Topological fate of edge states of single Bi bilayer on Bi(111)

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1 PHYSICAL REVIEW B 93, (26) Topoloical fate of ede states of sinle Bi bilayer on Bi() Han Woon Yeom,,2,* Kyun-Hwan Jin, 2 and Seun-Hoon Jhi 2 Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), 77 Cheonam-Ro, Pohan , Republic of Korea 2 Department of Physics, Pohan University of Science and Technoloy (POSTECH), Pohan , Republic of Korea (Received 7 November 25; revised manuscript received 6 January 26; published 26 February 26) We address the topoloical nature of electronic states of step edes of Bi() films by first-principles band structure calculations. We confirm that the dispersion of step-ede states reflects the topoloical nature of underlyin films, which become topoloically trivial at a thickness larer than eiht bilayers. This result clearly conflicts with recent claims that the step-ede state at the surface of a bulk Bi() crystal or a sufficiently thick Bi() film represents nontrivial ede states of the two-dimensional topoloical insulator phase expected for a very thin Bi() film. The trivial step-ede states have a iantic spin splittin of one-dimensional Rashba bands and substantial intermixin with electronic states of the bulk, which miht be exploited further. DOI:.3/PhysRevB I. INTRODUCTION The hallmark of a topoloical insulator (TI) is the existence of chiral ede mode(s) alon edes of the material. In its two-dimensional (2D) version, the chiral ede mode is hihly interestin, representin a quantum spin Hall (QSH) channel [ 4]. Very recently, such QSH ede states were reported to be directly probed by scannin tunnelin microscopy and spectroscopy (STM/STS) for sinle Bi bilayers (BLs) rown on Bi 2 Te 2 Se [5,6] and raphene [7], followin an earlier theoretical prediction [8,9]. The QSH phase of Bi ultrathin films was predicted to be stable for films of a few bilayers [] and a thicker film would become topoloically trivial as expected for bulk Bi crystals []. On the other hand, a recent STS study found ede-localized states, possibly spin polarized, alon step edes at the surface of a Bi() crystal and related them to the topoloical ede state of the 2D topoloically insulatin (or semimetal) sinle Bi BL [2]. This claim was followed by a more recent work on thick Bi() films rown on a Si substrate [3]. That is, these works interpret the surface layer of a Bi() crystal as a 2D topoloical material in apparent contradiction to the current knowlede of a Bi crystal as a topoloically trivial metal. This leads to important questions of whether the topoloically trivial Bi() crystal is covered with a nontrivial 2D layer and how the interaction of the surface Bi layer with its bulk preserves (or affects) its topoloical properties. In this work, we explicitly check the D band structure of step edes of Bi() films as a function of the film thickness. We unambiuously clarify that the Bi() surface layer on top of a sufficiently thick film becomes a trivial 2D metal, reflectin the topoloical phase transition of the Bi() film, and its step ede has trivial spin-split bands of the D Rashba type. II. CALCULATIONS Ab initio calculations were carried out in the planewave basis within the eneralized radient approximation * yeom@postech.ac.kr for the exchane-correlation functional usin the VASP packae [4,5]. A cutoff enery of 4 ev was used for the plane-wave expansion and a k-point mesh of 5 for the Brillouin zone samplin. The spin-orbit couplin was considered as implemented within VASP at the second variational step usin the scalar-relativistic eienfunctions as a basis [6]. In order to investiate ede states, we carried out calculations for sinle BL Bi nanoribbons (BNRs) of 9 or 5 Bi ziza chains on top of 6 or 7 9 BL slabs, respectively, as illustrated in Fi.. This mimics step edes with the ziza structure of 2 BL films. The atoms of the Bi slabs and the nanoribbons on top were fully relaxed until the Hellmann- Feynman force was less than. ev/å. The topoloical invariants were calculated usin the well-established method of Soluyanov and Vanderbilt, which can handle a system without inversion symmetry [7]. III. RESULTS AND DISCUSSION For ultrathin floatin Bi films, detailed band structure calculations are already available, which show that the 2D TI nature of Bi bilayers is maintained up to 8 BLs []. Thicker films are expected to become metallic with the band ap closed, obtainin the bulk properties of a topoloically trivial metal. However, this topoloical phase transition was apparently not clarified in previous work. Since this point is critical in the followin discussion, we first check the thickness needed for the phase transition into a trivial phase. The finerprint of a topoloically nontrivial QSH phase is band inversion, which always involves the closin and reopenin of the band ap as the strenth of spin-orbit couplin is increased. This band-ap reopenin can be seen clearly for floatin Bi() films up to the thickness of 8 BLs [Fis. 2 2], which is consistent with the previous calculation []. However, from 9 BL, there exist two successive band-ap closins [Fis. 2(e) and 2(f)], which indicate a phase transition into a topoloical phase and a reentrant transition into a trivial phase. The trivial phase at the full strenth of SOC for 9 and BL films is confirmed unambiuously by the band character with no inversion and further by the null topoloical invariant calculated (Table I). That is, the trivial bulk metallic behavior is reached at the thickness of 9 BL /26/93(7)/75435(5) American Physical Society

2 HAN WOONG YEOM, KYUNG-HWAN JIN, AND SEUNG-HOON JHI PHYSICAL REVIEW B 93, (26) B A TABLE I. Calculated Z 2 number of Bi films with different thicknesses. Bi film BL BL BL BL BL BL BL BL BL BL 4BL Z 2 9BL FIG.. Structural models of ziza Bi nanoribbons on, 4, and 9 BL Bi() films. Type A and type B edes are marked by red and blue atoms, respectively, which undertake different types of reconstruction. In order to reduce the computational load, we used narrower nanoribbons on 9 BL films. In this case, one ede is passivated by H atoms to ensure the decouplin between the edes. We also examined the band structure of surface layers of these films. Surface layers of a thick Bi() film or the bulk Bi() have topoloically trivial 2D spin helical bands due to the Rashba-type spin-orbit interaction [8,9]. In contrast to SOC strenth SOC strenth. 6BL x 2BL 5BL.5..2 SOC strenth.5 (e) 9BL x x BL SOC strenth x BL 4BL 7BL 3BL.5. SOC strenth (f) QSH BL FIG. 2. (e) Calculated enery aps ( ) of BL Bi() films at the Ɣ point as the strenth of spin-orbit couplin (SOC) is varied. The SOC strenth. corresponds to that of the bulk Bi(). The ap-closin point (X ) is exponentially reduced as the film thickness is increased. There are two ap-closin points above 8 BL and the QSH phase chanes into the normal metallic (NM) phase. NM thin floatin Bi bilayers, the bands of a Bi() surface layer have no band ap and no band inversion near the Fermi enery. As also shown in previous work [], the surface layers of films become trivially metallic at a thickness larer than 4 BL, well below the thickness for the topoloical phase transition of the whole film. Therefore, there is no obvious reason apriorito expect any topoloical ede state on the ede of these surface layers. The oriin of the difference between the floatin Bi bilayer and the Bi() surface layer will be discussed further below. We then calculated the ede band structure explicitly by simulatin the step-ede structure of the surface layer on Bi() films of varyin thickness. As shown in Fi., we enerate nanoribbons of 9 5 unit cells wide on top of Bi() films up to 9 BL thickness. The thickness of 9 BL is confirmed to be sufficient to simulate the topoloically trivial bulk property as discussed above. Since all the atoms are freely relaxed, the ede atoms are substantially reconstructed to reduce the enery cost of the danlin bond creation. Two different ede confiurations and reconstructions are formed due to the BL structure; ede atoms are buckled down or up for so-called A- or B-type edes, respectively, as shown in Fi.. These two ede structures yield slihtly different ede state dispersions as shown in Fi. 3. These two edes have in common that the ede bands have Dirac-like crossins at the ede of the Brillouin zone (X) within the enlared band ap. These crossed bands have a helical spin texture (as confirmed in our spin-polarized calculations; data not shown here). This spin helicity and the Dirac-like crossins are indeed very similar to the QSH ede state of the 2D TI phase of a sinle BL film [Fi. 3]. However, an important chane can be noticed even for the 2 BL case, the nanoribbon on a sinle BL substrate [Fi. 3]. Even at this thickness, the band ap of the film is reduced substantially, especially at the center of the Brillouin zone (Ɣ), which closes down completely above the 8 BL substrate. An immediate consequence of the band-ap closure is that the topoloical nature of the step-ede state appears ambiuous even on the BL substrate; as the band ap closes, the upper and lower branches of the spin-split ede states et very close to each other at Ɣ while their Dirac-like crossin at X is preserved. If these two branches mere at Ɣ, then their dispersions become topoloically trivial and the Dirac-like crossins at X become trivial Rashba band crossins. This result is consistent with the previous calculation of the 2 BL ede with a much smaller nanoribbon width and with unrelaxed ede structures [2]. In fact, this chane is what is expected from the topoloical phase transition of the film at a thickness between 8 and 9 BLs. For a thicker substrate than 4 BLs, the band-ap narrowin is substantial in the whole Brillouin zone and the ede states hybridize with the substrate

3 TOPOLOGICAL FATE OF EDGE STATES OF SINGLE Bi... PHYSICAL REVIEW B 93, (26).6 Enery (ev) (e) (f) Enery (ev) Γ X Γ Γ X Γ Γ X Γ FIG. 3. Calculated band structure of a pristine floatin BL Bi nanoribbon and of BL Bi nanoribbons on top of, 2, 4, and (e),(f) 9 BL films. Ede states are distinuished by red and blue dots for the A- and B-type edes accordin to Fi.. Shaded reions are the projection of the bulk (whole film) bands of, 3, 5, and BL films, respectively. bands for a lare part of the Brillouin zone. Thus, the step-ede state is not truly localized alon the ede atoms except for a limited part of the Brillouin zone. Note that these chanes do not depend on the type of ede structures. As mentioned above, the crucial part for the topoloical nature of the ede-state dispersion is the band dispersion near Ɣ. We thus scrutinize the chane of the ede-state dispersions as shown in Fi. 4 for one type, type A, of the ede structure. As clearly shown here, the step-ede states on a sufficiently thick film have their spin-split bands mered at the Ɣ point, becomin trivial Rashba spin-split bands. The same conclusion is reached for the other type of edes. We confirm that this chane occurs concomitantly with the quantum phase transition from the 2D topoloical insulator to a 3D trivial semimetal. This can be shown by the disappearance of the band inversion between the bands of different p orbitals; as depicted in Fis. 4 and 4, the valence and conduction bands at Ɣ have a clear band inversion for the sinle floatin BL film but return back to normal on the 9 BL substrate. In contrast to the present work, the previous STS measurements appeared to show a topoloically nontrivial step-ede state on top of a bulk Bi() crystal [2]. What was likely measured is a part of the dispersion of the ede-localized state, which is indicated by an arrow in Fi. 4. This part of the band had spin-polarized nature, due to the Rashba spin splittin. The topoloical nature of the ede states is, however, determined not by the spin splittin but by their dispersions especially at Ɣ in the present case. While one can aree that these Rashba-type ede states evolve into QSH ede states at a very small thickness as already observed on sinle BL films on Bi 2 Te 2 Se and raphene [5 7], the bands in a thicker film and the bulk are obviously not topoloical ede states but D bands with Rashba spin splittin. The D chiral electron state itself with a lare Rashba splittin was previously introduced in a surface nanowire array [2]. In other words, the stron couplin of the electronic states of a sinle BL surface layer Enery (ev) Enery (ev) Γ 5BL ede state p x, p y p z X Γ 2BL BL FIG. 4. Band structures of Bi nanoribbons on various Bi() thin films. Ede states of a ziza-eded floatin Bi nanoribbon. Ede states of the type-a edes (the same type of ede as in Ref. [2]) on, 4, and 9 BL films. Shaded reions are the projection of the correspondin bulk (whole film) bands. For the floatin or surface layer of and BL films, the orbital characters of the valence and conduction band edes are indicated. X

4 HAN WOONG YEOM, KYUNG-HWAN JIN, AND SEUNG-HOON JHI PHYSICAL REVIEW B 93, (26) Enery (ev) Enery (ev) p x p y p z k ΓK (Å ) k ΓK (Å ) FIG. 5. Calculated band structures of a sinle Bi bilayer on top of 9 BL films at a distance of 3, 2,, and Å from the equilibrium distance, respectively. The states oriinatin from the Bi bilayer are marked by red (p x and p y orbitals) and blue (p z orbital). The in-plane lattice constant is fixed to be that of the equilibrium surface layer case. with those of its substrate destroys their topoloical nature. This substrate effect is detailed further in Fi. 5. We calculate the band dispersions of a sinle Bi BL at varyin distances from its equilibrium position as the surface layer of 9 BL film. Note that, even at a sufficiently lare distance from the bulk with a minimal interaction, the sinle BL film loses the insulatin property of a free-standin film. This is due to the substantial difference in lattice constant between the free-standin film and the bulk (about 5% expansion in the bulk). Even with the closed band ap, the noninteractin film maintains its topoloical properties with the band inversion near the Fermi level as in the free-standin film. When the interaction with the bulk is turned on at a shorter distance, the band inversion disappears, with bands of a sinle character (red colored bands) across the Fermi level and the valence bands of the noninteractin film (blue bands) hybridized stronly with those of the bulk. These results show clearly how the sinle BL film and the Bi() substrate interact. The existence of a spin-helical ede state is necessary for a TI but not a sufficient condition to dictate the TI property and there is a substantial chane of the electronic state of the Bi bilayer on the bulk substrate, not described in [2] IV. CONCLUSION We investiated explicitly the topoloical nature of electronic states of step edes of Bi() films by first-principles calculations. We showed that the dispersion of step-ede states reflects the topoloical nature of the underlyin films, rather than indicatin the existence of topoloical ede states alon step edes on surfaces of a bulk Bi() crystal or a sufficiently thick Bi() film. The trivial step-ede states have spin splittin of one-dimensional Rashba bands. This D band with a hue Rashba splittin miht be interestin in the search for Majorana fermions but the existence of a stron intermixin with the substrate electronic state has to be considered. The topoloical character of an ultrathin 2D topoloical insulator put on a substrate has to be carefully examined, considerin the strain and electronic hybridization with the substrate. ACKNOWLEDGMENT This work was supported by Institute for Basic Science (Grant No. IBS-R5-D). [] B. A. Bernevi, T. L. Huhes, and S.-C. Zhan, Science 34, 757 (26). [2] M. Köni, S. Wiedmann, C. Brne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhan, Science 38, 766 (27). [3] C. Liu, T. L. Huhes, X.-L. Qi, K. Wan, and S.-C. Zhan, Phys. Rev. Lett., 2366 (28). [4] I. Knez, R.-R. Du, and G. Sullivan, Phys.Rev.Lett.7, 3663 (2) [5]S.H.Kim,K.-H.Jin,J.Park,J.S.Kim,S.-H.Jhi,T.-H.Kim, and H. W. Yeom, Phys.Rev. B89, (24) and references therein. [6] C. Sabater, D. Gosálbez-Martinez, J. Fernández-Rossier, J. G. Rodrio, C. Untiedt, and J. J. Palacios, Phys. Rev. Lett., 7682 (23). [7] Y. Lu, W. Xu, M. Zen, G. Yao, L. Shen, M. Yan, Z. Luo, F. Pan, K. Wu, T. Das, P. He, J. Jian, J. Martin, Y. P. Fen, H. Lin, and X.-s. Wan, Nano Lett. 5, 8 (25). [8] S. Murakami, Phys. Rev. Lett. 97, (26). [9] M. Wada, S. Murakami, F. Freimuth, and G. Bihlmayer, Phys. Rev. B 83, 23 (2). [] Z. Liu, C.-X. Liu, Y.-S. Wu, W. H. Duan, F. Liu, and J. Wu, Phys. Rev. Lett. 7, 3685 (2). [] A. Takayama, T. Sato, S. Souma, T. Ouchi, and T. Takahashi, Phys. Rev. Lett. 4, 6642 (25). [2] I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Pere, H. Ji, R. J. Cava, B. A. Bernevi, and A. Yazdani, Nat. Phys., 664 (24). [3] N. Kawakami, C.-L. Lin, M. Kawai, R. Arafune, and N. Takai, Appl. Phys. Lett. 7, 362 (25). [4] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 69 (996). [5] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (996). [6] D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B 62, 556 (2). [7] A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B 83, 2354 (2)

5 TOPOLOGICAL FATE OF EDGE STATES OF SINGLE Bi... PHYSICAL REVIEW B 93, (26) [8] Y. M. Koroteev, G. Bihlmayer, J. E. Gayone, E. V. Chulkov, S. Blüel, P. M. Echenique, and Ph. Hofmann, Phys. Rev. Lett.93, 4643 (24). [9]Y.M.Koroteev,G.Bihlmayer,E.V.Chulkov,andS.Blüel, Phys. Rev. B 77, (28). [2] H. Kotaka, F. Ishii, M. Saito, T. Naao, and S. Yainuma, Jpn. J. Appl. Phys. 5, 252 (22). [2] J. Park, S. W. Jun, M.-C. Jun, H. Yamane, N. Kosui, and H. W. Yeom, Phys. Rev. Lett., 368 (23)