their strong SOC, a precondition which is indispensable for realizing robust topological insulators at high temperature. Thin bilayer films
wo-dimensional (2D) topological insulators (TIs), characterized with quantum spin-Hall (QSH) states,1,2 are emerging as a new state of quantum matter. They have demonstrated a fundamental interplay between electronic structure and topology, thus providing a platform for potentially promising applications in spintronics. The unique fingerprint of TI materials is their gapless boundary state inside a bulk energy gap, which is spin-locked due to the protection of the time-reversal symmetry, namely the propagation direction of surface electrons is robustly linked to their spin orientation.3,4 Accordingly, all the scatterings of electrons in the presence of nonmagnetic impurities in 2D TIs are totally forbidden, leading to dissipationless transport edge channels.5 Furthermore, Majorana Fermions will appear when the material gets in contact with a superconductor.6 Besides their fundamental physical importance Majorana Fermions may play an important role in topological quantum computation schemes.7 Graphene was the first system proposed to be a QSH insulator through the spin−orbital coupling (SOC) effect.8 Unfortunately, this proposal is practically useless because the SOC is too weak to produce an observable effect under realistic conditions, as the gap opened by the spin−orbit interaction turns out to be on the order of 10−3 meV.9 Subsequently, quite a few compounds have been found to be 2D TIs, such as the © XXXX American Chemical Society
Received: October 21, 2014 Revised: December 18, 2014
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DOI: 10.1021/nl504037u Nano Lett. XXXX, XXX, XXX−XXX
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Figure 1. (a) Side and (b) top views of the schematic atomic structures of Me-Bi, Me-Sb, and Me-Pb. Red, blue, and orange balls denote Bi (respectively, Sb or Pb), C, and H atoms, respectively. Solid lines in (b) present the unit cell. Phonon band dispersions of (c) Me-Bi, (d) Me-Sb, and (e) Me-Pb films.
In this work, we provide a systematical investigation on the electronic and topological properties of methyl-functionalized Bi, Sb, and Pb thin bilayer films (Me-Bi, Me-Me-Sb, and MePb) by means of first-principles calculations. We demonstrate that these three films are 2D TIs with a large nontrivial gap. Especially for Me-Bi and Me-Pb, the nontrivial bulk gaps reach nearly 1 eV. The topological characteristics of these systems are confirmed by the direct calculation of the Z2 topological invariant and the nontrivial topological edge states. In addition, the QSH effect in these thin films is robust against biaxial strain. The giant band gap of Me-Bi and Me-Pb and the robust topology against strain are attractive features for potential applications of these thin films in spintronics. Me-Bi and Me-Pb are indeed special systems, calculations on Me-Hg and Me-Tl confirm that these systems are either instable or they are not 2D TIs. The density-functional calculations are performed using the plane wave basis Vienna ab initio simulation pack (VASP) code.39,40 We use the generalized gradient approximations (GGA)41 of Perdew−Burke−Ernzerhof (PBE)42 for electron− electron interactions and the projector augmented wave (PAW) method43,44 for electron−ion interactions. The kinetic energy cutoff is 500 eV and the convergence threshold for energy is 10 −6 eV. For the geometric and electronic structural calculations, the methyl-functionalized Bi, Sb, and Pb bilayer films are simulated by periodic slab geometries with a vacuum space of 18 Å between two layers to avoid spurious interactions between periodic images. The Brillouin zone integration is performed with a 17 × 17 × 1 k mesh for geometry optimization and self-consistent electronic structure calculations. All structures are fully optimized, including cell parameters and atomic coordinates, until the residual forces are less than 0.01 eV/Å. The SOC is included in self-consistent electronic structure calculations.45 We use the CASTEP code46,47 to calculate the phonon dispersion relations and the finite displacement method and PBE are employed. For the convenience of our discussion, we denote the methyl-functionalized Bi, Sb, and Pb thin bilayer films as Me-M (M = Bi, Sb, and Pb). Figure 1a and b present the schematic atomic structure of the methyl-functionalized bilayer films in their most stable configuration. As Figure 1a illustrates, the
based on the two elements have been the focus of attention in the studies of the QSH effect. Particularly, Bi (111) bilayer films are theoretically predicted to be large bulk-gap (about 0.2 eV) 2D TIs,20−22 and these materials have been synthesized and experimentally characterized recently.25−28 For Sb (111) bilayer films, a trivial insulating phase is predicted29 that could be transformed into 2D TI via appropriate strain engineering.15 Meanwhile, the successful production of Bi (111) bilayer films may stimulate the synthesis of related materials, such as Pb bilayer films. Note that, the bi-related three-dimensional (3D) materials have also attracted significantly attention.30,31 Recently, several halogenated and hydrogenated systems were observed to exhibit large bulk band gaps.10,15,16 Unfortunately, a recent experimental work32 has revealed that plasma fluorination and hydrogenation exhibit quick kinetics, with rapid increase of defects and lattice disorder even under short plasma exposures. Therefore, the achievement of fluorinated or hydrogenated systems of high quality is rather difficulty.32 Noteworthy is that the extensive existence of defects and lattice disorder would definitely and completely disrupt their potential applications. Fortunately, unlike the rapid destruction by plasma fluorination and hydrogenation, the reaction of methyl functionalization was observed to be much more moderate reaction kinetics.33 In this respect, the methyl group is particularly suitable for surface passivation over hydrogen and fluorine. Furthermore, it is well established that the hydrogen-functionalized systems are not stable long-term and are greatly air reactive,34 as there is extreme propensity to oxidize under ambient condition.35,36 In contrast, the methylfunctionalized surfaces are resilient toward oxidation,37,38 which is important for device application. Moreover, although regarding the bulk gap halogenation and hydrogenation can enable systems to work at high temperature in principle, the systems begin to amorphize at the relatively low temperature, thus limiting their applications at high temperature.33 On the other hand, replacing the halogenation and hydrogenation to methyl functionalization can considerably enhance the systems’ thermal stability, allowing for achieving large-bulk-gap’s full potential.33 Consequently, it becomes more interesting to realize large-bulk-gap topological phases in the methylfunctionalized Bi, Sb, and Pb thin bilayer films. B
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Nano Letters metal centers have a stable sp3 configuration analogous to the recently experimentally produced methyl-functionalized germanene,33 with the chemical functional groups bonding on both sides of the plane in an alternating way. The crystal structures of these thin films present a hexagonal Bravais lattice and can be described by the symmetry point group D3d, with one pair of M atoms and methyl groups in the unit cell. All these systems thus possess inversion symmetry. The optimized lattice constants of Me-Bi, Me-Sb, and Me-Pb are 5.497, 5.253, and 5.026 Å, respectively, with the buckling [d, as defined in Figure 1a], measured by the spacing in the bismuth, antimony, or lead planes, being 0.054, 0.032, and 0.855 Å, respectively. To evaluate the energetic stability of these systems, we investigate their formation energy, defined as Ef = Etotal − nMEM − nCEC − nHEH, where Etotal is the total energies of Me-M thin films, nM, nC, and nH are the numbers of M, C, and H atoms in the unit cell, respectively. EM, EC, and EH are the chemical potentials of M, C, and H atoms using bulk Bi (bulk Sb or bulk Pb), diamond, and H2 molecules, respectively, as reference systems. The calculated formation energies for Me-Bi, Me-Sb, and MePb are −0.52 eV, −0.65 eV, and −0.21 eV, respectively, suggesting that all of the systems predicted here would be stable. We further confirmed the dynamic stability of these thin films by calculating their phonon spectra, and the corresponding results are displayed in Figure 1c−e. All phonon branches are positive in Me-Bi, Me-Sb, and Me-Pb, confirming the dynamic stability of these three thin films. The band structures of Me-Bi and Me-Sb in absence of SOC are shown in Figure 2a and b, respectively. The results in Figure
Similar results on halogenated Bi bilayers have been reported recently by the Yao group.15 Interestingly, in addition to the Dirac points, we occasionally observe a very different band composition of the states around the Fermi level between MeBi (or Me-Sb) and Bi (111) bilayer films. Detailed analysis shows that the SOC determines the band structures of Me-Bi and Me-Sb around their Fermi levels. As shown in Figure 2a and b, the SOC lifts the degeneracy of valence and conduction bands around Fermi level, and thus produces a semimetal-to-semiconductor transition in the band structures of both Me-Bi and Me-Sb. From Figure 2b, we can observe a direct band gap for Me-Sb, with the valence band maximum and the conduction band minimum both located at the K points. In contrast, for Me-Bi, the conduction band near Fermi level at the Γ point are downshifted, whereas the conduction band at the K point is upshifted when turning on SOC, yielding a global indirect band gap, as shown in Figure 2a. Consequently, Me-Bi and Me-Sb have insulating phases with an indirect and a direct band gaps, respectively. As observed in previously reported 2D TIs like silicene, germanene, and ZeTe5 monolayers,11,12 the SOC-induced deformation of the Dirac points is a strong indication of the existence of topologically nontrivial phases. This means that Me-Bi and Me-Sb are possible nontrivial 2D TIs, and we confirm this hypothesis by the Z2 index and edge state calculations in the following. For practical applications of the QSH effect, it is instructive to evaluate the bulk band gap of the two thin films. Because these two systems are semimetals in the absence of SOC, we can conclude that the gap opening at the Dirac points must originate from SOC. More remarkably, as shown in Figure 2a and b, extremely large band gaps of 0.934 and 0.386 eV are established for Me-Bi and Me-Sb, respectively, which are large enough for room-temperature applications. The giant magnitudes of bulk band gaps are among the largest counterparts ever found in 2D TIs or even in 3D TIs. Especially for Me-Bi, its bulk band gap is three times larger than that of the pure Bi (111) bilayer films (about 0.2 eV).20−22 QSH systems with such giant bulk band gaps are desirable for realizing many exotic phenomena and fabricating new quantum devices that can operate at high temperature. To firmly establish the nontrivial topological nature of Me-Bi and Me-Sb, we analyze the Z2 topological invariant (ν) of these two systems. According to the Z2 classification, ν = 1 characterizes a topologically nontrivial phase and ν = 0 means a topologically trivial phase.48,49 Due to the presence of structural inversion symmetry in Me-Bi and Me-Sb, the method developed by Fu and Kane48,49 is valid here: the Z2 topological invariant can be directly calculated based on the parity of the Bloch wave function for all the filled bands at all the time-reversal-invariant momenta (TRIM). For Me-Bi and Me-Sb, there are four TRIM points (Ki) in the Brillouin zone, namely one at Γ point and three at M points. Accordingly, the topological indexes ν are established by
Figure 2. Electronic band structures of (a) Me-Bi, (b) Me-Sb, and (c) Me-Pb. The red lines correspond to band structures without SOC, and the blue lines correspond to band structures with SOC. The Fermi energy is set to 0 eV.
2a and b clearly indicate the band structure of Me-Bi is almost the same as that of Me-Sb. For both thin films, a semimetal nature can be observed with the valence and conduction bands meeting in a single point of the Brillouin zone, the Fermi level is located exactly at the cross point. Examination for the states near the Fermi level reveals that the slopes, that is, the first derivatives of the band energies, essentially remain unchanged when approaching the cross points. The results suggest that the cross points of valence and conduction bands are Dirac points, which are equivalent to those of graphene and located at the same positions in the Brillouin zone, the K and K′ points.
N
δ(K i) =
∏ ξ2im , (−1)ν m=1
4
=
∏ δ(K i)=δ(Γ)δ(M)3 i=1
where δ is the product of parity eigenvalues at the TRIM points, ξ = ±1 are the parity eigenvalues and N is the number of the occupied bands. For Me-Bi and Me-Sb, there are 12 spindegenerate bands for 24 valence electrons. Thus, we calculate the parity eigenvalues of the Bloch wave function for the 12 occupied spin-degenerate bands at all TRIM points in the C
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Nano Letters Table 1. Parities of Occupied Spin-Degenerate Bands at the TRIM Points for Me-Bi, Me-Sb, and Me-Pba Me-Bi Me−Sb Me−Pb
Γ 3M Γ 3M Γ 3M
+ − + − + −
− + − + − +
+ − + − + −
− + − + − +
+ + + + + +
+ − + − + −
− − − − − −
− + − + − +
− + − + + +
+ − + − − −
+ − + − + −
+ + + +
(−) (+) (−) (+) (−) (+)
a
Here, we show the parities of 12 occupied spin-degenerate bands for Me-Bi and Me-Sb and 11 occupied spin-degenerate bands for Me-Pb. Positive and negative signs denote even and odd parities, respectively. The sign in parentheses is the product of the parity eigenvalues of the occupied spindegenerate bands.
Figure 3. Calculated electronic band structures of the zigzag nanoribbons of (a) Me-Bi, (b) Me-Sb, and (c) Me-Pb with SOC. The Fermi energy is set to 0 eV.
Pb owing to the heavy SOC strength of Pb. As illustrated in Figure 2c, a giant band gap of about 1.115 eV opens at the Γ point when SOC is turned on. Because the valence band maximum of Me-Pb locates slightly off the Γ point, this system displays a global indirect band gap of 0.964 eV. We then also investigate the Z2 topological invariant of Me-Pb by evaluating the parity eigenvalues of its 11 filled spin-degenerate bands at the TRIM points. It is found that Me-Pb is indeed QSH insulators with a nontrivial topological invariant ν = 1. This extremely large band gap would ensure Me-Pb being a promising platform for realizing QSH effect. The nontrivial topological nature of Me-Bi, Me-Sb, and MePb should support an odd number of topologically protected gapless edge states, which will connect the valence and conduction bands of each system at certain k-points. To see the topological features of the three systems explicitly, we calculated the edge states using a zigzag nanoribbon structure with symmetric edges, where we passivate the dangling bonds of all edge atoms with hydrogen atoms. The nanoribbon structures of Me-Bi, Me-Sb, and Me-Pb adopted here contain 46 Bi/Sb/Pb atoms, 46 C atoms and 140 H atoms (see Supporting Information for more detail). All the widths of the nanoribbons exceed 10 nm, which is sufficient to avoid the interaction between their two edges. The calculated edge states with SOC for all three systems are shown in Figure 3. For the three nanoribbons, one can clearly see the topological edge states with a single Dirac-type crossing at M or Γ points. Such edge states are important for the applications in spintronics
Brillouin zone, the corresponding results are listed in Table 1. For Me-Bi, as expected, the product of the parity eigenvalues at the Γ point is −1, whereas it is +1 at M point, which thus yields a nontrivial topological invariant ν = 1, that is, a topological nontrivial state. For Me-Sb, the products of the parity eigenvalues at these two TRIM points are totally different, leading to a nontrivial topological invariant ν = 1. Consequently, both the studied thin films are indeed nontrivial topological insulators, suggesting that the QSH effect can be realized in both systems. Figure 2c shows the band structures of Me-Pb without and with SOC, respectively. In the case without SOC, it is clearly seen in Figure 2 that the main characteristics of the band structures of Me-Pb and Me-Bi (or Me-Sb) are close to each other, despite the fact that the Fermi level is downshifted in Me-Pb as compared to Me-Bi (or Me-Sb). Such a downshifting is because that Me-Pb is less than Me-Bi (or Me-Sb) two electrons per unit cell. Taking a look at the bands of Me-Bi (or Me-Sb) around −3 eV, one can notice that there are two bands touching in one point at the Γ point in the case without SOC, and SOC could open a large energy gap at the touching point (see Figure 2). As shown in Figure 2c, for Me-Pb without SOC, such downshifting leads Fermi level to locate exactly at the touching points at the Γ point. In other words, for Me-Pb, the valence band maximum and conduction band minimum meet in one point at the Γ point when excluding SOC. Therefore, similar to Me-Bi and Me-Sb, it is also expected that the SOC effect could open a giant band gap at the touching point in MeD
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Figure 4. Strain dependencies of the global bulk band gap of (a) Me-Bi, (d) Me-Sb, and (g) Me-Pb with SOC. The corresponding band structures of (b and c) Me-Bi, (e and f) Me-Sb, and (h and i) Me-Pb under the strain (6% and −6%) with SOC.
that the change of band components near the Fermi level is induced by surface decoration, which removes the pz orbits and leads to the giant band gap in Me-Bi. Similar situation is also observed in Me-Sb and Me-Pb, as the bands around the Fermi level are also mainly contributed by px and py orbitals of Sb or Pb. Although three bilayers (Sb, Pb, and Bi) without surface decoration share a similar geometric structure, their electronic and topological properties are distinctly different. For example, Pb bilayer is a metal while Sb (111) bilayer is a semiconductor. Meanwhile, although Me-Sb is a nontrivial 2D TI as we demonstrated, Sb (111) bilayer is actually a normal insulator with a trivial gap.20 The equilibrium lattice constant of Sb (111) bilayer is 4.01 Å, which is significantly smaller that that of MeSb with the value of 5.253 Å. In previous work,20 it has been demonstrated that the topological properties in Sb (111) bilayer is closely related to the Sb−Sb bond strength and nontrivial topological phase can thus be realized in Sb (111) bilayer under a tensile strain. With this idea in mind, we can then expect that some features of Me-Sb would correspond to those of a tensile strained Sb (111) bilayer. This arises an important question: is the topological nontrivial state in Me-Sb stable against strain? Based on the fact that the lattice constant of Me-Sb is larger than that of Sb (111) bilayer by about 30%, the topological nature in Me-Sb would be very stable against strain in principle although the detailed characteristic may depends sensitive on strain, as we will demonstrate. Strain engineering is a powerful approach to modulate the electronic properties and topological natures in 2D materials, and thus it is interesting to study these effects also in these materials. Previous studies indicated that the trivial semiconductor β-InSe could enter into a topologically nontrivial state at a reasonable biaxial lattice expansion.50 Many other similar examples based on this idea have been confirmed, such as Sb (111) bilayer, GaS and GaSe.14,51 Therefore, one main concern is to check the strain tolerance of nontrivial topology of these three systems. In what follows, we investigate the effect of biaxial strain on the topological properties of Me-Bi, Me-Sb, and Me-Pb. We impose biaxial strain on the 2D planes of these systems by turning the planar lattice parameter. After applying strain, the atomic structure is reoptimized. The magnitude of strain is described by ε = Δa/a0. Here, a0 and a = Δa + a0
considering their robustness against scattering. For these three systems, the symmetric nanoribbon model leads to one energetically degenerate crossing point located at opposite sides. Due to the nontrivial topology, the edge states always exist, though their details may depend explicitly on the edge. Thus, the edge-state calculation confirms conclusively the topologically nontrivial nature of these three materials. Besides, we also studied the corresponding properties of MeSn which is the counterpart of Me-Pb. We find that, in contrast to Me-Pb, Me-Sn is a semiconductor with a band gap when excluding SOC, and introduction of SOC does not cause a band inversion. Therefore, Me-Sn is a trivial semiconductor. This can be rationalized as the topological properties of Me-Sn and Me-Pb are closely related to the bond lengths between the M atoms. Although Me-Sn shares similar atomic structure as Me-Pb, the Sn−Sn bonds are relatively small due to the strong bonding between the adjacent Sn atoms. As a result, the splitting between the s and p orbitals would be very large, leading to a band gap in Me-Sn rather than a meeting point without SOC. The strength of SOC in Me-Sn is not large enough to induce band inversion, which thus yields a trivial topological invariant ν = 0, that is, a topological trivial state. The above results show that Me-Bi, Me-Sb, and Me-Pb are nontrivial 2D TIs. Particularly for Me-Bi and Me-Pb, the nontrivial band gap can amount to 0.934 and 0.964 eV, respectively. Given the fact that the band gap created by SOC in Bi (111) bilayer films is only about 0.2 eV,20−22 it is interesting to note that from Bi (111) bilayer films to Me-Bi, only groups with small SOC are added, but the SOC-induced band gaps are significantly enhanced. This crucial feature can be understood by the analysis of band components. For both MeBi and Bi (111) bilayer film, they can be described by the symmetry point group D3d. The symmetry character reflects the fact that p orbitals around the Fermi level are split into two groups, namely pz orbitals and px and py orbitals, respectively. It is important to mention that from pz orbitals to px and py orbitals, the strength of SOC can be enhanced manifold. Interestingly, by projecting the bands onto different atomic orbitals, we find that the energy spectrum of Me-Bi near the Fermi level is derived mostly from px and py orbitals of Bi atoms, whereas the energy spectrum of Bi (111) bilayer films is predominantly pz orbitals. Thus, it is reasonable to conclude E
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denotes the lattice parameters of the unstrained and strained systems, respectively. We find that for Me-Bi, Me-Sb, and MePb, the strain can reach up to at least ±8% without destroying the feature that products of the parity eigenvalues at the two TRIM points are distinct. This suggests that these three systems maintain their nontrivial topological phase within the strain range of ±8%. Such robust topology against lattice deformation makes it easier for experimental realization and characterization. On the other hand, for potential applications, it is also obviously of particular interest to obtain an effective way to tune the topological properties of these new 2D TIs by external factors, such as biaxial strain. As shown in Figure 4, the magnitude of nontrivial global bulk band gap can be modified significantly by strain, implying the interatomic coupling can modulate the topological natures of these systems. To gain a deeper insight into the strain effect on the band structure, the band structures of these three systems under strain 6% and −6% with SOC are presented in Figure 4. The tunability of nontrivial band gap offers a promising route to engineer the topology by biaxial strain for the benefit of spintronics. We wish to point out that although the strain may decrease the value of band gap for these three systems, their nontrivial bulk band gaps within the strain range of ±8% are still very large, especially for Me-Bi and Me-Pb. This can be reasonably understood because of the extremely large band gaps in their native structures. We wish to emphasize that, similar to graphene8 and silicene,13 the inclusion of SOC could not induce band inversion between the valence and conduction bands at the TRIM points (M and Γ points) in Me-Bi and Me-Sb, the effect of SOC in these systems is rather to open an energy gap at the K point around the Fermi level. For Me-Pb, the mechanism is similar to the case of surface functionalized stanene.10,52 Meanwhile, although Me-Bi and Me-Sb share similar band structures to graphene and silicene around the Fermi level, their band component of the states around the Dirac points are significantly different. In detail, the states around the Dirac points are mainly contributed by the px and py orbitals for MeBi and Me-Sb, whereas those of graphene and silicene are composed by pz orbitals. Such band component of the Dirac points in Me-Bi and Me-Sb was also reported in hydrogenated Bi bilayer.15 In summary, we show through first-principles calculations that Me-Bi, Me-Sb, and Me-Pb are nontrivial 2D TIs in their native structures. These three systems exhibit extraordinary large topological energy gaps of 0.934 eV, 0.386 eV, and 0.964 eV, respectively. These giant magnitudes of the bulk gaps are among the largest counterparts ever found in 2D TIs or even in 3D TIs. There are two reasons to ensure such extraordinary band gaps in Me-Bi and Me-Pb: (1) the Bi and Pb are two of the main group elements that has the strongest SOC and (2) the bands around the Fermi level are mainly contributed by the px and py orbitals rather than pz orbitals and the SOC within the px and py orbitals are strong. The direct calculation of Z2 topological invariant and nontrivial topological edge states confirm the topological characteristic of these systems. Further, we find that these three systems maintain their nontrivial topological phase within the strain range of ±8% although the nontrivial bulk gap can be effectively tuned by strain, indicating their topology is robust against strain. All of these render that Me-Bi, Me-Sb, and Me-Pb are of fundamental scientific interest and promising applications.
Letter
ASSOCIATED CONTENT
S Supporting Information *
Atomic structures of Me-Bi, Me-Sb, and Me-Pb nanoribbons used to calculate the edge states. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected] (Y.M.). *E-mail:
[email protected] (T.H.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Financial support by the European Research Council (ERC, StG 256962) and the National Science foundation of China under Grant 11174180 are gratefully acknowledged.
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REFERENCES
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DOI: 10.1021/nl504037u Nano Lett. XXXX, XXX, XXX−XXX
