Topological fate of edge states of Bi single bilayer on Bi(111)
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Topological fate of edge states of Bi single bilayer on Bi(111)
Han Woong Yeom,
1, 2, ∗ Kyung-Hwan Jin, and Seung-Hoon Jhi Center for Artificial Low Dimensional Electronic Systems,Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang 790-784, Republic of Korea Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Republic of Korea Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea (Dated: October 15, 2018)We address the topological nature of electronic states of step edges of Bi(111) films by firstprinciples band structure calculations. We confirm that the dispersion of step edge states reflectsthe topological nature of underlying films. This result unambiguously denies recent claims that thestep edge state on the surface of a bulk Bi(111) crystal or a sufficiently thick Bi(111) films representsnon-trivial edge states of the two dimensional topologcial insulator phase expected for a very thinBi(111) film. The trivial step edge states have a gigantic spin splitting of one dimensional Rashbabands and the substantial intermixing with electronic states of the bulk, which might be exploitedfurther.
PACS numbers: 71.10.Hf, 71.20.Be, 71.27.+a, 71.30.+h
I. INTRODUCTION
The hallmark of a topological insulator (TI) is the exis-tence of chiral edge mode(s) along edges of the material.In its two dimensional (2D) version, the chiral edge modeis highly interesting, representing a quantum spin Hall(QSH) channel.
Very recently, such QSH edge stateswere reported to be directly probed by scanning tunnel-ing microscopy and spectroscopy (STM/STS) for Bi sin-gle bilayers (BL) grown on Bi Te Se and graphene, following an earlier theoretical prediction. The QSHphase of Bi ultrathin films was predicted to be stableup to 8 BL thickness and a thicker film becomes topo-logically trivial as expected for bulk Bi crystals. Onthe other hand, a recent STS study found edge-localizedstates, possibly spin-polarized, along step edges at thesurface of a Bi(111) crystal and attributed them as thetopological edge state of the 2D topologically insulating(or semimetal) Bi single BL. This claim was followed bya more recent work on a thick Bi(111) films grown on aSi substrate. That is, these works interpret the surfacelayer of a Bi(111) crystal as a 2D topological materialin apparent contradiction to the current knowledge of aBi crystal as a topologically trivial metal. This leads toimportant questions of whether the topologically trivialBi(111) crystal is covered with a non-trivial 2D layer andhow the interaction of the surface Bi layer with its bulkpreserves (or affects) its topological property.In this work, we explicitly check the 1D band structureof step edges of Bi(111) films as a function of the filmthickness. We unambiguously clarify that the Bi(111)surface layer on top of a sufficiently thick film becomesa trivial 2D semimetal, reflecting the topological phasetransition of the Bi(111) film, and its step edge has trivialspin-split bands of the 1D Rashba type. (a)(b)(c)
B A
FIG. 1. (color online) Structural models of zigzag Bi nanorib-bons on (a) 1, (b) 4, and (c) 9 BL Bi(111) films. Type A andtype B edges are marked by red and blue atoms respectively,which undertake different types of reconstructions. In orderto reduce the computational load, we used narrower nanorib-bons on 9 BL films. In this case, one edge is passivated by Hatoms to ensure the decoupling between.
II. CALCULATIONS
Ab initio calculations were carried out in the plane-wave basis within the generalized gradient approximationfor the exchange-correlation functional . A cut-off en-ergy of 400 eV was used for the plane-wave expansion anda k -point mesh of 15 × × ∼ ∼ III. RESULTS AND DISCUSSION
For floating Bi films, detailed band structure calcu-lations are already available, which tell that the 2D TInature of Bi bilayers (BL) is maintained only up to 8BL. Thicker films become metallic with the band gapclosed, obtaining the bulk property of a topologicallytrivial metal. The closure of the band gap in this casedoes not involve the band inversion, which is anotherhallmark of TI. We reproduce this result consistentlywith our own calculations. We also examine the bandstructure of surface layers of these films. Surface layersof a thick Bi(111) film or the bulk Bi(111) have topologi-cally trivial 2D spin helical bands due to the Rashba-typespin-orbit interaction.
In contrast to thin floating Bibilayers, the bands of a Bi(111) surface layer have noband gap and no band inversion near the Fermi energy.Since the band inversion is required to have a topologi-cally non-trivial edge state, there is no obvious reason apriori to expect any topological edge state on the edge ofthese surface layers. The origin of the difference betweenthe floating Bi bilayer and the Bi(111) surface layer willbe discussed further below.We then calculated the edge band structure explicitlyby simulating the step edge structure of the surface layeron Bi(111) films of varying thickness. As shown in Fig.1, we generate nanoribbons of 9-15 unitcell wide on top ofBi(111) films up to 9 BL thickness. The thickness of 9 BLis confirmed to be sufficient to simulate the topologicallytrivial bulk property. Since the whole atoms are freelyrelaxed, the edge atoms are substantially reconstructedto reduce the energy cost of the dangling bond creation.Two different edge configurations and reconstructions areformed due to the BL structure; edge atoms are buckleddown or up for so called A or B type edge, respectively, asshown in Fig. 1. These two edge structures yield slightlydifferent edge state dispersions as shown in Fig. 2. Com-monly for these two edges, the edge bands have Dirac-likecrossings at the edge of Brillouin zone (X) within the en-larged band gap. These crossed bands have a helical spintexture (as confirmed in our spin-polarized calculations,data not shown here). This spin helicity and the Dirac-like crossings are indeed very similar to the QSH edgestate of the 2D TI phase of a single BL film [Fig. 2(a)].However, an important change can be noticed even forthe 2 BL case, the nanoribbon on a single BL film sub-strate [Fig. 2(b)]. Even at this thickness, the band gapof the film reduces substantially, especially at the centerof the Brillouin zone (Γ), which closes down completelyabove 8 BL substrate. An immediate consequence of theband gap closure is that the topological nature of thestep edge state becomes apparently ambiguous even onthe 1 BL substrate; as the band gap closes, the upperand lower branches of the spin split edge states get veryclose to each other at Γ while their Dirac-like crossing at X is preserved. If these two branches merges at Γ,then their dispersions become topologically trivial andthe Dirac-like crossings at X becomes trivial Rashba bandcrossings. In fact, this change is what is expected fromthe topological phase transition of the film at a thicknessaround 8 BL. For a thicker substrate than 4 BL, the bandgap narrowing is substantial in the whole Brillouin zoneand the edge states hybridize with the substrate bandsfor a large part of the Brillouin zone. Thus, the step edgestate is not truly localized along the edge atoms exceptfor a limited part of the Brillouin zone. Note that thesechanges do not depend on the type of the edge structures.As mentioned above, the crucial part for the topolog-ical nature of the edge state dispersion is the band dis-persion near Γ. We thus scrutinize the change of edgestate dispersions as shown in Fig. 3 for one type, typeB, of the edge structure. As clearly shown here, the stepedge states on a sufficiently thick film have their spin-splitbands merged at the Γ point, becoming a trivial Rashbaspin-split bands. The same conclusion is reached for theother type of edges. We confirm that this change occursconcomitantly with the quantum phase transition fromthe 2D topological insulator to a 3D trivial semimetal.This can be shown by the disappearance of the band in-version between the bands of different p orbitals; as de-picted in Figs. 3(a) and 3(d), the valence and conductionbands at Γ have the clear band inversion for the singlefloating BL film but return back to normal on the 9 BLsubstrate.In contrast to the present work, the previous STS mea-surements claimed the topologically non trivial step edgestate on top of a bulk Bi(111) crystal. What was ac-tually measured is a part of the dispersion of the edgelocalized state, which is indicated by an arrow in Fig.3(d). They claimed the spin polarized nature of this partof the band, which is obviously true due to the Rashbaspin splitting. The topological nature of the edge state is,however, determined not by the spin splitting but by theirdispersions especially at Γ in the present case. While onecan agree that this Rashba-type edge states evolve intoQSH edge states at a very small thickness as already ob-served on single BL films on Bi Te Se and graphene, the bands on a thick film are obviously not topologicaledge states but 1D bands with the Rashba spin splitting.The 1D chiral electron state itself with a large Rashbasplitting was previously introduced in a surface nanowirearray. In other words, the strong coupling of the elec-tronic states of a single BL surface layer with those ofits substrate destroys their topological nature. This sub-strate effect is detailed further in Fig. 4. We calculate theband dispersions of a Bi single BL at a varying distancefrom its equilibrium position as a surface layer of 9 BLfilm. Note that, even at a sufficiently large distance fromthe bulk with a minimal interaction, the single BL filmloses the insulating property of a free standing film. Thisis due to the substantial difference in the lattice constantbetween the free standing film and the bulk (about 5 %expansion in the bulk). Even with the closed band gap,
Γ X Γ (b) (c)(d)(a) (e) (f)-0.6-0.300.30.6 E ne r g y ( e V ) -0.6-0.300.30.6 E ne r g y ( e V ) Γ X Γ Γ X Γ
FIG. 2. (color online) Calculated band structure (a) pristine floating 1 BL Bi nanoribbon and 1 BL Bi nanoribbons on top of(b) 1, (c) 2, (d) 4, (e)-(f) 9 BL films. Edge states are distinguished by red and blue dots for the A and B type edges accordingto Fig. 1. Shaded regions are the projection of the bulk (whole film) bands of 1, 3, 5 and 10 BL films, respectively. -0.6-0.300.30.6 E ne r g y ( e V ) Γ X-0.6-0.300.30.6 E ne r g y ( e V ) Γ X1BL5BL 10BL edge statep , x p y p z FIG. 3. (color online) Band structures of Bi nanoribbons onvarious Bi(111) thin films. (a) Edge states of a zigzag-edgedfloating Bi nanoribbon. (b)-(d) Edge states of the type Bedges (the same type of edge as in Ref. 12) on 1, 4 and 9 BLfilms. Shaded regions are the projection of the correspondingbulk (whole film) bands. For the floating or surface layerof 1 and 10 BL films, orbital characters of the valence andconduction band edges are indicated. the non-interacting film maintains its topological prop-erty with the band inversion near the Fermi level as in thefree standing film. When the interaction with the bulkis turned on at a shorter distance, the band inversion disappears with bands of a single character (red coloredbands) across the Fermi level and the valence bands ofthe non-interacting film (blue bands) hybridized stronglywith those of the bulk. These results show clearly how thesingle BL film and the Bi(111) substrate interact. Theexistence of a spin-helical edge state is necessary for a TIbut not a sufficient condition to dictate the TI property.Unfortunately, this simple point was ignored by the pre-vious STS work as well as the substantial change of theelectronic state of the Bi bilayer on the bulk substrate. IV. CONCLUSION
We investigate explicitly the topological nature of elec-tronic states of step edges of Bi(111) films by first prin-ciples calculations. We show that the dispersion of stepedge states reflects the topological nature of underlyingfilms, unambiguously denying the existence of topolog-ical edge states along step edges on surfaces of a bulkBi(111) crystal or a sufficiently thick Bi(111) film. Thetrivial step-edge states have spin splitting of one dimen-sional Rashba bands. This 1D band with a huge Rashbasplitting might be interesting for the search of MajoranaFermion but the existence of a strong intermixing withthe substrate electronic state has to be considered.
ACKNOWLEDGMENTS
This work was supported by Institute for Basic Science(Grant No. IBS-R015-D1). p z p x p y -1-0.500.51 E ne r g y ( e V ) -0.6 -0.3 0 0.3 0.6 k ( Å ) -1ΓK -0.6 -0.3 0 0.3 0.6 k ( Å ) -1ΓK -1-0.500.51 E ne r g y ( e V ) (a) (b)(c) (d) FIG. 4. (color online) (a)-(d) Calculated band structures ofa sing Bi bilayer on top of 9 BL films at a distance of 4, 3,1, and 0 ˚ A from the equilibrium distance, respectively. Thestates originating from the Bi bilayer are marked by red ( px and py orbitals) and blue ( pz orbital). The in-plane latticeconstant is fixed with that of the equilibrium surface layercase. ∗ [email protected] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science , 1757 (2006). M. K¨ o nig, S. Wiedmann, C. Brne, A. Roth, H. Buhmann,L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science ,766 (2007). C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang,Phys. Rev. Lett. , 236601 (2008). I. Knez, R.-R. Du, and G. Sullivan, Phys. Rev. Lett. ,136603 (2011) S. H. Kim, K.-H. Jin, J. Park, J. S. Kim, S.-H. Jhi, T.-H.Kim, and H. W. Yeom, Phys. Rev. B , 155436 (2014)and references therein. C. Sabater, D. Gosa?lbez-Mart?nez, J. Fern?andez-Rossier, J. G. Rodrigo, C. Untiedt, and J. J. Palacios,Phys. Rev. Lett. , 176802 (2013). Y. Lu, W. Xu, M. Zeng, G. Yao, L. Shen, M. Yang, Z. Luo,F. Pan, K. Wu, T. Das, P. He, J. Jiang, J. Martin, Y. P.Feng, H. Lin, and X.-s. Wang, Nano Lett. , 80 (2015). S. Murakami, Phys. Rev. Lett. , 236805 (2006). M. Wada, S. Murakami, F. Freimuth, and G. Bihlmayer,Phys. Rev. B , 121310 (2011). Z. Liu, C.-X. Liu, Y.-S. Wu, W.-H.i Duan, F. Liu, and J.Wu, Phys. Rev. Lett. , 136805 (2011). A. Takayama, T. Sato, S. Souma, T. Oguchi, and T. Taka-hashi, Phys. Rev. Lett. , 066402 (2015). I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Perge,H. Ji, R. J. Cava, B. A. Bernevig, and A. Yazdani, NaturePhys. , 664 (2014). N. Kawakami, C.-L. Lin, M. Kawai, R. Arafune, and N.Takagi, Appl. Phys. Lett. , 031602 (2015). G. Kresse and J. Furthm¨ u ller, Phys. Rev. B , 11169(1996). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). Y. M. Koroteev G. Bihlmayer, J. E. Gayone, E. V.Chulkov, S. Bl¨ u gel, P. M. Echenique, and Ph. Hofmann,Phys. Rev. Lett. , 046403 (2004). Y. M. Koroteev G. Bihlmayer, E. V. Chulkov, and S.Bl¨ u gel, Phys. Rev. B , 045428 (2008). J. Park, S. W. Jung, M.-C. Jung, H. Yamane, N. Kosugi,and H. W. Yeom, Phys. Rev. Lett.110