Stable nontrivial Z2 topology in ultrathin Bi (111) films: a first-principles study
Zheng Liu, Chao-Xing Liu, Yong-Shi Wu, Wen-Hui Duan, Feng Liu, Jian Wu
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Stable nontrivial Z topology in ultrathin Bi (111) films: a first-principles study Zheng Liu,
1, 2
Chao-Xing Liu, Yong-Shi Wu,
4, 5
Wen-Hui Duan,
1, 6
Feng Liu, ∗ and Jian Wu
1, 6, † State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China Institute for Advanced Study, Tsinghua University, Beijing 10084, China Physikalisches Institut, Universit¨at W¨urzburg, D-97074 W¨urzburg, Germany Department of Physics and Astronomy, University of Utah, Salt Lake City, 84112 Department of Physics, Fudan University, Shanghai 200433, China Department of Physics, Tsinghua University, Beijing 10084, China Department of Materials Science and Engineering, University of Utah, Salt Lake City, 84112 (Dated: September 5, 2018)Recently, there have been intense efforts in searching for new topological insulator (TI) materials.Based on first-principles calculations, we find that all the ultrathin Bi (111) films are characterizedby a nontrivial Z number independent of the film thickness, without the odd-even oscillation oftopological triviality as commonly perceived. The stable nontrivial Z topology is retained by theconcurrent band gap inversions at multiple time-reversal-invariant k-points and associated with theintermediate inter-bilayer coupling of the multi-bilayer Bi film. Our calculations further indicatethat the presence of metallic surface states in thick Bi(111) films can be effectively removed bysurface adsorption. PACS numbers: 73.21.Ac, 71.70.-d, 73.20.At
As a new insulating phase in condensed matter, thetopological insulator (TI) has recently attracted a greatdeal of attention[1]. The TI is distinguished from theconventional insulator by its unique gapless surface statesresiding in the middle of band gap as a consequence ofthe so-called Z topology encoded in the wavefunctions.There has been an intensive search for TI materials. Al-though quite a few compounds have been found to be3D TIs[2–6], up to now only the HgTe quantum well isverified to be a 2D TI (or a quantum spin Hall insulator)experimentally[7]. Recently, Murakami predicted a sin-gle bilayer (BL) Bi (111) film to be an elemental 2D TI[8, 9], and further speculated the multi-BL Bi (111) filmto exhibit an odd/even oscillation of topological trivialitywith film thickness[8] by considering the multi-BL film asa stack of BLs with no or very weak inter-BL coupling[10]. Generally speaking, the 2D TI phase in thin films iscommonly perceived to depend sensitively on film thick-ness, as shown for the Bi Se and Bi T e ultrathin films[11]. The requirement of fine tuning the thickness makesthe experimental fabrication of 2D TIs rather difficult.Therefore, it is desirable to search for new materials ornew schemes to obtain a 2D TI.In this Letter, based on the first-principles calculationsof band structures and wavefunction parities, we find thatin fact all the Bi (111) ultrathin films are characterizedby a nontrivial Z number independent of the film thick-ness. The films with 1- to 4-BL are intrinsic 2D TIs,while those with 5- to 8-BL are 2D TIs sandwiched withtrivial metallic surfaces that can be extrinsically removedby surface adsorption. This finding is in direct contrastto the odd/even oscillation of topological triviality spec-ulated for the Bi films [8], as well as to the thickness-dependent topology shown for other 2D films [11]. Thesurprisingly stable 2D TI phase in the Bi(111) films are found to be retained by a concurrent band gap inver-sion at multiple time-reversal-invariant k-points (TRIKs)when the film thickness is increased. A detailed analysisof the 2- and 3-BL Bi films indicates that the intermedi-ate inter-BL coupling plays an important role in definingtheir unique topological property.Bismuth is one of the main group elements that hasthe strongest spin-orbit coupling (SOC), a fundamentalmechanism to induce the Z topology. For this reason,many 3D TIs make use of Bi, even though 3D bulk Biitself is topologically trivial [2]. The electronic propertiesof Bi, such as the bulk band structure [12], the surfacestates [13] and the semimetal-to-semiconductor transi-tion [14, 15], have been well established in the literature.The outermost shell of Bi has the electron configuration6 s p . It tends to form three bonds to close the shell.The single-crystal Bi has a bi-layered structure, with anABC stacking sequence along the (111) direction (Fig.1a). Within each BL, every Bi atom forms three σ bondswith its nearest neighbors in a trigonal pyramidal ge-ometry. Projecting onto the (111) plane, the BL formsa hexagonal lattice with two atoms per unit cell (Fig.1b). There are three key structural parameters to definethe lattice: the in-plane lattice constant a , the intra-BL bond angle α (or the intra-BL height difference d ),and the inter-BL spacing d . Our calculated Bi crystalstructural parameters as shown in Fig. 1a and b agreewell with previous calculations [13]. The ultra-thin Bi(111) thin films, consisting of a few number of stackedBLs, have slightly relaxed structural parameters relativeto the bulk values. Specifically, α approaches 90 o and d increases by about 6%. This implies a slightly enhancedp-orbital feature of the intra-BL bonds and a weakeningof the inter-BL coupling.To identify the 2D TI phase, a single Z topological FIG. 1: (a) The hexagonal unit cell of single-crystal bismuth.(b) The top view of the Bi lattice. (c) The first Brillouin zoneof the hexagonal lattice. number ( ν ) is used as the “order parmeter” [10]: ν = 1indicates a topologically nontrivial phase; ν = 0 indicatesa trivial phase. The calculation of the Z number can bedramatically simplified by the so-called “parity method”[2], if the system is space inversion invariant, as the caseof the Bi (111) film. Accordingly, the Z number of Bifilms can be obtained from the wavefunction parities atfour TRIKs ( K i ), one Γ and three M s, as δ ( K i ) = N Y m =1 ξ i m ( − ν = Y i =1 δ ( K i ) = δ (Γ) δ ( M ) where ξ = ± is the parity eigenvalues and N is the num-ber of the occupied bands.Single-electron wavefunction parities are calculatedwithin the density functional theory using the plane wavebasis, as implemented in the ABINIT package [16]. Weemploy the local density approximation (LDA) [17] andthe Hartwigsen-Goedecker-Hutter pseudopotential [18],which is generated on the basis of a fully relativistic all-electron calculation and tested to be accurate for heavyelements like Bi. The spin-orbit coupling is included inthe self-consistent calculations as described in [19]. Tomodel the thin film, a supercell of slab is used with peri-odic boundary conditions in all three dimensions with a10 ˚ A thick vacuum layer in the (111) direction to elim-inate the inter-slab interaction. A plane wave cutoff of24 Ry and a Γ-centered k-point mesh of 10 × × Z numbers for 1- to 8-BL films. Surprisingly, all the films we calculated arecharacterized by the nontrivial Z number ( ν = 1), indirect contrast to the oscillation of topological trivialityas commonly perceived. We notice that the total par-ity δ (Γ) and δ ( M ) simultaneously change their signs forevery 2 BLs. This kind of parity oscillation, which orig- TABLE I: The total parity at the Γ and M points and the Z number of Bi (111) films with different thickness δ (Γ) + + - - + + - -3 δ ( M ) - - + + - - + + ν inates from the inverted band gap of a 3D TI, has beenreported in Bi Se and Bi T e ultrathin films, but onlyat the Γ point [11]. The uniqueness of Bi film is the in-verted band gaps both at the Γ and M points due to thestrong SOC of Bi [2]. As a consequence, the parity oscilla-tion under the quantum confinement occurs concurrentlyat all TRIKs (one Γ and three Ms) as the film thicknessincreases. Being the product of δ (Γ) and δ ( M ), the Z number shows up as the “beat” of two oscillations, whichmakes the 2D TI phase in Bi ultrathin films much morestable than in the Bi Se and Bi T e films. We expectthat the Z number of Bi films will eventually change atsome point when the phase difference between the oscil-lations at Γ and M accumulates to π , but this requirescalculations of much thicker Bi films possibly beyond thecurrent computational capability.In Fig. 2a, we plot both the direct and indirect bandgaps as a function of the film thickness. Within the calcu-lated film thickness range, the direct gap always remainsopen, which is essential for a well-defined Z number.Below 4 BLs, the film is a semiconductor having a non-trivial Z number, and hence representing an intrinsic2D TI. However, the indirect band gap becomes nega-tive above 4 BLs, leading to a semiconductor-semimetaltransition[15]. The semi-metallization arises from twooverlapping bands around the Fermi level, as shown inFig. 2b for the 5-BL film as an example. It has beenpointed out [13, 15] that these two bands have evidentsurface band features and tend to become gapless at the Γand M points in the limit of a semi-infinite system. Fromthis view, the films from 5- to 8-BL can be regarded as a2D TI sandwiched between two trivial metallic surfaces(top panel, Fig. 2c). The meaning of “trivial” here istwo-fold. On the one hand, if we consider the surface asan individual 2D system, its Z number is 0 (trivial), asobtained from the surface band parities at the Γ and Mpoints (see Fig. 2b). Therefore, the surface bands haveno contribution to the nontrivial Z number of the film.On the other hand, in the limit of semi-infinite system,because bulk Bi is a 3D Z topologically trivial insula-tor, these trivial metallic surface bands are not robustas those of a 3D strong TI and hence, in principle, canbe easily removed by surface defects or impurities, e.g.via surface adsorption. To test this idea, we have ter-minated the two surfaces with H atoms as schematicallydepicted in the bottom panel of Fig. 2c and repeatedthe calculation. We find that upon surface adsorption,the two surface bands are separated apart opening a gaparound the Fermi level, as shown in Fig. 2d. Meanwhile,the H atoms introduce additional occupied bulk bands,which are found to be topologically trivial, so that the Z number remains nontrivial (see Fig. 2d). Thus, bythe extrinsic effects of H surface adsorption, the thickerfilms above 5 BLs are effectively converted into 2D TIssimilar to those ultrathin films below 4 BLs. We notethat because LDA is known to underestimate the bandgap, the actual transition thickness is likely to be thickerthan the 4-to-5 BLs, but the overall trend of behavior weshow here should remain valid.It has been predicted that the 1-BL Bi (111) film isa 2D TI [8, 9]. If we could regard the n-BL film as astack of these nontrivial 1-BL films and without inter-BL coupling, all the bands would have n-fold degeneracyand there would be naturally an odd-even oscillation of Z topology: ν = 1 for the odd-BL stacks and ν = 0for the even-BL stacks [10] as every additional BL flipsthe Z number. Such odd-even oscillation of Z topologyfrom the zero inter-BL coupling limit can be extended tothe weak inter-BL coupling regime under the adiabaticapproximation, which was speculated to be applicable tothe Bi(111) films [8]. However, our first-principles resultsshow that the Bi (111) film may not be adiabaticallyconnected to the zero inter-BL coupling limit.The Bi (111) film actually represents a special inter-esting class of films having an intermediate inter-BL cou-pling strength. The inter-BL bond energy is calculatedto be 0.3-0.5 eV per bond, which is noticeably largerthan the typical values of weak interfacial bonds, suchas Van der Waals bond, but smaller than the values oftypical chemical bonds. This intermediate inter-BL cou- FIG. 3: The band structure of (a) 2-BL and (b) 3-BL underdifferent inter-BL spacing. (c) The energy level at the Γ pointof (left) 2-BL and (right) 3-BL as a function of the inter-BLspacing. The red (circle) lines indicate even-parity levels. Theblue (square) lines indicate odd-parity level. pling may have a significant influence on the topologicalproperty. To reveal the influence, we have calculateda set of “model” Bi films with their inter-BL couplingtuned gradually from the real intermediate regime to thehypothetical weak coupling regime that could be adia-batically connected to the zero coupling limit. This isdone by artificially increasing the inter-BL distance fromthe equilibrium value d with an increment of ∆ d , usingthe supercell technique. Here we take 2-BL (3-BL) filmas an example of even (odd) number of BL films. Fig-ure 2a and 2b show the evolution of band structures as afunction of ∆ d for the 2-BL and 3-BL films, respectively.For ∆ d = 0, the figures show the realistic band struc-tures of the 2-BL and 3-BL films, while for ∆ d = 4˚ A ,the corresponding band dispersion of hypothetical filmsshows a weak inter-BL coupling case that is adiabaticallyconnected to the zero coupling limit, based on topologi-cal analysis. We note that as the ∆ d is tuned from 4˚ A to 0˚ A , the direct gap at Γ point undergoes a closing andre-opening process, indicating that the realistic band dis-persion of Bi films may not be adiabatically connected tothe zero coupling limit.To trace the change of topology with the inter-BL cou-pling, we analyze the parity property of the energy levelsat Γ point as a function ∆ d in Fig. 2c for the 2-BL(left) and 3-BL (right) Bi films, respectively. When ∆ d is reduced, the n-fold degeneracy for the zero couplinglimit as mentioned above is lifted by the inter-BL cou-pling. Although the variation of band gaps is similarfor the 2-BL and 3-BL films, the change of total parityof all the occupied bands shows quite different behav-iors. For the 2-BL film, each doubly degenerate levelis split into one odd-parity sub-level with higher energyand one even-parity sub-level with lower energy. Conse-quently, the lowest unoccupied level and the highest oc-cupied level have the opposite parities (left panel of Fig.2c). As ∆ d is reduced, a level crossing and hence a par-ity exchange happens at ∆ d = 2˚ A due to the oppositeparities of the two crossing levels, leading to a changeof ν from 0 (for the non-coupling even-BL films) to 1,and hence converting the Z number of 2-BL film fromtrivial to nontrivial. The situation, however, is differ-ent for the 3-BL film. Now, each 3-fold degenerate levelis split into three sub-levels, always with one odd-paritysub-level sandwiched by two even-parity sub-levels (rightpanel of Fig. 2c). Consequently, both the lowest unoc-cupied and highest occupied levels have the same evenparity. As ∆ d is reduced, a level anti-crossing insteadof crossing happens and the number of odd- and even-parity sub-levels for both the occupied and unoccupiedbands remain unchanged, resulting in no change to thenon-trivial Z topology of the 3-BL film.In the above simple picture, we have implicitly usedthe prerequisite condition that the level crossing or anti-crossing happens only at the Γ point but not at the Mpoint and involves only a few levels close to the bandgap, which can only be satisfied by an inter-BL cou-pling that is not too strong. Also, the coupling cannot be too weak in order to move away from the weakcoupling regime that is adiabatically connected to thezero coupling limit. Therefore, the intermediate couplingstrength is a mandatory condition for the non-trivial Z topology. Such ”intermediate coupling principle” may beutilized in search for the 2D TI phase in other materials.As a summary, the film below 4 BLs is an intrinsic2D TI with the band structure consisting of ”molecularorbital” levels without distinction of surface bands frombulk bands, as shown in Fig. 3a (left panel) and 3b (leftpanel) for the 2-BL and 3-BL film, respectively. Above4 BLs, the band structure is made of surface bands su-perimposed onto a 2D projected bulk band, as shown inFig. 2a for the 5-BL film. The projected 2D bulk bandskeep the non-trivial topology of a 2D TI with a sizablegap, but the surface bands gradually appear in the mid-dle of the projected bulk band gap with the increasingfilm thickness (see also [15]), leading to a semiconductor(1- to 4-BL) to a semimetal (5- to 8-BL) transition. Thetrivial metallic surface states can be removed by surfaceH adsorption (Fig. 2d), which effectively converts the Bifilms into true 2D TIs.Our finding of all the ultrathin Bi (111) films being2D TIs independent of thickness may provide a possi-ble explanation of the recent observation of 1D topolog-ical metal on the Bi (114) surface[20], and will stimu-late more experimental interest in this intriguing system.The Bi (111) films above 6-BL have already been success- fully grown via molecular beam epitaxy [21], and hope-fully even thinner Bi films can be grown in the near fu-ture. The physical mechanism we identified for retainingthe stable nontrivial Z topology has broad implications.Most importantly, tuning the inter-layer coupling to theintermediate regime, so as to remove the odd-even oscil-lation of topological triviality, can be applied as a generalstrategy to obtain the TI phase.We thank Xiaofeng Jin for stimulating discussions andfor sharing with us their unpublished experimental dataon the Bi thin films. The work in Tsinghua University issupported by the Ministry of Science and Technology ofChina (Grants No. 2011CB606405, 2011CB921901, and2009CB929401) and NSF of China (Grants No. 10974110and 11074139). CXL acknowledges the support fromHumboldt foundation, YSW is supported in part by USNSF grant PHY-0756958, and FL acknowledges supportsfrom the DOE-BES program. ∗ Electronic address: fl[email protected] † Electronic address: [email protected][1] J. E. Moore, Nature , 194 (2010); M. Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010); X. L. Qi, S.C. 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