Conductance of quantum spin Hall edge states from first principles: the critical role of magnetic impurities and inter-edge scattering
CConductance of quantum spin Hall edge states from first principles: the critical role ofmagnetic impurities and inter-edge scattering
Luca Vannucci, ∗ Thomas Olsen, and Kristian S. Thygesen
1, 2 CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark Center for Nanostructured Graphene (CNG), Technical University of Denmark, 2800 Kgs. Lyngby, Denmark (Dated: March 20, 2020)The outstanding transport properties expected at the edge of two-dimensional time-reversal in-variant topological insulators have proven to be challenging to realize experimentally, and have sofar only been demonstrated in very short devices. In search for an explanation to this puzzlingobservation, we here report a full first-principles calculation of topologically protected transport atthe edge of novel quantum spin Hall insulators — specifically, Bismuth and Antimony halides —based on the non-equilibrium Green’s functions formalism. Our calculations unravel two differentscattering mechanisms that may affect two-dimensional topological insulators, namely time-reversalsymmetry breaking at vacancy defects and inter-edge scattering mediated by multiple co-operatingimpurities, possibly non-magnetic. We discuss their drastic consequences for typical non-local trans-port measurements as well as strategies to mitigate their negative impact. Finally, we provide aninstructive comparison of the transport properties of topologically protected edge states to thoseof the trivial edge states in MoS ribbons. Although we focus on a few specific cases (in terms ofmaterials and defect types) our results should be representative for the general case and thus havesignificance beyond the systems studied here. I. INTRODUCTION
In a pair of groundbreaking papers published in 2005,Charles Kane and Eugene Mele argued that graphene be-comes a two-dimensional topological insulator (2D TI) atsufficiently low temperature . In other words, a flake ofgraphene cooled down to cryogenic temperature will es-sentially behave as a band insulator everywhere excepton its boundaries, where special metallic edge states willappear. Such states are immune to scattering againstimpurities or disorder, and therefore realize a perfectdissipationless conductor with great potential for futuretechnological applications. The reason behind this strik-ing robustness is rooted in the mathematical concept oftopology, hence the name topological insulators .Unfortunately, the band gap opened by spin-orbit cou-pling (SOC) in graphene is actually too small to give riseto any measurable effect . To realize the first 2D TI —or quantum spin Hall insulator (QSHI) — it was there-fore necessary to resort to quantum-well heterostruc-tures, which indeed showed the much anticipated sig-natures of topologically protected transport in non-localmulti-terminal measurements . With the rising aware-ness that the realm of monolayer 2D materials is ac-tually much larger than initially thought, several newQSHIs have been reported in other 2D monolayers thangraphene in recent years .It is, however, not fully understood to what extent theideas of topological protection can materialize into thenext generation of electronic devices, due to some incon-sistency between theory and experiments. Time-reversalsymmetry (TRS) forbids electron backscattering on theedge of 2D TIs, since counter-propagating modes haveopposite spin polarization — they are therefore termed helical edge states . The defining feature of such a pairof protected states is a well defined quantized conduc- tance plateau at G = 2 e /h , which is, however, difficultto attain in the lab. The few successful attempts areall limited to very low temperature ( ∼
1K in quantum-well heterostructures ) or very short channels ( ∼ ). Attempts to under-stand this discrepancy at the model level have focusedon many diverse backscattering mechanisms driven byelectron-electron interactions, charge puddles, embeddednuclear spins, coupling to phonons and electromagneticnoise . Nonetheless, the question is still much de-bated and deserves a careful analysis from a different andmore realistic point of view.Here we report for the first time a full first-principlesstudy of topologically protected transport at the edge ofnovel QSHIs. We use newly developed computational 2Dmaterials databases , containing existing structuresas well as hitherto unknown monolayers, to identify afamily of large-gap QSHIs ideally suited for the currentstudy. We then explore the electronic band structure ofsuch candidates at the level of density functional theory(DFT), both as infinite bulk monolayers and in differ-ent nanoribbon geometries. For nanoribbons, we high-light the emergence of robust metallic states whose eigen-values cross the region of the bulk gap, and investigatethe robustness of their transport properties in the frame-work of the non-equilibrium Green’s functions (NEGF)formalism , with full account of spin-orbit interactions.Our calculations show that naturally occurring native de-fects at the edge can spontaneously acquire a magneticmoment, thereby violating TRS and leading to a sup-pression of edge transport. This result reveals the mech-anism that is most likely to affect edge conduction in 2Dtopological insulators. Interestingly, we find that chem-ical saturation of vacancy defects (e.g. by Hydrogen) issufficient to remove the magnetic moment, thereby pro-viding a strategy to restore topological protection at the a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r edge. Perhaps more surprisingly, even non-magnetic im-purities may be detrimental for transport. We indeedshow that multiple non-magnetic impurities may createa channel for inter-edge scattering between opposite edgestates in relatively wide ribbons, thereby affecting trans-port properties even when they do not represent a threatto transport individually.The structure of this paper is as follows. In Section IIwe illustrate the family of topological materials studied inthis work and present their band structure in the zigzagnanoribbon geometry. In the subsequent Sections III andIV we calculate the transport properties of topologicalnanoribbons in the presence of impurities, and discussthe defect-induced intra-edge and inter-edge backscatter-ing mechanisms. A similar formalism is then applied tozigzag MoS nanoribbons in Section V, thereby providingan insightful comparison with the transport propertiesof topologically trivial edge states. Finally, Section VIsummarizes the main results of this paper. Appendix Ais dedicated to technical details about DFT calculations. II. ELECTRONIC STRUCTURE OFTOPOLOGICAL NANORIBBONS
The aim of this work is to address the problem of topo-logically protected transport by going beyond the sim-ple model approximation, and to perform full transportcalculations of realistic topological compounds from firstprinciples. Thanks to the application of automated high-throughput methods in the context of material science,the portfolio of theoretically proposed 2D materials isnowadays expanding at a remarkable pace . This haslead to the identification of several new candidates for 2Dtopological insulators , some of which are now beingtested in the lab .We choose here to focus on Bismuth and Antimonyhalides, i.e. binary compounds BiX and SbX with X =(F, Cl, Br, I), whose topological nature has already beeninvestigated in earlier work . The reason for ourchoice is threefold:(i) They are simple binary compounds of Bi/Sb and ahalogen element, with a rather small number of va-lence electrons per unit cell. This allows us to dealwith bigger devices without extreme computationaleffort (compared to other candidates).(ii) They are dynamically stable and thermodinami-cally meta-stable, in the sense that their heat of for-mation is negative with respect to the pure elemen-tal form and not much larger than other compet-ing phases (see Ref. for further discussion aboutthermodynamic stability of 2D materials). Thismeans that the compound might be synthesizable,although this is not an essential aspect of our work.Indeed, we expect our conclusions to be universallyapplicable to any QSHI. (iii) Their bandgap is predicted to be in the range0 . − . graphane . Lattice parameters andband structures for infinite 2D monolayer are all reportedin the Supplemental Material .Being band insulators with non-trivial topological in-variant Z = 1 , the interface between BiX/SbX anda trivial insulator (e.g. vacuum) should host a pair of he-lical edge states. We confirm this by investigating theelectronic structure of zigzag-terminated ribbons of dif-ferent widths, such as the one shown in Fig. 1. Sincematerials with the same group-15 element but differenthalogens behave in a qualitatively similar fashion, we willfocus on BiBr and SbBr, and show results for the remain-ing materials in the Supplemental Material .Figure 1 shows the band structure of BiBr and SbBrzigzag nanoribbons across half of the 1D Brillouin zone.In stark contrast with infinite 2D structures, all nanorib-bons are gapless and show robust metallic states lyingthe region of the bulk gap, which is of the order of 0.8 eVfor BiBr and 0.4 eV for SbBr. Such states, highlightedin green in Fig. 1, are remarkably stable as the widthof the ribbon is increased, as opposed to the remainingvalence and conduction states which become more andmore dense and are displaced in energy as we move to-wards larger ribbons. We thus conclude that metallic in-gap states are localized on the edge, a fact which is alsoconfirmed by inspecting the corresponding wavefunction(see Fig. S4 in Supplemental Material ). Note that thespectrum is spin degenerate owing to the presence of in-version symmetry, but metallic bands with opposite spinare located on opposite edges.It is worth noticing that the presence of edge statesis attributed to the non-trivial topological character ofthe bulk bands, and not to the particular edge termina-tion. Indeed, we observe qualitatively similar features inboth zigzag and armchair nanoribbons, with the latter re-ported in the Supplemental Material for completeness . III. BREAKDOWN OF EDGE TRANSMISSIONDUE TO MAGNETIC EDGE DEFECTS
To calculate edge transport properties of Bismuth andAntimony halides we make use of the software packageQuantumATK , which allows to simulate transportdevices in the framework of the NEGF formalism . Wewill focus on the zero-bias transmission spectrum (TS) of Figure 1. Top: a Bismuth or Antimony halide zigzag nanoribbon of width W = 8. Dark purple dots denote the positions ofBi or Sb atoms, while halogen atoms (F, Cl, Br or I) are denoted in red. The shaded region corresponds to the unit cell used torepresent the infinite ribbon. Bottom: Band structure of BiBr and SbBr zigzag nanoribbons of different width W . Topologicaledge states are highlighted in green. Energy is measured with respect to the Fermi energy. a two-terminal device over an energy window of 3 eV thatincludes the bulk gap. This is straightforwardly linked tothe two-terminal conductance of a real device through thewell known Landauer formula G = e h T ( E F ) .Our setup is illustrated in Fig. 2 and is made as follows.For any given material, we create two identical, semi-infinite, pristine electrodes by repeating the nanoribbon unit cell shown in Fig. 1. We then create a central scat-tering region by considering a finite-length portion of TInanoribbon, and remove one or more halogen atoms toaccount for the presence of vacancy defects (denoted asV X , X being the missing atom). By connecting togetherleft electrode, central scattering region and right elec-trode we obtain a two-terminal device setup for transportcalculations.Let us first focus on the case of a halogen vacancy onthe edge (hereafter named edge defect ), as shown in Fig.2. Transmission spectra for W = 8 zigzag nanoribbonswith edge defects are reported with red dashed lines inFig. 2, where we also report the band structure and TSof pristine ribbons.As shown in Fig. 2b, introducing an edge defect inBiBr does not lead to any observable effect in the regionof edge states dispersion, where the transmission exactlyequals the number of bands. On the other hand, it doesgive a partial suppression of transmission for bulk va-lence and conduction states. Such a behavior is indeedthe hallmark of 2D TIs, whose edge states conduction isprotected even in the presence of disorder as long as timereversal symmetry is not broken. However, the same con-sideration does not hold for the case of SbBr (see Fig. 2f),which shows an unexpected and well pronounced anti-resonance in the TS around 0.4 eV and therefore a fail-ure of topological protection. A similar feature, althoughmuch less pronounced, can be also observed around − . . µ B which is exactly localized at the Sb siteunderneath the Br vacancy. On the other hand, the BiBrnanoribbon have a negligible small magnetic moment atthe Br vacancy. We thus attribute the suppression ofedge conductance in SbBr nanoribbons to the sponta-neous magnetization of the edge impurity, which inval-idates topological protection and allows for intra-edgeback-scattering. This is a recurrent feature for all An-timony halides, for which we systematically observe ananti-resonance in a narrow energy window around 0.2–0.5eV due to the formation of localized magnetic moments(see Supplemental Material ).It is interesting to notice that a similar mechanismhas been proposed very recently in the framework ofthe Kane-Mele-Hubbard model in graphene, where thebreakdown of time reversal symmetry at vacancy defectsis shown to lead to a corresponding breakdown of conduc-tance quantization . The present ab-initio calculationssupport this picture and also bear similarities with simpletheoretical models in which magnetic scatterers, such asmagnetic adatoms or ferromagnetic gates, are introducedin 2D TIs .The magnetization of edge defects originates from thepresence of a dangling bond at the vacancy site, whichmakes the configuration chemically unstable and drivesthe formation of a localized magnetic moment, as re-ported in earlier work . We thus conjecture that thechemical saturation of the dangling bond with a suitableelement should eliminate any magnetic structure at thedefect and restore a perfect transport at the edge. In-deed, we find that saturation with Hydrogen restores theperfect transmission, as shown in Figs. 2c and 2g. We have also calculated the TS for a non-magneticconfiguration of the halogen vacancy (we manually setall magnetic moments to zero), which we report in Figs.2d and 2h. As expected, there’s no backscattering in thiscase since time reversal symmetry is not violated. How-ever, we find rather small energy difference between themagnetic and non-magnetic configurations, which mightbe beyond the accuracy of DFT calculations.There is however one last puzzling question emergingfrom Fig. 2, which is the unexpected back-scattering ob-served in SbBr nanoribbons between − . T ( E ) = 6 (see panels f and h). We at-tribute this result to the following mechanism. Due tothe presence of non-monotone energy dispersion of theedge states, the nanoribbon actually hosts three pairs ofmetallic states per edge in this energy range, whose di-rection of motion can be easily inferred from the slopeof the bands. ? Thus, it becomes possible for an electronto scatter into a state with same spin but different direc-tion of motion, as schematically depicted in Fig. 3. Thismechanism can never lead to the total suppression of edgeconductance, as there will always be a pair of helical edgestates which are not accompanied by the correspondingcounter-propagating states. In other words, in the pres-ence of impurities or disorder, whatever odd number ofedge states pairs is practically equivalent to a single pair,which is a manifestation of the binary nature of the Z topological invariant. This implies that the TS in theregion between − . T ( E ) = 4as the impurities become more and more abundant, sincethe three transport channels on the bottom edge will beaccompanied by only one surviving pair on the disorderedtop edge. To check this, we have investigated differentnon-magnetic configurations of SbBr nanoribbons withmultiple edge defects, which are all reported in Fig. 3 to-gether with the single-impurity configuration previouslydiscussed. Indeed, the resulting TS never drops below 4in the aforementioned region, supporting our interpreta-tion.In passing, it is worth noting that we have also calcu-lated the TS for smaller ribbons ( W = 4) and for all re-maining materials in the presence of similar edge defects,obtaining qualitatively similar results — that is, sharpanti-resonance in the conductance due to the formation ofmagnetic defects, and intra-edge backscattering when thestructure carries three helical pairs per boundary. Theseresults are shown in the Supplemental Material . IV. INTER-EDGE SCATTERING MEDIATEDBY NON-MAGNETIC BULK DEFECTS
We now turn our attention to halogen vacancies lo-cated away from the edge, which will be denoted bulkdefects . As for the case of edge defects, we have studiedboth magnetic and non-magnetic configurations. How-ever, we will only present results for non-magnetic bulkdefects in BiBr for the sake of clarity.
Figure 2. Top: Sketch of the setup used for transport calculations. Two semi-infinite electrodes (in gray) are connected to thescattering region in the middle (light green). The latter is a finite-size nanoribbon with one or more vacancy defects. A centralregion with an edge defect is shown to the right. Bottom: Transmission spectrum (TS) of BiBr and SbBr zigzag nanoribbonsof width W = 8 in presence of one of the following edge defects: magnetic edge defect (b and f); Hydrogen-saturated edgedefect (c and g); non-magnetic edge defect (d and h). The TS for a pristine ribbon is also shown for comparison in each panel,and its band structure is reported in panels a and e from Fig. 1. The energy region of insulating bulk is highlighted in yellow. We have explored a scenario where two halogen atomsare simultaneously removed from a BiBr nanoribbon ofwidth W = 8, leaving a couple of bulk vacancy defectsin the central region. As shown in Fig. 4, both vacanciesare symmetrically placed at a distance d = √ a from theedge, with the defect-defect distance being d = 5 a/ √ d = 9 .
46 ˚A and d = 15 .
76 ˚A for BiBr).The TS of such a configuration is reported in Fig. 4.While both impurities do not affect transport proper-ties in the bulk gap region individually, which is demon-strated by the perfect TS in the region − . . E . . E ≈ Figure 3. Transmission spectrum (TS) of an SbBr zigzag nanoribbon of width W = 8 in presence of multiple non-magneticedge defects as shown in the bottom panel: single edge defect, reported from Fig. 2h (b); double edge defect (c); triple edgedefect (d). The TS for a pristine ribbon is also shown for comparison in each panel, and its band structure is reported fromFig. 1 in panel a. The energy region in which the nanoribbon bears 3 pairs of edge states in highlighted in yellow, while theregion 4 ≤ T ( E ) ≤ purities. Finally, it tunnels into the bottom edge states,where it propagates back towards the left electrode with-out having to flip the spin polarization. This analysis isfurther confirmed by the behavior of the local density ofstates shown in Fig. 5, which clearly demonstrates thatthe impurity states occupy a large transverse portion ofthe nanoribbon, while also having a substantial overlapbetween them. In this scenario, it is the co-operation be-tween multiple impurities that creates a path for inter-edge scattering, even in such cases where they would notrepresent any threat to transport if considered individu-ally. Such a mechanism has been frequently neglected inthe literature, which rather focus on the effect of singleimpurities on the transport properties of 2D TIs. Never-theless, it could play a major role when the defect concen-tration exceeds a certain threshold , or when oppositeedge states are deliberately funneled through a narrowconstriction .It is worth noting that chemical saturation of the dan- gling bonds with Hydrogen removes the energy levels ofthe defects from the bulk gap region. The correspondingTS, which we show in Fig. 4d, is basically unaffected, thussuggesting a strategy to minimize the impact of impurity-mediated inter-edge scattering on transport.Finally, we mention that we have observed signaturesof inter-edge scattering for different disordered configu-rations and nanoribbon widths, as reported in the Sup-plemental Material . In particular, opposite edge statesin narrow nanoribbons ( W = 4) can be coupled togetherby one single bulk impurity, since the spatial extensionof the defect wavefunction becomes comparable with thenanoribbon width . Figure 4. Transmission spectrum (TS) of a BiBr zigzag nanoribbon of width W = 8 in presence of multiple non-magneticbulk defects as sketched in the bottom panel: single bulk defect (b); double bulk defect (c); Hydrogen-saturated double bulkdefect (d). The TS for a pristine ribbon is also shown for comparison in each panel, and its band structure is reported in panela from Fig. 1. The energy region of insulating bulk is highlighted in yellow. a) b) c) Figure 5. Local density of states (LDOS) for the configuration in Fig. 4. Side and top views of the surface LDOS( x, y, z ) =0 .
02 ˚A − eV − are shown in panels a and b respectively. Panel c shows the LDOS in the transverse direction y averaged overthe xz plane, with the spatial region spanned by the edge states highlighted in cyan. V. COMPARISON WITH NON-TOPOLOGICALEDGE STATES
So far, we have discussed two mechanisms that leadto the failure of conductance quantization in 2D TIs. Inthis section, we illustrate how the results presented up tonow are clearly related to topology, and how edge stateswould behave in the absence of topological protection.We thus focus here on the transport properties of aMoS nanoribbon, whose crystal structure is shown inFig. 6. Such structures are known to host metallic edgestates at the zigzag termination which are however notdue to topology, but rather to polar discontinuity at theinterface between MoS and the vacuum . We willtherefore call these trivial edge states , to indicate thatthey are not generated as boundary states between ma-terials with different topological invariants.In Fig. 6a we show the band structure of a pristineMoS nanoribbon of width W = 8. The metallic edgestates, indicated with a different color in the figure, areclearly visible. We then create two different defect con-figurations by removing an atom on the edge (either Moor S). We calculate the TS for such structures neglect-ing the contribution from SOC, which is known to havea negligible effect in this case, and compare them to theTS of a pristine nanoribbon.Figures 6b and 6c show that the presence of an edgedefect has a dramatic consequence for the transmissionproperties of the nanoribbon. Generally, we observe amuch larger effect of backscattering as compared to thecase of topologically protected materials. The conduc-tance is basically halved over the entire energy range0 . − . . − . nanoribbons are muchmore prone to backscattering than the topologically pro-tected edge states in the Bi (Sb) halides.Finally, we have also calculated the TS for the caseof Oxygen-saturated S vacancy, which is shown in Fig.6d. Once again, chemical saturation seems to be bene-ficial to edge transport, as we recover an almost perfecttransmission. However, the TS still deviates by 3% fromthe pristine value in the region 0 . − . VI. CONCLUSIONS
In summary, we have performed ab-initio transportcalculations of two-terminal TI nanoribbons using theNEGF formalism. By accounting for the presence of bothedge and bulk defects, we have pinpointed two sources ofbackscattering which lead to the breakdown of conduc-tance quantization: (i)
Intra-edge scattering due magnetic edge impurities.
The dangling bond originated by the removal ofone atom at the edge may in some cases drivethe formation of a localized magnetic moment atthe impurity site. This local breaking of TRS al-lows for backscattering events involving spin-flip(as sketched in Fig. 7a). This basically blocks elec-trical conduction through one of the edges, whileleaving the opposite one unperturbed. The corre-sponding transmission function drops from 2 to 1when the energy of the incoming electron resonateswith the defect level.(ii)
Inter-edge scattering due to (multiple) bulk impu-rities.
In a narrow nanoribbon, a single non-magnetic bulk impurity can open a backscatter-ing channel between opposite edge states withoutbreaking TRS (Fig. 7b). Although this can be obvi-ously avoided by increasing the nanoribbon width,large nanoribbons will be still affected by inter-edgescattering above a certain threshold of defect con-centration, when multiple bulk impurities generatea backscattering path across the structure (Fig. 7c).Finally, it’s worth mentioning that our results are by nomeans limited to the particular class of materials chosenin this work. Rather, we expect them to be relevant forall QSHIs. We also anticipate that similar mechanismsmay deteriorate surface conduction of three-dimensionaltime-reversal invariant TIs.
ACKNOWLEDGMENTS
We thank Nicola Marzari for stimulating discussions,Mads Brandbyge and Tue Gunst for useful discussionsand technical help. The research leading to these re-sults has received funding from the European Union’sHorizon 2020 research and innovation program under theMarie Sk lodowska-Curie grant agreement No. 754462.(EuroTechPostdoc). KST acknowledges funding fromthe European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovationprogramme (Grant No. 773122, LIMA). The Center forNanostructured Graphene is sponsored by the DanishNational Research Foundation, Project DNRF103.
Appendix A: Methods
All structures are obtained directly from the C2DBdatabase , where they are relaxed with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation func-tional using the DFT code GPAW and the softwarepackage ASE . For further details about this first stepwe refer to Ref. .To create a nanoribbon, we define a rectangular unitcell as shown in Fig. 1, including a different number of Figure 6. Top: Top and lateral view of an MoS zigzag nanoribbon of width W = 8. Atoms removed to create defects aredenoted with different colors. Bottom: Transmission spectrum (TS) of MoS2 zigzag nanoribbons of width W = 8 in presenceof one of the following edge defects shown in Fig. 6: Mo vacancy (b); S vacancy (c); Oxygen-saturated S vacancy (d). The TSfor a pristine ribbon is also shown for comparison in each panel, and its band structure is shown in panel a. The energy regionof insulating bulk is highlighted in yellow. atoms according to the width W . Each ribbon is sep-arated from its periodic replicas by including a suitableamount of vacuum in the unit cell, both in the out-of-plane and the non-periodic in-plane directions.We then use the atomistic simulation toolkitQuantumATK to calculate the DFT band struc-ture of pristine nanoribbons. For QuantumATK bandstructure calculations we resorted to an LCAO basis us-ing the SG15 pseudo-potentials — with the only ex-ception of BiCl and SbCl structures, where we use theOpenMX package . We sample the Brillouin zone ofthe nanoribbon with a 1 × ×
16 Monkhorst–Pack (MP)grid to ensure convergence, while the density mesh cut-off controlling the real-space grid is set to 100 Hartree.We also checked the band structure with GPAW by usinga plane-wave basis set with an energy cutoff of 400 eVand an identical MP grid, obtaining an excellent agree- ment with QuantumATK calculations. In both cases weuse the PBE functional with the inclusion of spin-orbitcoupling, which is a crucial ingredient here.To calculate the transmission spectrum at zero bias weuse the NEGF formalism as implemented in Quantu-mATK. We define a transport setup by creating identicalleft and right electrodes and a central scattering region,as shown in Fig. 2. We create pristine electrodes by us-ing the same material as in the central region, and makesure that electrodes are well screened by repeating theunit cell in the transport direction ( z axis) a suitableamount of times — only once for BiCl and SbCl, threetimes for MoS and twice for the remaining materials.One or more defects in the central scattering region arecreated by removing one or more halogen atoms. Notethat structures obtained in such a way are not optimized,so that we neglect reconstruction effects. A special care0 a)b) c) Figure 7. Graphical representation of different backscatter-ing mechanisms: intra-edge scattering due to magnetic edgeimpurities (a); inter-edge scattering through a single non-magnetic bulk defect (b); inter-edge scattering mediated bymultiple bulk impurities (c). Metallic edge states with op-posite spin polarization are represented with different colors(red and blue). is taken in including a suitable portion of pristine mate-rial at both sides of the central scattering region, so thatthe latter is smoothly connected to the electrodes.Electrodes for NEGF calculations are sampled with a1 × ×
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1, 2 CAMD, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark Center for Nanostructured Graphene (CNG), Technical University of Denmark, 2800 Kgs. Lyngby, Denmark (Dated: March 20, 2020) a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r CONTENTS
I. Band structure of Antimony and Bismuth halides 3A. 2D monolayers 3B. Zigzag nanoribbons 4C. Armchair nanoribbons 7II. Transmission spectra for large ( W = 8) and narrow ( W = 4) ribbons 8A. Topological insulator nanoribbons 8B. Non-topological (MoS ) nanoribbons 13III. Disordered configurations 13IV. Local density of states 14V. Magnetic moments of edge and bulk impurities 15References 16 Sb, BiF, Cl, Br, I Hexagonal cell y zx zy x aa’ Monoclinic cellBrillouin zone k y k z k y k z Figure S1. Lattice structure of monolayer BiX and SbX (X = F, Cl, Br, I). The unit cell is marked with a shaded region. Thecorresponding Brillouin zone in shown on the right, and the path used for band structure calculations is reported with dashedcolored lines. We use the conventional ΓMKΓ path for materials with hexagonal unit cell, while the path ΓYHH XΓ is usedfor the case of monoclinic cell (such as SbI).material a (˚A) a’ (˚A) d (˚A) h (˚A) ∆
GPAW (eV) ∆
ATK (eV)BiF 5.32 2.09 0.42 0.99 1.02BiCl 5.45 2.52 0.21 0.90 0.90BiBr 5.46 2.67 0.17 0.86 0.83BiI 5.48 2.87 0.10 0.78 0.86SbF 5.10 1.97 0.28 0.36 0.34SbCl 5.22 2.40 0.11 0.43 0.40SbBr 5.24 2.57 0.09 0.44 0.41SbI 5.05 5.36 2.78 0.02 0.49 0.43Table S1. Lattice parameters and band gap of monolayer BiX and SbX (X = F, Cl, Br, I). a: hexagonal lattice constant; a’:2nd lattice constant (for materials with monoclinic structure); d: Bi-X or Sb-X distance; h: buckling height; ∆
GPAW : electronicband gap obtained with GPAW ; ∆ ATK : electronic band gap obtained with QuantumATK . I. BAND STRUCTURE OF ANTIMONY AND BISMUTH HALIDESA. 2D monolayers
The crystal structure of monolayer Bismuth (Antimony) halides consists of a low-buckled layer of Bismuth (Anti-mony) sandwiched between two layers of halogen atoms, which are disposed in an alternated fashion with respect tothe central layer (see Figure S1). Seven structure out of 8 have hexagonal unit cell, with both inversion symmetryand a three-fold rotational symmetry. SbI is the only exception, having monoclinic structure with broken rotationalsymmetry.Structural parameters for all materials — such as lattice constants a and a , Bi(Sb)-halogen distance d , bucklingheight h — are reported in Table S1, together with the corresponding 2D band gap. Results are in good agreementwith previous work .The band structure of 2D monolayers is reported in Fig. S2. Here we have calculated electronic bands both withoutand with spin-orbit coupling (SOC). In the absence of SOC, all materials are semi-metals with a Dirac cone at theK point (slightly shifted for the monoclinic structure SbI). However, SOC has an essential role here as it opens agap of 0 . − . . − . . M K k points3210123 E ( e V ) BiF monolayer with SOC without SOCM K k points SbF monolayer
M K k points3210123 E ( e V ) BiBr monolayer with SOC without SOCM K k points SbBr monolayer
M K k points3210123 E ( e V ) BiCl monolayer with SOC without SOCM K k points SbCl monolayer
Y H H X k points3210123 E ( e V ) BiI monolayer with SOC without SOCY H H X k points SbI monolayer
Figure S2. Band structure of bulk 2D monolayers along the path shown in Fig. S1. We plot the eigenvalues both without andwith the inclusion of spin-orbit coupling (SOC). Note that BiI has actually a hexagonal unit cell, but we use the monoclinicpath for a better comparison with SbI.
B. Zigzag nanoribbons
Here we report the band structure of zigzag nanoribbons for all materials that have not been shown in the maintext. All structures are gapless with a pair of metallic in-gap states that is quite stable with respect to the width W .The fact that such states are localized along the edge is confirmed by looking at their associated wavefunction,which we show in Fig. S4. The figure clearly shows that Bloch eigenstates for the in-gap states are sharply localizednear the edge of the nanoribbon, while electronic states deep into the conduction band are spread over the entirewidth of the ribbon. E ( e V ) W =4 bulk bands edge bands W =8 W =12 BiF W =160 / a k x E ( e V ) a k x a k x a k x SbF E ( e V ) W =4 bulk bands edge bands W =8 W =12 BiCl W =160 / a k x E ( e V ) a k x a k x a k x SbCl E ( e V ) W =4 bulk bands edge bands W =8 W =12 BiI W =160 / a k x E ( e V ) a k x a k x a k x SbI
Figure S3. Band structure of BiX and SbX zigzag nanoribbons (X = F, Cl, I) of different width W . Topological edge statesare reported in green. The energy scale is referred to the Fermi energy. Figure S4. Wavefunctions of an SbBr nanoribbon of width W = 8 for the electronic states indicated in the left panel. Thefigure shows the surface of constant wavefunction amplitude | ψ ( x, y, z ) | = 0 . − / , while the color scale indicates the phase. W y z Figure S5. Top view of a Bismuth or Antimony halide armchair nanoribbon of width W = 8. Dark purple dots denote thepositions of Bi or Sb atoms, while halogen atoms (F, Cl, Br or I) are denoted in red. The shaded region corresponds to theunit cell. C. Armchair nanoribbons
In this section we show the electronic band structure for armchair nanoribbons of different width, whose crystalstructure is depicted in Fig. S5. As reported in Figure S6, all structures show the presence of additional metallicedge states with respect to the infinite bulk monolayer. Note that the spectrum is still gapped for small armchairnanoribbons ( W = 4 − W are actually narrower by a factor √ W . However, it is still interesting to note that the magnitude of the gap does not seem to decreaselinearly with increasing W , especially for Antimony-based materials. E ( e V ) W =4 bulk bands edge bands W =8 W =12 W =16 W =20 BiF W =240 / a k x E ( e V ) a k x a k x a k x a k x a k x SbF E ( e V ) W =4 bulk bands edge bands W =8 W =12 W =16 W =20 BiCl W =240 / a k x E ( e V ) a k x a k x a k x a k x a k x SbCl E ( e V ) W =4 bulk bands edge bands W =8 W =12 W =16 W =20 BiBr W =240 / a k x E ( e V ) a k x a k x a k x a k x a k x SbBr E ( e V ) W =4 bulk bands edge bands W =8 W =12 W =16 W =20 BiI W =240 / a k x E ( e V ) a k x a k x a k x a k x a k x SbI
Figure S6. Band structure of BiX and SbX armchair nanoribbons (X = F, Cl, Br, I) of different width W . Topological edgestates are reported in green. The energy scale is referred to the Fermi energy. II. TRANSMISSION SPECTRA FOR LARGE ( W = 8 ) AND NARROW ( W = 4 ) RIBBONSA. Topological insulator nanoribbons In this section we report the transmission spectrum of all BiX and SbX zigzag nanoribbons of width W = 4 and W = 8 in the presence of either edge or bulk defects — some of which have already been shown in the main text. Forall materials considered, we report three transmission spectra corresponding to the following three configurations: • V X , ( µ = 0): single edge/bulk defect with (possibly) non-vanishing magnetic moments. • V X + H , ( µ = 0): Hydrogen saturated edge/bulk defect, with no magnetic moment. • V X , ( µ = 0): single edge/bulk defect with all magnetic moments set to zero.Figures S7 and S8 pertain to edge defects in nanoribbons of width W = 8 and W = 4 respectively. Similarly, Figs.S9 and S10 consider bulk defects in nanoribbons of width W = 8 and W = 4 respectively.Data shown here do not add new physics, but further demonstrate the phenomena already discussed in the maintext for the case of BiBr and SbBr. In particular: E ( e V ) a) Pristine b) V F ( 0) pristine with defectc) V F +H ( =0) BiF W = 8d) V F ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbF W = 8h) 1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V Cl ( 0) pristine with defectc) V Cl +H ( =0) BiCl W = 8d) V Cl ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbCl W = 8h)1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V I ( 0) pristine with defectc) V I +H ( =0) BiI W = 8d) V I ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbI W = 8h) Figure S7. Transmission spectrum (TS) of BiX and SbX (X = F, Cl, I) zigzag nanoribbons of width W = 8 in presence of oneof the following edge defects: simple edge defect (b and f); Hydrogen-saturated edge defect (c and g); edge defect with zeromagnetic moment (d and h). The TS for a pristine ribbon is also shown for comparison in each panel, and its band structureis reported in panels a and e. The energy region of insulating bulk is highlighted in yellow. (i) SbX nanoribbons with edge defects show a partial suppression of the transmission in the form of a localizedanti-resonance, due to the formation of a magnetic moment at the vacancy — see panels f in Figs. S7 and S8for nanoribbons of width W = 8 and W = 4 respectively.(ii) Bulk defects do not affect transport for W = 8, but lead to inter-edge scattering for W = 4 which is driven bythe partial overlap of the impurity state with the edge modes — see Figs. S9 and S10.(iii) For the case of Antimony-based materials, transport is not fully protected in the energy range where three pairsof edge states form at each interface. This is a manifestation of the underlying Z invariance, as we discuss inthe main text. Note that the transmission spectrum never drops below 4.(iv) Chemical saturation of the dangling bonds with Hydrogen is generally sufficient to restore topological protection,since spurious magnetic moments are removed and the energy level of the defect are moved away from the bulkgap region.0 E ( e V ) a) Pristine b) V F ( 0) pristine with defectc) V F +H ( =0) BiF W = 4d) V F ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbF W = 4h) 1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V Cl ( 0) pristine with defectc) V Cl +H ( =0) BiCl W = 4d) V Cl ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbCl W = 4h)1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V Br ( 0) pristine with defectc) V Br +H ( =0) BiBr W = 4d) V Br ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbBr W = 4h) 1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V I ( 0) pristine with defectc) V I +H ( =0) BiI W = 4d) V I ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbI W = 4h) Figure S8. Transmission spectrum (TS) of all zigzag nanoribbons of width W = 4 in presence of one of the following edgedefects: simple edge defect (b and f); Hydrogen-saturated edge defect (c and g); edge defect with zero magnetic moment (dand h). The TS for a pristine ribbon is also shown for comparison in each panel, and its band structure is reported in panels aand e. The energy region of insulating bulk is highlighted in yellow. W =8 E ( e V ) a) Pristine b) V F ( 0) pristine with defectc) V F +H ( =0) BiF W = 8d) V F ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbF W = 8h) 1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V Cl ( 0) pristine with defectc) V Cl +H ( =0) BiCl W = 8d) V Cl ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbCl W = 8h)1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V Br ( 0) pristine with defectc) V Br +H ( =0) BiBr W = 8d) V Br ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbBr W = 8h) 1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V I ( 0) pristine with defectc) V I +H ( =0) BiI W = 8d) V I ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbI W = 8h) Figure S9. Transmission spectrum (TS) of all zigzag nanoribbons of width W = 8 in presence of one of the following bulkdefects: simple bulk defect (b and f); Hydrogen-saturated bulk defect (c and g); bulk defect with zero magnetic moment (dand h). The TS for a pristine ribbon is also shown for comparison in each panel, and its band structure is reported in panels aand d. The energy region of insulating bulk is highlighted in yellow. A sketch of the structure with the position of the defectis shown in the upper panel. W =4 E ( e V ) a) Pristine b) V F ( 0) pristine with defectc) V F +H ( =0) BiF W = 4d) V F ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbF W = 4h) 1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V Cl ( 0) pristine with defectc) V Cl +H ( =0) BiCl W = 4d) V Cl ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbCl W = 4h)1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V Br ( 0) pristine with defectc) V Br +H ( =0) BiBr W = 4d) V Br ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbBr W = 4h) 1.51.00.50.00.51.01.5 E ( e V ) a) Pristine b) V I ( 0) pristine with defectc) V I +H ( =0) BiI W = 4d) V I ( =0)0 / a k x E ( e V ) e) T ( E ) f) T ( E ) g) T ( E ) SbI W = 4h) Figure S10. Transmission spectrum (TS) of all zigzag nanoribbons of width W = 4 in presence of one of the following bulkdefects: simple bulk defect (b and f); Hydrogen-saturated bulk defect (c and g); bulk defect with zero magnetic moment (dand h). The TS for a pristine ribbon is also shown for comparison in each panel, and its band structure is reported in panels aand d. The energy region of insulating bulk is highlighted in yellow. A sketch of the structure with the position of the defectis shown in the upper panel. a k x E ( e V ) a) Pristine 0 2 4 6 8 10 T ( E )b) V Mo pristine with defect T ( E )c) V S T ( E ) MoS W = 4d) V S + O Figure S11. Transmission spectrum (TS) of MoS2 zigzag nanoribbons of width W = 4 in presence of one of the followingedge defects: Mo vacancy (b); S vacancy (c); Oxygen-saturated S vacancy (d). The TS for a pristine ribbon is also shown forcomparison in each panel, and its band structure is shown in panel a. The energy region of insulating bulk is highlighted inyellow. B. Non-topological (MoS ) nanoribbons In Fig. S11 we report the transmission spectrum for an MoS zigzag nanoribbon of width W = 4. As for the caseof W = 8, both Mo and S vacancies lead to a very strong suppression of the transmission spectrum, while chemicalsaturation with Oxygen restores the transport properties of pristine nanoribbons, although not perfectly. III. DISORDERED CONFIGURATIONS
In this section we compare the transmission function of different nanoribbon configurations with a pair of non-magnetic bulk defects. The structure in Fig. S12a corresponds to the one discussed in the main text, while panels b,c, and d pertain to different disordered configurations and nanoribbon widths. In all cases a single bulk impurity isnot sufficient to generate inter-edge backscattering. Panels a, b and d show instead that multiple impurities generatea dip in the transmission function around E ≈ a)b) c)d) Figure S12. Transmission spectrum (TS) of BiBr zigzag nanoribbons of different widths in presence of multiple non-magneticbulk defects. Positions of the impurities (denoted d1 and d2) are reported in the corresponding structures. In all cases, wecompare the TS in the presence of d1 only, d2 only, and d1 and d2 simultaneously.
IV. LOCAL DENSITY OF STATES
To compare the spatial extension of edge and defect states, we study here the local density of states (LDOS) fordifferent device configurations. As demonstrated by Fig. S13a and S13b, a single bulk defect state cannot overlapsignificantly with both edge states in a wide nanoribbon, so that impurity-mediated backscattering through one singlevacancy is not permitted. On the other hand, a significant overlap is present in the cases of multiple impurities (panelc) or narrow structures (panel d).5 a)b)c)d)
Figure S13. Local density of states (LDOS) for different nanoribbon widths and disordered configurations according to thelegend. For each panel, the first and second columns show side and top views of the surface LDOS( x, y, z ) = 0 .
02 ˚A − eV − respectively. The third column corresponds to the LDOS in the transverse direction y averaged over the xz plane, while thelast column shows the LDOS along three different lines as reported on the left. The spatial region spanned by the edge statesis highlighted in cyan in the last two cases. V. MAGNETIC MOMENTS OF EDGE AND BULK IMPURITIES
Here we report the magnetic moments of all the magnetic device configurations we have found.We have extracted the magnetic moments from a Mulliken population analysis by subtracting the number of spindown electrons from the number of spin up electrons at each site. We interpret the number obtained in this way asthe local magnetic moment in units of the Bohr magneton µ B . Note that the Mulliken population for this particularcase of non-collinear spin calculation is a four component spin tensor, which is separately diagonalized at each site togive a local spin direction.Generally, magnetic moments are zero everywhere except in the immediate vicinity of the defect, as shown in FigureS14. Therefore, we only show the magnitude of the magnetic moment at the Bimsuth (Antimony) site closest to the6 Figure S14. Magnetic moments of an SbBr zigzag nanoribbon of width W = 8 calculated from a Mulliken population analysis.Sb atoms are in purple, while Br atoms are in red. A yellow arrow denotes the local spin direction and the magnitude of themagnetic moment. material W µ edge defect ( µ B ) µ bulk defect ( µ B )BiF 4 0.015 0.042BiF 8 0.017 0.072BiCl 4 0.018 0.370BiCl 8 0.018 0.401BiBr 4 0.001 0.298BiBr 8 0.000 0.361BiI 4 0.010 0.289BiI 8 0.015 0.319SbF 4 0.658 0.491SbF 8 0.651 0.539SbCl 4 0.960 0.710SbCl 8 0.945 0.661SbBr 4 0.933 0.752SbBr 8 0.916 0.702SbI 4 0.860 0.689SbI 8 0.857 0.627Table S2. Magnetic moments calculated at the Bi (Sb) sites closest to the vacancy defect. Two different nanoribbon widthsare considered here ( W = 4 and W = 8). vacancy. The corresponding results are given in Table S2.We note that Antimony-based structures always show a sizable magnetic moment of the order of 0 . − . µ B . Onthe other hand, Bismuth-based structures behave quite differently depending on the position of the impurity. Theyare almost non-magnetic in the edge defect configuration, with µ ≈ . µ B , while they bear a magnetic moment µ = 0 . − . µ B in the bulk defect configuration — the only exception being a bulk defect in BiF, with a much smallermoment µ = 0 . − . µ B . ∗ [email protected] J. Enkovaara, C. Rostgaard, J. J. Mortensen, J. Chen, M. Du lak, L. Ferrighi, J. Gavnholt, C. Glinsvad, V. Haikola, H. A.Hansen, H. H. Kristoffersen, M. Kuisma, A. H. Larsen, L. Lehtovaara, M. Ljungberg, O. Lopez-Acevedo, P. G. Moses,J. Ojanen, T. Olsen, V. Petzold, N. A. Romero, J. Stausholm-Møller, M. Strange, G. A. Tritsaris, M. Vanin, M. Walter,B. Hammer, H. H¨akkinen, G. K. H. Madsen, R. M. Nieminen, J. K. Nørskov, M. Puska, T. T. Rantala, J. Schiøtz, K. S.Thygesen, and K. W. Jacobsen, Electronic structure calculations with GPAW: a real-space implementation of the projectoraugmented-wave method, J. Phys. Condens. Matter , 253202 (2010). S. Smidstrup, D. Stradi, J. Wellendorff, P. A. Khomyakov, U. G. Vej-Hansen, M.-E. Lee, T. Ghosh, E. J´onsson, H. J´onsson,and K. Stokbro, First-principles Green’s-function method for surface calculations: A pseudopotential localized basis setapproach, Phys. Rev. B , 195309 (2017). S. Smidstrup, T. Markussen, P. Vancraeyveld, J. Wellendorff, J. Schneider, T. Gunst, B. Verstichel, D. Stradi, P. A.Khomyakov, U. G. Vej-Hansen, M.-E. Lee, S. T. Chill, F. Rasmussen, G. Penazzi, F. Corsetti, A. Ojanper¨a, K. Jensen,M. L. N. Palsgaard, U. Martinez, A. Blom, M. Brandbyge, and K. Stokbro, QuantumATK: an integrated platform of elec-tronic and atomic-scale modelling tools, J. Phys.: Condens. Matter , 015901 (2020). Z. Song, C.-C. Liu, J. Yang, J. Han, M. Ye, B. Fu, Y. Yang, Q. Niu, J. Lu, and Y. Yao, Quantum spin Hall insulators andquantum valley Hall insulators of BiX/SbX (X= H, F, Cl and Br) monolayers with a record bulk band gap, NPG Asia Mater. , e147 (2014). X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys.83