Topologically Protected Metallic States Induced by a One-Dimensional

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Topologically-protected metallic states induced by a onedimensional extended defect in the bulk of a 2D topological insulator Erika Nascimento Lima, Tome M Schmidt, and Ricardo W. Nunes Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b00521 • Publication Date (Web): 10 Jun 2016 Downloaded from http://pubs.acs.org on June 14, 2016

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Topologically-protected metallic states induced by a one-dimensional extended defect in the bulk of a 2D topological insulator Erika N. Lima,†,¶ Tome M. Schmidt,‡ and Ricardo W. Nunes∗,† Departamento de Física, ICEx, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, MG, Brazil, and Instituto de Física, Universidade Federal de Uberlândia, 38400-902, Uberlândia, MG, Brazil E-mail: [email protected] Phone: +55 (0)31 34095655. Fax: +55 (0) 31 34095600

Abstract We report ab initio calculations showing that a one-dimensional extended defect generates topologically-protected metallic states immersed in the bulk of two-dimensional topological insulators. We find that a narrow extended defect, composed of periodic units consisting of one octagonal and two pentagonal rings (a 558 extended defect), embedded in the hexagonal bulk of a bismuth bilayer, introduces two pairs of onedimensional counterpropagating helical-fermion electronic bands with the opposite spinmomentum locking characteristic of the topological metallic states that appear at the ∗

To whom correspondence should be addressed Universidade Federal de Minas Gerais ‡ Universidade Federal de Uberlândia ¶ Current address: Universidade Federal de Mato Grosso, Departamento de Matemática, Rondonópolis, Mato Grosso, Brazil †

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edges in two-dimensional topological insulators. Each one of these pairs of helicalfermion bands is localized, respectively, along each one of the zigzag chains of bismuth atoms at the core of the 558 extended defect, and their hybridization leads to the opening of very small gaps (6 mev or less) in the helical-fermion dispersions of these defect-related modes. We discuss the connection between the defect-induced metallic modes and the helical-fermion edge states that occur on bismuth bilayer ribbons.

Keywords: topologically-protected metallic states, one-dimensional extended defect, first principles calculations, bismuth bilayers

Introduction Topological insulators (TIs) are a new class of materials theoretically predicted to exist in 2005, 1,2 with the first experimental confirmation in HgTe/CdTe quantum wells reported in 2007. 3–5 TIs have since been subject of intensive theoretical 4,5 and experimental 6 studies due to the coexistence of an insulating bulk band structure with a non-trivial topology that, when interfaced with a topologically trivial insulator such as the vacuum, gives rise to timereversal-protected metallic surface states, with helical-fermion dispersions spanning the bulk band gap in the one-dimensional (1D) edges in two-dimensional (2D) TIs. Existence of the edge states is a requirement imposed by the different topologies of the band structures across the interface. In these edge states, the spin quantization axis and the momentum direction are locked-in, implying that the metallic edge states are protected from backscattering, rendering their electronic conductance robust against the presence of disorder. Robust conduction and spin polarization may allow the manipulation of edge modes of TIs in many applications such as spintronics 7 and quantum computation. 4,5 While topological-insulating band structures are also found in three-dimensional (3D) systems, manipulation of the metallic surface modes in 3D TIs is commonly hampered by the difficulty in tuning the Fermi level to achieve sufficiently low (ideally null) levels of bulk

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Figure 1: In all geometries, atoms in the upper (lower) Bi triangular sub-lattice are shown as green (blue) circles. (a) Top and side view of a pristine Bi bulk bilayer. (b) Electronic band structure and Brillouin zone of a pristine Bi bilayer. (c) Top and side view of a Bi bilayer with an embedded 558 defect (BiBL+558 in the text). Supercell-vector dimensions are 77.40 Å along the x-axis and 8.66 Å along the y-axis. The latter is the period of the 1D cell of the extended defect. (d) Helical-fermion linear dispersions of the defect-related electronic modes, crossing the band gap of the bulk bilayer. Defect-related bands are doubly degenerate, with one pair of counterpropagating modes on each one of the zigzag chains of bismuth atoms bordering the bismuth dimers at the geometric center of the defect. (e) Top and side view of a Bi bilayer with an embedded quadruple-pentagon double-octagon defect (BiBL+Q5D8 in the text). Supercell-vector dimensions are 72.20 Å along the x-axis and 8.66 Å along the y-axis. (f) Band structure of Bi+Q5D8. carriers, and the metallic surface carriers are significantly outnumbered by bulk carriers in most 3D TI samples. 8,9 Hence, 2D TIs can be advantageous in transport applications, 3


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because the 2D bulk is fully exposed to chemical manipulation and, besides, the bulk Fermi level can also be tuned by proper gating. When an insulating bulk is achieved, electrons can conduct only along the edges in these structures. This has motivated several studies to find candidate 2D TI systems. In this work, we show, by means of ab initio density functional theory (DFT) calculations, the formation of “edge-like” 1D metallic electronic states localized on the core of an extended 1D defect embedded in the bulk of a single (111)-oriented bismuth bilayer (BiBL). We show that, in this system, the extended defect generates two pairs of topologically-protected helical-fermion modes, each pair localized on each one of the two zigzag chains of Bi atoms that are connected by a line of interstitial dimers at the core of the 1D extended defect. The helical-fermion pairs are shown to couple and hybridize only weakly, and to have opposite chiralities, which in principle allows for backscattering of carriers propagating in these defectinduced modes. Furthermore, we also introduce an alternative crystalline form of a Bi bilayer, consisting entirely of pentagonal and octagonal rings, which we refer to as the pentaoctite form of a BiBL. Below, we show that the pentaoctite BiBL, with a formation energy that is only 0.045 eV/atom larger than the bulk-derived hexagonal pristine layer, is also a 2D topological insulator. Our calculations show that the occurrence of spin-polarized helical-fermion electronic states localized on the core of the extended line defect, which is fully immersed in the bulk of the 2D TI BiBL, is closely related to the emergence of metallic states along the edges of sufficiently large zigzag-terminated bismuth nanoribbons (ZBiNR). The 1D extended defect in our study is a buckled version of the so-called 558-defect, composed of periodic units consisting of one octagonal and two pentagonal rings, as shown in Fig.1(c). A flat version of the 558-defect has been shown experimentally to occur in graphene monolayers, 10,11 and theoretically to display magnetic quasi-1D electronic states in n-doped layers. 12,13 While the possible occurrence of metallic fermionic modes along the core of 1D extended

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defects (dislocations) in 3D TIs has already been predicted theoretically for a model tightbinding Hamiltonian, 14,15 to the best of our knowledge this work provides the first demonstration, by ab initio calculations, of the emergence of topologically-dictated helical-fermion states along the core of a 1D extended defect immersed in the bulk of a 2D TI. In the following discussion, we provide strong numerical evidence that the helical modes generated by the 558 defect, in the bulk of the 2D bismuth bilayer, do not belong to the class of dislocation-related helical modes predicted to occur on grain boundaries in 2D and 3D TIs. 14–16 Rather, the 558-defect states are closely related to the topological edge modes that occur at the zigzag edges of a bismuth bilayer ribbon.

Methodology Structural and electronic properties of extended 1D defects in a BiBL, in the present work, are computed using the DFT scheme implemented in the VASP code. 17 The projectoraugmented-wave method 18 is used to describe the ionic core-valence electron interactions. The generalized gradient approximation (GGA) is employed to describe the exchange and correlation potential. 19 For one of the systems in our study, we employed the PBE0 hybrid functional to obtain a more realible value of the gap between valence and conduction bands. The PBE0 functional 20,21 mixes the PBE (75%) and Hartree-Fock exchange functionals (25%) along with the full PBE correlation. Spin-orbit coupling is included in all electronic structure calculations. Electronic wave functions are expanded in plane waves with energy cutoff of 300 eV. Convergence with respect to the energy cutoff was carefully checked from calculations with cuttofs in the 200-400 eV range. Geometries are optimized until the forces on each atom are less than 0.03 eV/Å. In our BiBL supercells containing extended defects, the bilayer is on the xy plane, and a 6x6x1 k-point sampling is used for the Brillouin zone (BZ) integrations. In a few selected cases, convergence of the BZ integrations with respect to BZ sampling was

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also verified by means of calculations employing 5x5x1 and 7x7x1 samplings. An electronictemperature smearing of 0.05 eV was employed for the occupation of the electronic states.

Results and discussion It is well established that a single (111) Bi bilayer in its pristine form, shown in Fig.1(a), is a 2D TI (a quantum spin Hall system). 22–27 Metallic edge states are found at the borders of a BiBL finite sheet, with a pair of counterpropagating helical-fermion states with opposite spin textures (or spin-momentum locking) on each border. 28,29 The lower panel in Fig.1(a) shows that the BiBL is formed from two Bi 2D triangular sublattices displaced by 1.74 Å along the (111) direction (the z-axis in our supercells). Figure 1(b) shows the insulating band structure of an infinite pristine BiBL (thus devoid of edges) along the Γ-K and Γ-M lines in the BZ, with an indirect band gap of ∼0.5 eV, within the GGA approximation. The top of the valence band at the Γ point is ∼0.1 eV below the absolute maximum of the valence band, so the direct gap at Γ is ∼0.6 eV. The structure of the 558-defect embedded in a (111) BiBL is illustrated in Fig. 1(c). The 558-defect is a zigzag-oriented (along the y-axis of the cell) translational grain boundary between two crystalline domains shifted with respect to each other by a third of the armchairdirection lattice period, along the x-axis of the cell. The relative shift leaves a seam between the two domains that is filled with a roll of Bi dimers, thus forming the two-pentagonone-octagon periodic unit of the defect. The period of the 558-defect is twice the lattice constant of the pristine BiBL (a = 4.33 Å). In the following, we shall refer to the BiBL with the embedded 558-defect as BiBL+558. After relaxation, the BiBL+558 system retains the buckled structure of the pristine BiBL. The Bi-Bi bond length in a pristine BiBL is 3.04 Å. In the pentagonal and octagonal rings of the BiBL+588 system, bond-length values range from 3.03 Å to 3.11 Å. Figure 1(d) shows the band structure of the supercell for the BiBL+558 system. Inclusion

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of the 558-defect leads to the emergence of two degenerated pairs of gap-crossing bands, with one pair of bands localized on the zigzag chain of Bi atoms on the left of the dimers at the center of the defect, and the other pair localized on the zigzag chain on the right of the dimers, as shown by the charge density plots in the bottom panels in Fig. 2. The defect-related bands span the band gap of the bulk BiBL, and show helical-fermion linear dispersions with the characteristic spin textures of 2D TI edge modes, as also shown in Fig. 2. Figure 2 shows that the two pairs of helical-fermion bands at the 558 defect have opposite spin textures, with spin ’up’ (’down’) states moving ’up’ (’down’) on the left side of the dimers and ’down’ (’up’) on the right side of the dimers. Hence, the two zigzag chains of Bi atoms at the core of the 558 defect behave as right and left ribbon edges, preserving the “handedness” (i.e., the circulation of the counterpropagating spin channels) of each edge, which explains the inversion of the spin texture between the two sides of the Bi dimers. This is an important point, concerning the spin transport of the defect modes along the core of the 558-defect, since due to the inverted spin-momentum locking between the helical-fermion bands on the two sides of the defect, backscattering is possible by a process where the charge carriers are scattered to the other side of the defect and propagate backwards, without flipping the spin. The possible formation of 2D helical-fermion modes on 2D boundaries (domain or grain boundaries) in 3D TIs has been discussed previously for TB hamiltonians. 14,30 We anticipate that the preservation of the handedness of the boundary modes may prove true also in these cases. Hence, from our result for a 1D boundary in a 2D TI, we expect inversion of the spin texture and the possibility of backscattering between helical-fermion modes on the two sides of 2D boundaries in 3D TIs. Below, we address numerically the hybridization between the two sets of helical-fermion modes on each side of the 558-defect, and come to the conclusion that the line of dimers at the core of the defect strongly reduces the coupling between the modes. It is thus conceivable that the possible backscattering between the two sides of the defect, we alluded to above, may be inhibited by the reduced hybridization between the two sets of helical bands at the

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Figure 2: Spin-momentum locking and charge densities of the two pairs of helical-fermion modes at the core of the 558 extended defect in Bi bilayers. (a) Helical-fermion bands localized on the left side of the dimers, as shown by an isosurface of the related charge density, in the lower panel. (b) Helical-fermion bands localized on the right side of the dimers, with an isosurface of the related charge density in the lower panel. defect. We propose the following interpretation for the appearance of the helical-fermion states 8


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Table 1: Energy gap (Eg in eV) for 2D Bi ribbons (ZBiNR) as a function of the ribbon width (L in Å). Last two entries show Eg values for Bi layers with the extended 1D defect (BiBL+558), for two values of the distance between the 1D defect and its periodic images. L (Å) 12.5 20.0 27.6 35.1 50.7 58.2 99.45 57.4 (BiBL+558) 77.4 (BiBL+558)

Eg (eV) 0.452 0.237 0.129 0.071 0.026 0.011 0.001 0.022 0.005

in the BiBL+558 system: the region occupied by the line of dimers at the geometric center of the 558 defect behaves topologically as a trivial insulating material, akin to the vacuum surrounding the edges of a BiBL ribbon. Hence, the two pairs of helical-fermion modes we observe at the core of the 558 extended defect are those expected for two zigzag edges connected by the line of dimers at the center of the defect. Furthermore, previous works 28,29 have established that by saturating the edges of a ZBiNR with H atoms, the helical-fermion bands of topological edge states move from the 1D BZ borders (the ±π/a points), in the case of unsaturated edges, to the Γ point in the H-saturated-edge case. (Note that the ±π/a and Γ points of the 1D defect BZ are the points onto which the M points of the BIBL 2D BZ project.) We observe the same behavior in the metallic states along the core of the 558-defect in a BiBL: the crossing point of the helical-fermion branches occurs at the Γ point. Hence, the line of Bi dimers at the core of the defect plays a role of saturation likewise that of hydrogens in a ZBiNR. We will expand on the above interpretation in the following, but let us first analyze the nature of the hybridization between the two pairs of helical-fermion modes on the core of the 558-defect. We start from the behavior of the gap of the helical-fermion edge modes of a ZBiNR, due to the quantum tunneling between the topological states with the same spin alignment in 9


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the two edges of the ribbon. The gap in the edge helical-fermion dispersions in the ZBiNR reflects the fact that the edge states penetrate into the bulk, with an exponential decay, and the coupling between the states from the two edges leads to the opening of a bondingantibonding gap. 31 As shown in Fig. 3 and Table 1, as we increase the width of the ribbon the gap decreases, as expected. Figure 3(e), shows an exponential fitting of the ribbon gap (Eg ) as a function of the ribbon width (L). For a ∼57 Å-wide ribbon the gap is 11 meV. (a)

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periodic images, one to the right and the other to the left of the defect in the home cell. The last two entries in Table 1 show the gap of the defect-related modes for the BiBL+558, for two different widths of the pristine region of the supercell. The gap for the supercell with a ∼57 Å-wide pristine region is 22 meV, twice the value for a ribbon this wide, as included in Table 1, and the gap for a ∼74 Å-wide supercell is only 5 meV. The above results are suggestive of a weak coupling between the two sets of helicalfermion modes in the core of the 558-defect, mediated by the line of dimers in the the defect core. In order to investigate the degree of hybridization between the defect helical modes, we compute the gap in the edge states for two H-saturated ribbons, coupling through the vacuum, in the same geometric configuration as in the BiBL+558 system, i.e., we removed the line of Bi dimers and kept the distance between the edges of the ribbons unchanged, at 5.35 Å, which is the distance between the zigzag chains on the 558 defect core in the BiBL+588. In this case, the gap increased by an order of magnitude to 134 meV. This result indicates that in the BiBL+558 the line of dimers strongly supresses the coupling between the helical-fermion edge states. This is further confirmed by a calculation of the gap for a ∼77 Å-wide BiBL+558 system that yields a value that falls on top of the exponential fitting for an isolated H-saturated ribbon of the same width, as shown by the blue square in Fig. 3. We must stress that gaps in the helical-fermion bands in the BiBL+558 are due to two different couplings: (1) The coupling between the 558 defect and its periodic images, due to our use of periodic boundary conditions in the x-axis, where lattice translational invariance is broken by the introduction of the defect. This is a supercell-size effect, since for sufficiently large cells this coupling can be made as negligible as needed. (2) The coupling between the two pairs of helical modes localized on the defect. This is a physical effect that is due to the hybridization between the two pairs of helical-fermion bands, through the line of dimers, as shown by the charge density isosurfaces in Fig. 2.

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In order to have an estimate of the effect (1) above in our supercells, we employ the exponential fitting of the ZBiNR gaps in Fig. 3(e), and obtain the following gaps for ribbons with the sizes we employed for the two BiBL+558 supercells included in Table 1: L = 57.42 Å→ Eg = 16 meV and L = 77.42 Å→ Eg = 4 meV. Note that these values are only an estimate of the cell-size effect, since in our BiBL+558 cells the zigzag chains at the 558 defect are coupled to periodic images on both sides, while in the ribbon cells the zigzag chain at one edge couples only to the chain on the other edge (large vacuum regions in the ribbon supercells strongly suppress the coupling with the periodic images of the ribbon). The above values allow us to estimate that the physical coupling (2) above, of the two helical modes on the 558 defect, due to hybridization, is of the order of 1-6 meV only, i.e., smaller than thermal energy fluctuations at room temperature. Let us go back now to our interpretation for the emergence of topological edge states along the core of the 558-defect. It rests on the assumption that the defect region acts as a portion of a trivial insulator. In order to confirm this picture, we compute the Z2 topological invariant for a pristine BiBL and for two “defective” Bi 2D layers. For a pristine BiBL, the Z2 has been obtained by looking at the parity of the occupied bands at the time-reversal-invariant momenta (TRIMs). We obtain Z2 = 1, confirming its non-trivial topological character, with band inversions at the M points. 23 For a 2D bulk composed by a sequence of 558-defects connected by a stripe of single hexagons, as shown in Fig. 4(a), from the parity of the bands at the TRIMs we obtain Z2 = 0, showing that, indeed, this stripe of a hypothetical penta-hexa-octo form of 2D Bi bilayers is a trivial insulator. The band structure for the penta-hexa-octo Bi bilayer is shown in Fig. 4(b). One may be tempted to consider the possibility that the helical-fermion bands we observe in the 558 extended defect could be derived, somehow, from individual 558 point defects, acting as dislocations in the 2D BiBL, and binding a Kramers pair of π-flux related topological zero modes, deep in the bulk band gap. These midgap zero-mode states would then form the helical-fermion bands of extended electronic states by hybridization, when the 558

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Figure 4: (a) and (b) show a top of view and the band structure of a penta-hexa-octo form of a 2D Bi layer, respectively. line defect would form as an accumulation of such point defects. Several arguments point otherwise: A first argument is the fact that the 2D BiBL belongs to the Γ phase in the topological classification discussed in Ref. 16 The reason is that the minimal direct gap is located at or very near the Γ point as seen in Fig. 1(a). So, the gap function that determines the nature of the Skyrmion lattice in a 2D TI has a minimum at, or very near, the Γ point, which means that the 2D BiBL belongs to the Γ phase in the topological classification in Ref. 15 In Γ phase 2D TIs, a dislocation does not bind a Kramers pair of zero-mode midgap states. Furthermore, even in an M phase 2D TI, a dislocation would bind either one or three Kramers pairs of zero-mode states, depending on the number of band inversions that give rise to the TI phase. In the 558 extended defect, we have actually two pairs of helical-fermion bands, one on each side of the defect, and this strongly supports our interpretation of such states as being generated by two zigzag interfaces of the 2D BiBL with a "trivial insulator" portion at the core of the 558 defect. Note that the spin-momentum locking we obtain for the two branches of defect-related helical states, shown in Fig. 2, is precisely what one would expect for two Bi ribbons with zigzag edges meeting the line of dimers to form the 558 extended defect. Moreover, in the honeycomb lattice, there is no finite (or point) defect with the ring structure of the extended 558 defect, since any finite version of this defect would be bound by heptagons at both ends. Further confirmation of the topological nature of the defect bands of the 558 defect in 13


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Fig. 2 can be provided by a calculation with some kind of disorder in the defect region. The spin texture and the linear dispersion of the topologically-dictated modes can be destroyed by a magnetic impurity, due to the breaking of the required time-reversal symmetry. We have performed a calculation where we include an adsorbed Mn atom atop a Bi atom in the defect region. We add one Mn atom at every defect periodic unit, a rather large concentration, and observe the development of a parabolic dispersion in one of the formely linear dispersions of the topological defect states, and also the spin texture of the helical modes is lost due to the coupling with the magnetic moment of the Mn adatom. A full account of the effects of magnetic and non-magnetic impurities of the helical modes of the 558 defect will be published in a forthcoming publication. A corollary of our interpretation for the formation of the 558 defect helical bands is that if we inserted a stripe of a topological insulator as an extended 1D defect in a BiBL, we would see no helical-fermion bands of topologically-protected states along the core of such a defect. In order to test this idea, we insert another extended defect, the quadruple pentagon-double octagon (Q5D8) defect, shown in Fig. 1(e), inside the BiBL. The Q5D8-defect is formed by two adjacent stripes of the 558-defect. As shown in Fig. 1(f), the Q5D8-defect does not introduce helical-fermion modes in the band gap of the BiBL, and a band gap is always present in the electronic structure of the BiBL+Q5D8 supercell. It follows that the Q5D8defect can be seen as a finite portion of a Bi structure that belongs to the same non-trivial topological class as the pristine BiBL. Hence, our result for the electronic structure of the BiBL+Q5D8 system suggests a striking conclusion, that an extended crystalline form of the Q5D8-defect should be a topological insulator. Based on that observation, we decided to investigate the topology of the electronic structure of a pentaoctite form of a BiBL, as shown in Fig. 5(a). In this geometry, the bilayer consists of buckled pentagons and octagons only. Figure 5(a) shows a 2x2 cell of the pentaoctite BiBL. The two unit cell vectors at the lower-left of the figure delimit the 12-atom unit cell for this periodic structure. Indeed, this pentaoctite BiBL is an insulator,

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with a small gap of 0.035 eV, at the GGA level. For the calculation of the Z2 topological invariant of the Bi pentaoctite, we found it more expedient to employ the real-space methodology introduced by Soluyanov and Vanderbilt. 32 We obtain Z2 = 1, confirming our expectation that the pentaoctite BiBL is a 2D topological insulator. Given the rather small GGA gap we obtain for the pentaoctite BIBL, we compute also the gap for this system using the PBE0 20,21 hybrid functional. We obtain a value of 0.45 eV, showing that the pentaoctite BiBL is a robust insulating phase. (b)

(a)

0.50 Energy (eV)

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0.25 0.0 −0.25 −0.50 Y

Γ

X

Γ

T

Figure 5: (a) and (b) show a top of view and the band structure of a pentaoctite form of a 2D Bi layer, respectively. The difference between the two cases can be understood from their bonding pattern. Although the 558-defect and the Q5D8 are both formed by pentagons and octagons, there is a crucial structural difference between the two defects, as follows. The topology of the bands is associated with a competition between the pz orbitals and the planar px and py orbitals around the band gap edges. In the pristine Bi bilayer, the non-trivial band topology is due to band inversions between the p bands at the M points in the 2D BZ. Without the inclusion of spin-orbit coupling, the bottom of the conduction band comes mostly from pz orbitals. By turning on the spin-orbit coupling, these pz orbitals will dominate the character of the valence bands and the px and py orbitals will give rise to empty conduction bands. The above scenario applies for the pristine BiBL, that consists of two Bi triangular sublattices, labeled A and B, which are shifted by 1.74 Å in the perpendicular direction. In this geometry, each Bi atom in the “upper” (B) sublattice is bonded to three Bi atoms in the “lower” (A) sublattice, and vice-versa, as shown in Fig. 1(a). When we insert the

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Q5D8-defect in a pristine BiBL, all Bi-Bi bonds along the direction perpendicular to the defect are the same as in the pristine BiBL [a sequence of BiB -BiA bonds, see Fig. 1(e)]. The same observation applies for the pentaoctite form of the BiBL, and as consequence, this system belongs to the same topological class as the BiBL itself. On the other hand, the BiBL+558 and penta-hexa-octo BiBL systems [see Fig. 1(c) and Fig. 4(a)] presents planar BiB -BiB and BiA -BiA bonds, which brake the A-B sublattice bonding pattern. The presence of these planar bonds leads to a stabilization of the bands with px and py character, that shift to lower energies, while the pz bands will shift to higher relative energies. This will push the pz bands inwards into the conduction band, which supresses the band inversion even in the presence of spin-orbit coupling. This bonding pattern is present in the penta-hexa-octo form of the Bi bilayer, that hence lacks the required band inversion, and thus belongs to the topological class of trivial insulators. The fact that a hypothetical pentaoctite form of a Bi bilayer is a topological insulator naturally leads us to examine its possible experimental realization. In Table 2, we show a comparison of the formation energy and atomic density of the bulk-derived hexagonal BiBL and of the pentaoctite BiBL. The energy of the hexagonal layer is set to zero, for reference. The formation energy of the pentaoctite BiBL is only 0.045 eV/atom above that of the hexagonal BiBL, i.e., the energy difference between the two phases is of the order of the thermal energy fluctuations per atom, at room temperature. Thus, from the energetic point of view the pentaoctite BiBL is a viable crystalline phase. Table 2 also shows the atomic density of the two phases. The pentaoctite BiBL is a less dense phase than the hexagonal layer, which indicates that a BiBL may undergo a phase transition to the pentaocite form under a tensile strain. The pathway for a hexagonal-BiBL → pentaoctite-BiBL transition will be the subject of a forthcoming publication.

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Table 2: Formation energy (in eV/atom) and atomic density (in atom/Å2 ) of a bulk-derived hexagonal bismuth bilayer (BiBL) and of a pentaoctite form of the BiBL. Formation energy of the hexagonal layer is set to zero. structure pristine pentaoctite

formation energy 0.000 0.045

atomic density 0.1231 0.1174

Conclusions In conclusion, ab initio DFT calculations indicate the formation of helical-fermion metallic states localized on the core of an extended 1D defect embedded in the bulk of a single (111)oriented bismuth bilayer (BiBL). The formation of topologically-protected metallic states on the core of the 1D defect is closely related to the emergence of such metallic states along the edges of sufficiently large Bi ribbons. The core of the extended defect in our study is composed of two zigzag chains of Bi atoms connected by a roll of Bi dimers, forming the periodic unit of the defect, consisting of one octagonal and two pentagonal rings. The two zigzag lines of Bi atoms on the core of the 1D defects act as two zigzag edges, each edge hosting a pair of helical-fermion metallic modes with linear dispersions. Numerical DFT results show that the coupling and hybridization between the helicalfermion modes on the two sides of the defect is strongly reduced by the roll of Bi dimers at the geometric center of the 1D defect core. The handedness (or circulation) of the helicalfermion modes at the two “zigzag edges” meeting at the core of the 1D defect is preserved, which leads to an inversion of the spin-momentum locking between the two pairs of helicalfermion modes localized on the 1D defect: spin ’up’ (’down’) modes propagate up (down) on one side of the defect and down (up) on the other side, which leads to the possibility of backscattering between the helical-fermion modes, induced by disorder in the region of the 1D extended-defect core. Moreover, we also introduce an alternative crystalline form of a Bi bilayer, consisting entirely of pentagonal and octagonal rings, which we refer to as the pentaoctite form of a Bi bilayer. The formation energy of the pentaoctite Bi bilayer is only 0.045 eV/atom larger than 17


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that of the bulk-derived hexagonal layer, and its density is slightly larger than that of the hexagonal layer, indicating a possible structural transition between the two 2D crystalline phases at moderate tensile biaxial strains. Computation of the Z2 topological invariant shows that the pentaoctite BiBL is a 2D topological insulator.

Acknowledgments This work was supported by the Brazilian agencies CAPES, CNPq, FAPEMIG, and INCT de Nanomateriais de Carbono-MCT, and the computational work was done at the PhysicsDepartment-UFMG and the Cenapad-Unicamp computational facilities. R. W. Nunes acknowledges illuminating discussions with Vladimir Juričić.

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Topologically Protected Metallic States Induced by a One-Dimensional

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