Topological edge states in single- and multi-layer Bi$_{4}$Br$_{4}$

Topological edge states at the boundary of quantum spin Hall (QSH) insulators hold great promise for dissipationless electron transport. The device application of topological edge states has several critical requirements for the QSH insulator materials, e.g., large band gap, appropriate insulating substrates, and multiple conducting channels. In this paper, based on first-principle calculations, we show that Bi$_{4}$Br$_{4}$ is a suitable candidate. Single-layer Bi$_{4}$Br$_{4}$ was demonstrated to be QSH insulator with sizable gap recently. Here, we find that, in multilayer systems, both the band gaps and low-energy electronic structures are only slightly affected by the interlayer coupling. On the intrinsic insulating substrate of Bi$_{4}$Br$_{4}$, the single-layer Bi$_{4}$Br$_{4}$ well preserves its topological edge states. Moreover, at the boundary of multilayer Bi$_{4}$Br$_{4}$, the topological edge states stemming from different single-layers are weakly coupled, and can be fully decoupled via constructing a stair-stepped edge. The decoupled topological edge states are well suitable for multi-channel dissipationless transport.

Topological edge states at the boundary of quantum spin Hall (QSH) insulators hold great promise for dissipationless electron transport. The device application of topological edge states has several critical requirements for the QSH insulator materials, e.g., large band gap, appropriate insulating substrates, and multiple conducting channels. In this paper, based on first-principle calculations, we show that Bi 4 Br 4 is a suitable candidate. Single-layer Bi 4 Br 4 was demonstrated to be QSH insulator with sizable gap recently. Here, we find that, in multilayer systems, both the band gaps and low-energy electronic structures are only slightly affected by the interlayer coupling. On the intrinsic insulating substrate of Bi 4 Br 4 , the single-layer Bi 4 Br 4 well preserves its topological edge states. Moreover, at the boundary of multilayer Bi 4 Br 4 , the topological edge states stemming from different single-layers are weakly coupled , and can be fully decoupled via constructing a stair-stepped edge. The decoupled topological edge states are well suitable for multi-channel dissipationless transport. The hallmark of quantum spin Hall (QSH) insulators, also known as two-dimensional (2D) topological insulators, is the gapless helical edge states inside the bulk band gap 1,2 . Along a given edge of QSH insulator, a pair of edge states with opposite spins propagate in opposite directions, and they are topologically protected against backscattering from non-magnetic disorder. With this novel property, topological edge state promises its application to dissipationless transport, which has been demonstrated experimentally in HgTe/CdTe 3 and InAs/GaSb 4 quantum wells. Despite the great promise, the device application of topological edge state has been hampered by the lack of suitable materials that meet several critical requirements, e.g., large band gap for room temperature applications, and multiple conducting channels for high signal-to-noise ratio.
Inspired by the discovery of graphene, 2D materials with atomic thickness have become an emerging playground for exploring novel physics. The QSH effect was firstly predicted in graphene 5 , in which the band gap opened by spin-orbital couping (SOC) is extremely small 6 . Subsequently, some honeycomb-like materials with heavier elements were proposed to be QSH insulators with experimental accessible gaps, such as silicene 7 and Bi(111) bilayer 8 . However, the lack of appropriate insulating substrates becomes another crucial issue. The QSH phase of 2D materials may be destroyed due to its interaction with substrates 9 . Even the QSH phase survives, the hybridization between the topological edge states and the substrate's bulk states is disturbing 10,11 . Topological edge modes of Bi-bilayer on the surface of Bi single crystal were detected by STM recently 12 , thanks to that the edge states of certain edge type are only slightly hybridized with bulk Bi. Yet the metallic sura) Electronic mail: ygyao@bit.edu.cn face of bulk Bi is inadequate for edge state transport. Other newly proposed QSH insulators with sizable band gaps, such as single-layer Bi 4 Br 4 13 and transition metal dichalcogenides 14 , may break this obstruction because their bulk crystals are insulators.
In this paper, based on first-principles calculations, we study the effect of interlayer coupling on the electronic structures and edge states of multilayer Bi 4 Br 4 . We find that the band gaps of multilayer Bi 4 Br 4 hardly changes as the number of layers increasing, and the interlayer coupling has small impact on the low-energy electronic structures of multilayer Bi 4 Br 4 , which is attributed to the special orbital character of the band edges. Therefore, the surface of bulk Bi 4 Br 4 can be an intrinsic insulating substrate for the SL Bi 4 Br 4 . With this substrate, the Fermi velocity of topological edge states is slightly reduced compared to the freestanding case. Moreover, at the boundary of multilayer Bi 4 Br 4 , the topological edge states stemming from different single-layers are weakly coupled, which can be further decoupled by constructing a stair-stepped edge. The decoupled topological edge states can serve as multiple conducting channels. Our results indicate that the Bi 4 Br 4 is an excellent candidate for manufacturing multi-channel dissipathionless electron device.
First principle calculations are carried out using the projector augmented wave method 15 as implemented in the Vienna ab initio simulation package 16 . Both the Perdew-Burke-Ernzerof generalized gradient approximation (GGA) 17 and the Heyd-Scuseria-Ernzerhof hybrid functional (HSE06) 18 are used for the exchangecorrelation potential. The ionic position are relaxed employing the van der Waals (vdW) corrections 19,20 . Maximally localized Wannier functions (MLWFs) for the porbitals of Bi and Br atoms are constructed using the wannier90 code 21 . In the HSE06 calculations, the surface electronic structures are calculated using the combination of MLWFs and surface Green's function arXiv:1409.0943v1 [cond-mat.mtrl-sci] 3 Sep 2014 methods 22 . Bi 4 Br 4 has a layered structure 23,24 , as shown in Fig. 1(a). Within each single-layer (SL), one Bi atomic layer is sandwiched by two Bi/Br atomic layers. The normal and mirror-reflected SLs are stacked alternatively along the z direction with the interactions between adjacent SLs of weak vdW-type. A SL of Bi 4 Br 4 has a thickness of ∼7Å. It can be regarded as a parallel arrangement of one-dimensional (1D) infinite molecule chains[ Fig. 1(b)]. From the top view shown in Fig. 1(c), one can see that the SL structure belongs to the centered rectangular lattice, whose primitive unit cell is half size of its conventional cell. The conventional unit cell consists of two 1D chains, and the lattice constants a and b are 13.064Å and 4.338Å , respectively 23 .
The SL Bi 4 Br 4 has been predicted to be QSH insulator in our recent work 13 . In the absence of SOC, the top of valence band (TVB), dominated by Bi in -p x orbital, has odd parity under inversion symmetry; while the bottom of conduction band (BCB), dominated by Bi ex - p x orbital, has even parity 13 . After turning on SOC, as shown in Fig. 1(d), both the orbital character and parity of the band edges are inverted at the R point due to the strong SOC of Bismuth. This band inversion result in non-trivial topological phase (Z 2 = 1) in SL Bi 4 Br 4 .
From SL to multilayer system, the significant electronic properties may be altered by interlayer coupling, e.g., the direct to indirect band gap transition between monolayer and multilayer MoS 2 25 . To study the effect of interlayer couping on electronic structure of Bi 4 Br 4 , especially for the inverted band gap, we calculate the band structures of few-layer Bi 4 Br 4 using both GGA and HSE06 potentials. The calculated band gaps of Bi 4 Br 4 from SL to bulk systems are shown in Fig. 2(a). In the absence of SOC, the band gaps of HSE06 are obviously larger than those of GGA. This is because the GGA calculation usually underestimates the band gap 26 . In both GGA and HSE06 calculations, the band gap slightly decreases as the number of layers increasing, and the differences of band gaps are within ∼0.2 eV. When SOC is turned on, the band gaps of multilayer Bi 4 Br 4 are inverted in the similar way as SL system. The inverted band gaps are presented with negative values in Fig. 2(a). The dependence of band gap on the number of layers is further reduced. For HSE06 result, the band gap difference between SL and bulk systems is only within ∼30 meV, which is a very small value compared to other layered materials, e.g., black phosphorus (∼0.7 eV) 27 .
Apart from the band gaps, the low-energy electronic structures are also insensitive to the interlayer coupling. Fig. 2(b) shows the comparison between HSE06 band structures of SL and triple-layer Bi 4 Br 4 . The band features are essentially the same for both systems. Each band in SL Bi 4 Br 4 corresponds to three bands in triplelayer Bi 4 Br 4 , which are split due to the interlayer coupling. As can be seen in Fig. 2(b), The band splitting around Fermi-level is rather small, e.g., the band splitting of BCB (TVB) is about 40 (80) meV. The small band splittings can be attributed to the special orbital characters of the band edges. When SLs are stacked together to form a multilayer structure, the states dominated by orbitals with larger interlayer hopping usually have larger band splitting in multilayer systems. Obviously, the out-of-plane p z orbital has larger interlayer hopping compared to the in-plane p x/y orbitals, and the orbitals from Bi ex have larger interlayer hopping compared to those from Bi in . The band structure of SL Bi 4 Br 4 with orbital projected characters is plotted in Fig. 2(c). The second valence bands are dominated by Bi ex -p z orbital, thus have relatively large band splitting of a few hundred meV in triple-layer system [ Fig. 2(b)]. In contrast, the low-energy bands are dominated by the in-plane p x/y orbitals , mainly Bi in -p x [ Fig. 1(d)], therefore they are less affected by the interlayer coupling.
In the weak coupling limit, the multilayer system can be regarded as simple stacking of many isolated SLs which have topological edge states at the boundaries. We now focus on the evolution of these topological edges states when a weak interlayer coupling is introduced, such as the case in multilayer Bi 4 Br 4 . The edge electronic structures and spin polarizations for the SL, double-layer and triple-layer Bi 4 Br 4 are shown in Fig. 3. Since the coupling between adjacent 1D chains is much weaker than the intra-chain bonding 23 , atomically sharp edges along the 1D chain axis (y-direction) without dangling bond can be stabilized. We construct such edges as semiinfinite systems, for which the surface Green's functions are calculated. The energy and momentum dependent density of states, that are extracted from the imaginary part of the surface Green's function, are used to analyze the edge electronic structures. For SL Bi 4 Br 4 [ Fig. 3(a)], a single-Dirac-cone edges states linearly cross the bulk band gap. The Fermi velocity calculated by HSE06 is ∼ 6.5 × 10 5 m·s −1 , which is larger than the GGA result 13 . For double-layer Bi 4 Br 4 [ Fig. 3(b)], two pairs of topological edges states are weakly coupled, and a small gap of ∼20 meV is opened. The gapped edge states indicate topological trivial phase in double-layer Bi 4 Br 4 . For triple-layer Bi 4 Br 4 [Fig. 3(c)], three pairs of topological edges states are coupled. However, there is one pair of edge states cross the band gap without gap opening, which indicates topological non-trivial phase in triple-layer Bi 4 Br 4 . The difference of topological phase is because that the multilayer Bi 4 Br 4 with even (odd) number of layers has even (odd) times of band inversions. The even times of band inversions result in a trivial insulator with Z 2 = 0, while odd times of band inversions result in a QSH insulator with Z 2 = 1.
Since the topological edge states are weakly couped at the boundary of multilayer Bi 4 Br 4 , they can be decoupled by constructing rough edge, such as a stair-stepped edge. We construct a two-layer Bi 4 Br 4 film with stairstepped edge supported by bulk Bi 4 Br 4 surface, as plotted schematically in Fig. 4(b). With the step width of ∼5 nm, we calculate the energy spectrum of this system as shown in Fig. 4(a). The edge states linearly cross the bulk gap without gap-opening. Compared to the freestanding double-layer one[ Fig. 3(b)], the topological edge states from the two different layers are fully decoupled at the stair-stepped edge, and the two single-Dirac-cones are degenerate. Another observation is that, due to the weak interaction with substrate, the Fermi velocity of the decoupled edge states (∼ 5.6 × 10 5 m·s −1 ) is a little smaller than that of the free standing SL system [ Fig. 3(a)]. As the effect of interlayer coupling, the decreased Fermi velocity is also observed in the edge states of triple-layer systems.
To understand the essential physics, we develop a lowenergy effective k · p Hamiltonian for SL Bi 4 Br 4 by using the theory of invariants, from which the effective Hamiltonian for the topological edge states can be further derived. For the convenience to construct natural edges along the 1D chain axis, we adopt the conventional unit cell. Consequently the band edges are folded to the Γpoint, as illustrated in Fig. 1(f). Both the BCB and TVB are double degenerate, and the two degenerate states are related by time reversal symmetry. Since the BCB (TVB) has odd (even) parity under inversion symmetry, we can denote these four states as |−, ↑ (↓) , |+, ↑ (↓) . By analyzing the inversion, mirror σ h and time reversal symmetry, we can write down the low-energy effective Hamiltonian using the four states as basis (in the order of |+, ↑ , |−, ↓ , |+, ↓ , −|−, ↑ ) A 1 and A 2 are complex parameters, while the others are real parameters. By fitting the energy spectrum of the Hamiltonian with the HSE06 band structure [see Fig. 1(e)], we can determine these parameters as following: C = 0.0 eV, D 1 = 0.506 eV·Å 2 , D 2 = 4.82 eV·Å 2 , M 0 = 0.09 eV, B 1 = 3.86 eV·Å 2 , B 2 = 0.0032 eV·Å 2 , A 1 = −1.81 + 0.046i eV·Å, A 2 = −4.15 + 0.141i eV·Å. The band inversion can be produced by the fact of M 0 , B 1 , B 2 > 0. The form of the Hamiltonian is different from that of HgTe quantum well 28 , but similar to that of Bi 2 Se 3 29 . In a similar way as for Bi 2 Se 3 , we can derive the effective Hamiltonian for the topological edge states.
H edge (k y ) = |A 2 | k y σ x With the fitted value of A 2 , the Fermi velocity of the topological edge states is given by |A2| 6.3×10 5 m·s −1 , which is consistent with the HSE06 result [ Fig. 3(a)]. The value of A 2 is modified when SL system is supported by Bi 4 Br 4 surface. For the edge states of multilayer Bi 4 Br 4 , the effective Hamiltonian can be simply written by introducing coupling terms between H edge (k y ) of different SLs.
In summary, first-principle calculations demonstrate that both the band gaps and low-energy electronic structures of multilayer Bi 4 Br 4 are little affected by the interlayer coupling. When SL Bi 4 Br 4 is supported by the surface of bulk Bi 4 Br 4 , its topological edge states well survive except for a reduced Fermi velocity. Moreover, at the stair-stepped edge of multilayer Bi 4 Br 4 , the topological edge states from different SLs are fully decoupled. Our results indicate nano-fabrication on the cleaved surface of layered Bi 4 Br 4 single crystal is adequate to realize multiple dissipationless conducting channels 12,30 , hence Bi 4 Br 4 is an excellent platform for manufacturing QSH-based devices.