Proving Nontrivial Topology of Pure Bismuth by Quantum Confinement
S. Ito, B. Feng, M. Arita, A. Takayama, R.-Y. Liu, T. Someya, W.-C. Chen, T. Iimori, H. Namatame, M. Taniguchi, C.-M. Cheng, S.-J. Tang, F. Komori, K. Kobayashi, T.-C. Chiang, I. Matsuda
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Proving Nontrivial Topology of Pure Bismuth by Quantum Confinement
S. Ito, B. Feng, M. Arita, A. Takayama, R.-Y. Liu, T. Someya, W.-C. Chen, T. Iimori, H. Namatame, M. Taniguchi, C.-M. Cheng, S.-J. Tang,
4, 5
F. Komori, K. Kobayashi, T.-C. Chiang, and I. Matsuda Institute for Solid State Physics (ISSP), The University of Tokyo, Kashiwa, Chiba 277-8581, Japan Hiroshima Synchrotron Radiation Center (HSRC),Hiroshima University, Higashi-Hiroshima, Hiroshima 739-0046, Japan Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan National Synchrotron Radiation Research Center (NSRRC), Hsinchu, Taiwan 30076, Republic of China Department of Physics and Astronomy, National Tsing Hua University, Hsinchu, Taiwan 30013, Republic of China Department of Physics, Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, Japan Department of Physics and Frederick Seitz Materials Research Laboratory,University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (Dated: September 13, 2018)The topology of pure Bi is controversial because of its very small ( ∼
10 meV) band gap. Here weperform high-resolution angle-resolved photoelectron spectroscopy measurements systematically on14 −
202 bilayer Bi films. Using high-quality films, we succeed in observing quantized bulk bandswith energy separations down to ∼
10 meV. Detailed analyses on the phase shift of the confinedwave functions precisely determine the surface and bulk electronic structures, which unambiguouslyshow nontrivial topology. The present results not only prove the fundamental property of Bi butalso introduce a capability of the quantum-confinement approach.
PACS numbers: 73.20.-r, 79.60.-i
Semimetal bismuth (Bi) has been providing an ir-replaceable playground in condensed matter physics.Its extreme properties originating from the three-dimensional Dirac dispersion enabled the first observa-tions of several important phenomena such as diamag-netism [1] and the various effects associated with Seebeck[2], Ettingshausen and Nernst [3], Shubnikov and de Haas[4] and de Haas and van Alphen [5]. Even now, numbersof novel quantum phenomena have been intensively re-ported on this system [6–13]. In spite of the enormousamount of research, one fundamental property of Bi hasbeen controversial: its electronic topology. Because of itshuge spin-orbit coupling (SOC) [14], Bi has also been acentral element in designing topological materials such asBi − x Sb x , Bi Se , Na Bi, and β -Bi I [15–19]. A combi-nation of SOC and several symmetries produces topolog-ically protected electronic states with inherent spin split-ting. Despite the essential role in topological studies, apure Bi crystal itself had long been believed topologicallytrivial based on several calculations [20–26], which hadbeen considered to agree with transport [27] and angle-resolved photoelectron spectroscopy (ARPES) measure-ments [22, 28, 29]. However, a recent high-resolutionARPES result suggests the surface bands are actually dif-ferent from previously calculated ones and Bi possessesa nontrivial topology [30, 31]. New transport measure-ments also imply the presence of topologically protectedsurface states [32, 33].Nevertheless, the recent ARPES result has not yetbeen conclusive because it lacks clear peaks of bulk bands[30, 31]. In principle, surface-normal bulk dispersionscan be measured by changing the incident photon energy,where the momentum resolution is determined from theuncertainty relation ∆ z · ∆ k z ≥ / z is an escape depth of photoelectrons.) However, the Diracdispersion of Bi is so sharp against this resolution that hν -dependent spectra show no clear peak [29–31]. This isa serious problem because Bi has a very small ( ∼
10 meV[21, 26]) band gap and a slight energy shift in bulk bandscan easily transform a nontrivial case [Fig. 1(d)] into atrivial case [Fig. 1(e)]. In short, to unambiguously iden-tify the topology of Bi, one must precisely determine boththe surface and bulk electronic structures. One promisingapproach is using a thin film geometry, where quantum-well state (QWS) subbands are formed inside bulk bandprojections [35, 36]. Although QWSs originate from bulkstates, they possess a two-dimensional character and canbe clearly observed in ARPES measurements.In this Letter, we performed high-resolution ARPES (a) M K õ (b) T õ W XUKL M K õ [111] (c) M (L, X) ~10 meV (d)
Nontrivial Semimetal M õ SS1SS2 E F (f) Trivial Semimetal M õ E F (e) Trivial Semimetal M õ E F (g) Trivial Semimetal M õ E F E F FIG. 1. Schematic representation of (a) the bulk and sur-face Brillouin zone of Bi crystal in the [111] direction and (b)the Fermi surface. (c) Near- E F structure of the bulk pro-jections at ¯M . (d) − (g) Possible band structures along the¯Γ ¯M direction on the Bi(111) surface. The blue and red linesindicate the two spin-splitting surface bands, SS1 and SS2,respectively. measurements on Bi(111) films with thicknesses increas-ing from 14 to 202 BL (bilayer; 1 BL = 3.93 ˚A [21]). High-quality films enabled us to clearly observe the QWS sub-bands with energy separations down to ∼
10 meV. Afterwe confirmed the interaction between the top and bottomsurface states in the 14 BL film, we systematically fol-lowed the evolution of the electronic structures. Detailedanalyses on the phase shift of the QWS wave functionsprecisely determined the surface and bulk band disper-sions. The revealed electronic structures unambiguouslyshow that a pure Bi crystal has a nontrivial topology.The present results not only prove the fundamental prop-erty of Bi, but also highlight the QWS approach as apowerful tool to determine fine electronic structures.A surface of a p -type Ge wafer cut in the [111] di-rection was cleaned in ultrahigh vacuum by several cy-cles of Ar + bombardment and annealing up to 900 K.Bi was deposited at room temperature and annealedat 400 K [37]. The pressure was kept at ∼ × − Paduring the deposition. The film thickness was preciselymeasured with a quartz thickness monitor. The quali-ties of the substrate and the film were confirmed fromlow-energy electron diffraction measurements. ARPESmeasurements were performed at BL-9A of HSRC andBL-21B1 of NSRRC. In BL-9A a high-intensity unpo-larized Xe plasma discharge lamp (8.437 eV) was usedin addition to synchrotron radiation (21 eV). The mea-surement temperature was kept at 10 K, and the totalenergy resolution was 12 meV for 21 eV photons and 7meV for 8.437 eV photons. The first-principles calcu-lations were performed using the VASP computer code[38]. A free-standing slab was used based on previousreports [23, 36, 39]. (See the Supplemental Material [40],which includes Refs. [21, 23, 38, 41–44].)First we organize information regarding the Bi topol-ogy. For the (111) surface of Bi, two spin-splitting sur-face bands SS1 and SS2 bridge the ¯Γ and ¯M points. Al-though experimental and theoretical results agree thatboth bands connect to the valence band (VB) aroundthe ¯Γ point, a discrepancy lies in their connection aroundthe ¯M point [14, 22, 28–30]. Based on Kramers’s theo-rem, a spin-splitting band cannot exist at time-reversal-invariant momenta (TRIM) [15, 16]. Therefore, wecan limit the possible cases to those depicted in Figs.1(d) − hν = 21 eV. The shape is very closeto that of bulk Bi [28, 30, 45]. Figure 2(b) shows thecorresponding band structures along the ¯Γ ¯M directionwith calculated bulk projections. Two surface bands ex-ist inside the bulk band gap and QWS subbands insidethe bulk projection. The observed bands are consistentwith previous reports [36, 46, 47]. Figure 2(c) illustrates (b)(a) (d) õ M W ave nu m b e r [ Å - ] E l ec t r on d e n s i t y [ a r b . un i t] Bi layer [BL]
141 5 10
SS1SS2
DCBA topbottom surface-normal B i nd i ng E n e r g y [ e V ] (e)(f) M õ E F CBM+ SS1SS2 VBM M õ E F CBMSS2 VBMSS1 (c)
Wave number [Å -1 ] B i nd i ng E n e r g y [ e V ] SS1SS2
A B C D
QWS - FIG. 2. (a), (b) The Fermi surface and the band structuresmeasured along the ¯Γ ¯M direction in a 14 BL Bi(111) film at hν = 21 eV. Solid lines in (b) indicate bulk projections cal-culated by a tight-binding method [21]. (c) Band structuresobtained by the first-principles calculations for a 14 BL Bislab. (d) Plane-averaged electron densities within the filmcalculated at the four k points marked in (c). (e), (f) Possibleband assignments in an ultrathin Bi film. Gray areas illus-trate positions of the VB maximum (VBM) and CB minimum(CBM). the band structures obtained by the first-principles cal-culations. Although there is a slight discrepancy in theenergy positions, the overall structures show good quali-tative agreement.It is clear that the SS1 and SS2 bands are non-degenerate at ¯M , which appears to suggest that Bi istopologically nontrivial based on Figs. 1(d) − (g). How-ever, in an ultrathin Bi film whose thickness is as small asa decay length of the surface state, the top and bottomsurface states can interact with each other and modifytheir shape from the bulk limit [36, 39, 48]. Figure 2(d)shows plane-averaged electronic charge densities withinthe film calculated at the four k points marked in Fig.2(c). Although these states are actually localized on sur-faces near the center of the Brillouin zone ( A, B ), theygradually penetrate into the film and form bulklike statesin approaching ¯M point (
C, D ). Because the state C liesfar from bulk projections around ¯M , this bulklike behav-ior arises indeed from such a surface-surface interaction.These merged states possess even numbers of electronsand can exist inside a band gap at TRIM without vio-lating Kramers’s theorem. Therefore, in addition to thenontrivial scenario that SS1 connects to the conductionband (CB) at ¯M [Fig. 2(e)], it is also possible that SS1connects to the VB in the bulk limit but that it is pushedinto a gap in an ultrathin film by the surface-surface in-teraction [Fig. 2(f)] [30, 36]. Although Figs. 2(e) and2(f) depict SS2 hybridizing with the VB at ¯M as sug-gested by previous studies [22, 30], it must also be tested.To identify Bi topology, we have to follow the evolutionof SS1 and SS2. If they never cross each other even inthe bulk limit, there is no choice but the Fig. 2(e) [thatis, Fig. 1(d))] case, which unambiguously proves pure Biis topologically nontrivial.Figure 3(a) shows the wide-range band structures mea-sured along the ¯Γ ¯M direction at hν = 21 eV for 14, 18,and 79 BL films. Whereas quantized bands were clearlyobserved in 14 and 18 BL films, these bands became al-most continuous in a 79 BL film except for a region near E F around ¯M . To observe the area in more detail, weperformed ARPES measurements with higher energy res-olution at hν = 8.437 eV. Figure 3(b) shows ARPESimages taken inside the red box in Fig. 3(a). The thick-nesses of the films are systematically increased from 14 to202 BL. As the thickness increases, a QWS energy sepa-ration decreases from ∼
200 to ∼
10 meV. A series of QWSsubbands near ¯M gradually converges into the projectedVB and the intensity of the SS2 band drops abruptlywhen it crosses the edge. This implies SS2 around ¯Mstrongly hybridizes with bulk states and becomes a partof the QWSs.To test this hypothesis, we analyzed the QWS energypositions in more detail. Figure 4(a) shows the energydistribution curves (EDCs) extracted at ¯M for each thick-
14 BL 18 BL 79 BL õ M Wave number [Å -1 ] maxmin B i nd i ng E n e r g y [ e V ] (a)
155 BL 43 BL14 BL 18 BL79 BL 31 BL 35 BL58 BL 100 BL 202 BL
Wave number [Å -1 ] B i nd i ng E n e r g y [ e V ] (b) Wave number [Å -1 ] B i nd i ng E n e r g y [ e V ] M FIG. 3. (a) Wide-range band structures measured along the¯Γ ¯M direction in 14, 18, and 79 BL Bi(111) films at hν = 21 eV.The colored images were produced using a curvature methodfor better visualization [49]. (b) Near- E F band structuresmeasured at hν = 8.437 eV inside the red box in (a). Thethickness is systematically increased from 14 to 202 BL. ness. Peak positions were determined using Lorentzianfittings. These energy positions can be simply describedusing the phase accumulation model, which assumes elec-tronic waves propagating forward and backward acrossthe film and being reflected at the top and bottom sur-faces [35]. The model provides the expression2 k ⊥ ( E ) N ( E ) t + Φ( E ) = 2 π ( n −
1) (1)The first term represents the phase shifts in propagation,with k ⊥ ( E ) and N ( E ) denoting the surface-normal dis-persion and the number of bilayers, respectively, and t the thickness of one bilayer (3.93 ˚A [21]); Φ( E ) is thetotal phase shift at the top and bottom surfaces and n isa quantization number.To experimentally extract information concerning k ⊥ ,we note that some QWSs have the same binding energybut different N and n . Since the phase shift Φ can beregarded as only a function of E [35], we can derive k ⊥ , exp = πt n − n ′ N − N ′ (2)Figure 4(b) shows the E - k ⊥ , exp dispersion obtained usingthis relation [40]. The error bars are estimated by uncer-tainties in thicknesses and fitted peak positions. Here thesurface-normal direction at ¯M corresponds to LX [Fig.1(a)] and Bi has its Dirac dispersion along this direction.Figure 4(c) shows the tight-binding result [21]. The ex-perimental data are indeed perfectly fitted by the solidline in Fig. 4(b); the fitted result is E = αk ⊥ , exp + β ,where α = 3 . ± .
11 eV · ˚A and β = 0 . ± .
002 eV.Now that we have experimentally obtained k ⊥ ( E ), wecan derive a total phase shift using Eq. (1). For thispurpose, we used n = 1 and n = 2 QWS energy positionsand corresponding thicknesses. The result shown in Fig.4(d) exhibits an almost constant relation in this energyrange. The fitted value by a constant function is Φ exp =( − . ± . π , which is similar to those reported inultrathin Bi films on a Si substrate [46]. Furthermore,we compared the experimental and analytical results byplotting N against E (a structure plot) in Fig. 4(e). Thelatter is obtained using N ( E ) = 2 π ( n − − Φ exp k ⊥ , exp ( E ) t (3)It excellently reproduces the experimental data not onlyfor n = 1 and n = 2 QWSs but also for each of the other n values. The consistency of the entire analysis shows thatSS2 band around ¯M indeed becomes a part of QWSs,and also demonstrates the validity of the obtained phaseshift.As a final step we follow the evolution of the VB andSS1 bands at ¯M to identify Bi topology. Figure 5(a)shows EDCs magnified around a peak near E F . The peakbroadens as thickness increases and finally exhibits mul-tiple peaks. This is attributed to a tail of a QWS locatedabove E F . We noted the clear threshold between 43 and k õ , exp [Å -1 ]Binding Energy [eV] B i nd i ng E n e r g y [ e V ] (b) (e)(d) Th i ck n ess [ B L ] Binding Energy [eV] -2 (cid:139) - (cid:139) n = 4n = 3n = 2n = 1 B i nd i ng E n e r g y [ e V ] (c) LX X - k õ [Å -1 ] (a) I n t e n s i t y [ a r b . un i t] Binding Energy [eV]
202 BL155 BL100 BL79 BL58 BL43 BL35 BL31 BL n = 1n = 2n = 3 I n t e n s i t y [ a r b . un i t] Binding Energy [eV]
202 BL155 BL100 BL79 BL58 BL43 BL35 BL31 BL n = 1n = 2n = 3 I n t e n s i t y [ a r b . un i t] Binding Energy [eV]
202 BL155 BL100 BL79 BL58 BL43 BL35 BL31 BL n = 1n = 2n = 3 I n t e n s i t y [ a r b . un i t] To t a l ph se s h i ft FIG. 4. (a) EDCs extracted at ¯M ( k = 0 . − ). The trian-gles show peak positions fitted by Lorentzian functions [inset].(b) E - k ⊥ dispersion experimentally obtained using Eq. (2).The solid line represents a linear fit. (c) E - k ⊥ dispersion ob-tained from a tight-binding calculation [21]. (d) Total phaseshifts experimentally derived using Eq. (1). (e) A plot ofthe N - E relation in QWSs (a structure plot). Solid lines aredrawn using Eq. (3).
58 BL films and applied a specific fitting method for filmsabove 43 BL [40]. Extracted peak positions were plottedagainst an inverse thickness 1 /N along with the VBM( n = 1 QWS) peaks in Fig. 5(b). Using Eq. (1), an in-verse thickness 1 /N and a surface-normal wave number k ⊥ are simply connected by k ⊥ = − Φ / N t at the VBand CB edges ( n = 1). Since the total phase shift turnsout to be constant within this energy range, the VBMevolution is expressed as E = − α Φ exp t N + β (4)The gray solid line in Fig. 5(b) represents this linear func-tion, which perfectly reproduces the experimental data.The evolution of the SS1 peak also appears to fit alinear function, suggesting a hybridization between theCBM and SS1. To test it, we extended the phase anal-ysis for the VB to CB. A simple two-band model indi-cates that a total phase shift of a QWS wave functionis strongly affected by the parity and changes its valueby 2 π across the band gap [50]. The blue solid line inFig. 5(b) is a linear fit, whose gradient can be repro-duced by Eq. (4) when Φ CBM = Φ
VBM + 1 . π . Herewe used the same α value as for the VB based on com-pletely symmetric dispersions shown in Fig. 4(c). The E F M MMMVBM
CBM + SS1SS2
Binding Energy [eV] I n t e n s i t y [ a r b . un i t]
202 BL155 BL100 BL79 BL58 BL43 BL35 BL31 BL -1 ] B i nd i ng E n e r g y [ e V ] Bulk limit
VBMCBM + SS1 (b)
30 50 100 200
Thickness [BL] ∞ CBM + SS1 magnified around E F (a) (c) VBM
FIG. 5. (a) EDCs at ¯M magnified around E F . (b) Evolutionsof peak positions extracted in (a) and those of VBM ( n = 1QWS in VB) against an inverse thickness 1 /N . The meaningsof the solid lines are discussed in the text. (c) Schematicrepresentation of the evolution in electronic structures of Bifilms approaching the bulk limit. close correspondence with 2 π strongly suggests that thepeak near E F belongs to a QWS at the CBM that di-rectly hybridizes with SS1.The CBM and VBM values inthe bulk limit are 0.012 ± ± ± ∼ /N dependence. A latticestrain exhibits an exponential decay against the filmthickness [51], but the linear dispersion in Fig. 5(b) doesnot appear to fit an exponential decay. Moreover, anexponential function has downward convexity with 1 /N ,which further reduces the possibility that the VBM andCBM cross each other.In conclusion, we were able to unambiguously provethat pure Bi is topologically nontrivial. Although the in-teraction between the top and bottom surface states doesexist as revealed by calculations, the splitting betweenSS1 and SS2 is not a consequence of the interaction butrather the electronic structure unique to Bi. The presentresult provides an important insight for recent attemptsto detect novel quantum phenomena on pure Bi, wherethe three-dimensional massive Dirac fermion and its non-trivial topology can show an interesting connection. Fur-thermore, the topologically protected surface states witha giant spin splitting offer great potential in spintron-ics applications. Recent transport measurements haveshown Bi keeps its unique surface transport at ambientpressure [32, 33]. A possible application of Bi surfacestates to valleytronics was also recently reported [13].Finally we also emphasize the capability of the QWSapproach we used. Further advancing the establishedmethod [35, 46], we demonstrated that systematic anal-yses on QWSs can precisely assign and map surface andbulk bands even at ∼
10 meV scale and can reveal hy-bridizations between them. Novel topological materi-als recently predicted can have as small energy scalesas observed here in Bi [52, 53]. Precise determinationof surface and bulk electronic structures is indispensablein driving forward topological studies, where the presentmethod can be one of the most powerful tools.We acknowledge Y. Ohtsubo, K. Yaji, K. Kuroda Y.Ishida, S. Yamamoto, and P. Zhang for valuable discus-sions. We also thank C.-H. Lin, D.-X. Lin and F.-Z. Xiaofor their experimental help. The ARPES measurementswere performed with the approval of the Proposal As-sessing Committee of HSRC (Proposal No. 15-A-38) andthe Proposal Assessing Committee of NSRRC (ProjectNo. 2015-2-090-1). T.-C.C. acknowledges support bythe U.S. National Science Foundation under Grant No.DMR-1305583. S.I. was supported by JSPS through Pro-gram for Leading Graduate Schools (ALPS). [1] A. Brugmans,
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