aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec APS/123-QED
Electrons on the surface of Bi Se form a topologically-orderedtwo dimensional gas with a non-trivial Berry’s phase Y. Xia, L. Wray, D. Qian, D. Hsieh, A. Pal, H. Lin, A.Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan
1, 4 Joseph Henry Laboratories of Physics, Department of Physics,Princeton University, Princeton, NJ 08544, USA Department of Physics, Northeastern University, Boston, MA Department of Chemistry, Princeton University, Princeton, NJ 08544, USA Princeton Center for Complex Materials,Princeton University, Princeton, NJ 08544, USA ∗ (Dated: October 9, 2018) ∗ Electronic address: [email protected] ecent experiments and theories suggest that spin-orbit coupling (SOC) canlead to new phases of quantum matter with highly non-trivial collective quantumeffects. Two such phases are the quantum spin Hall insulator [1, 2] and thestrong topological insulator (STI) [3, 4, 5] both realized in the vicinity of aDirac point [3] but yet quite distinct from graphene [6]. The STI phase containsunusual two-dimensional edge states, which may provide a route to spin-chargeseparation in higher dimensions [7, 8] and may realize other novel magneticand electronic properties [9]. It has been suggested that the interface of an STIand a conventional superconductor can support exotic quasiparticle states whichcould be used in fault-tolerant computing schemes [10]. It is currently believedthat the Bi − x Sb x alloys realize the only known topological insulator phase inthe vicinity of a 3D Dirac point[3]. However, a particular challenge for thetopological insulator Bi − x Sb x system is that the insulating bulk gap is small andthe material contains a significant amount of alloying disorder which is difficultto gate for the manipulation and control of charge carriers to realize a device.The topological insulator Bi − x Sb x features five surface states of which only onecarries the topological quantum number. Therefore, there is an extensive world-wide search for similar topological phases in stoichiometric materials with noalloying disorder, a larger gap and fewer yet odd numbered surface states. Herewe present angle-resolved photoemission (ARPES) data and electronic structurecalculation which seems to suggest the existence of a Z topological phase with asurface Berry’s phase in the stoichiometric compound Bi Se . Our results takencollectively suggest that the undoped Bi Se can serve as the parent matrixcompound for the long-sought topological materials with only one surface statewhere in-plane carrier transport would be fully dominated by the Z topologicaleffects. Moreover, the undoped compound should exhibit topological quantumeffects at room temperature. Undoped Bi Se is a semiconductor which belongs to the class of materials Bi X (X=S,Se,Te) with a rhombohedral crystal structure (space group D d ( R ¯3 m )). The unitcell contains five atoms, with the quintuple layers ordered in the Se(1)-Bi-Se(2)-Bi-Se(1)fashion. These compounds have been studied extensively in connection to thermoelectricapplications [12]. Experimental measurements suggest a band gap of approximately 0.352V [13, 14, 15] where as the bulk band calculations estimate the gap to be in between 0.24to 0.32 eV [16, 17]. Various measurements and calculations have reported that althoughthe bulk of the material is a direct gap semiconductor [18] its electronic properties can varysignificantly with different sample preparation conditions [19], with a strong tendency to ben-type [18] due to impurity and vacancies.Single crystal of Bi Se was grown by melting stoichiometric mixtures of high purityelemental Bi and Se in a 4 mm inner diameter quartz tube. The sample was cooled over aperiod of two days, from 850 to 650 ◦ C, and then annealed at that temperature for a week.Single crystals were obtained and could be easily cleaved from the boule. High-resolutionARPES measurements were then performed using 17 to 45eV photons on beamline 5-4 at theStanford Synchrotron Radiation Laboratory (SSRL) and beamline 12.0.1 of the AdvancedLight Source at the Lawrence Berkeley National Laboratory. The energy and momentumresolutions were 15meV and 2% of the surface Brilloin Zone (BZ) respectively using a Scientaanalyzer. The samples were cleaved in situ between 10K and 55K under pressures of lessthan 5 × − torr, resulting in shiny flat surfaces. For the deposition-effect measurements,the Fe atoms were deposited using an e-beam heated evaporator at a rate of approximately0.12˚ A /min. We also carried our surface band calculations for the Bi Se (111) surface forcomparison with the experimental data. The calculations were performed with the LAPWmethod in slab geometry using the WIEN2K package [21]. GGA of Perdew, Burke, andErnzerhof [22] was used to describe the exchange-correlation potential. SOC was includedas a second variational step using scalar-relativistic eigenfunctions as basis after the initialcalculation is converged to self-consistency. The surface was simulated by placing a slab oftwelve quintuple layers in vacuum. A grid of 21 × × Se (111) along the¯ K − ¯Γ − ¯ M direction. The result with and without SOC are overlaid together for comparison.The bulk band projections are represented by the shaded areas in blue with SOC and greenwithout. In the bulk, time-reversal symmetry demands E ( ~k, ↑ ) = E ( − ~k, ↓ ) while spaceinversion symmetry requires that E ( ~k, ↑ ) = E ( − ~k, ↑ ). Therefore, all the energy bandsshould be doubly-degenerate. However, space inversion symmetry is broken at the surface,so the degeneracy of the SS can be broken by spin-orbit interactions. Nevertheless, Kramer’sTheorem dictates that spin degeneracy should be preserved at some high symmetry points3 x k y k z Γ M K L F Z Γ (111) M M M Γ k x k y (a) (b)(c) E F no socsocno socsoc BulkSurface
FIG. 1: Spin-orbit interaction induced surface Fermi surface: (a) A schematic of the bulk 3D BZof Bi Se and the 2D BZ of the projected (111) surface. (b) LDA band-structure of the 2D surfacestates along the ¯ K − ¯Γ − ¯ M k-space cut. Bulk band projections are represented by the shadedareas. The band-structure results with spin-orbit coupling (SOC) is presented in blue and thatwithout SOC is in green. No pure surface band is observed within the insulating gap without SOC(black lines). One pure gapless surface band crossing E F is observed when SOC is included (redlines). (c) The corresponding surface FS is a single ring centered at ¯Γ. The band responsible forthis ring is singly degenerate. The time-reversal-invariant momenta (TRIM) on the (111) surfaceBZ : the ¯Γ and the three ¯ M points are marked by red dots. of the surface BZ that are time reversal invariant momenta (TRIM). In the Bi Se (111)surface BZ, these are given by ¯Γ and three ¯ M points, located 60 ◦ away from each other andin the middle of two ¯Γ points (Fig. 1(c)). The calculated result without SOC shows a cleargap between the valence and conduction bulk bands, with no pure surface bands crossingE F . SOC drastically changes the band structure of the SS. One finds two singly degeneratesurface bands emerging from the bulk projection which are paired together at the ¯Γ point(one of the Kramers’ points). The top surface band forms an electron pocket at E F , giving aring-like surface Fermi surface (FS). We emphasize that while doubly degenerate states mayappear as two overlapping bands in the calculated band structure for some k-values, the twosurface bands each appear only once, and represent one eigenstate at a given momentumand most importantly, the central Fermi surface is singly degenerate.The calculated valence band dispersion is in good correspondence while making a com-parison with the experimental data taken in both the ¯Γ − ¯ M and ¯Γ − ¯ K directions (Fig. 2).The first and twelfth quintuple blocks are taken as the surfaces of the slab. The fraction of4 .0 0.4 0.8 k y (Å ) -1 (a) E ( e V ) B ΓM M Γ M (b) k x (Å ) -1 ΓK K Γ
ARPES ARPES
FIG. 2: Experimental band dispersions are compared with theoretical calculations: The bandsecond-derivative images and first-principles calculation results along (a) ¯Γ − ¯ M and (b) ¯Γ − ¯ K directions are presented. The color of the calculated bands represents the fraction of electroniccharge residing in the surface layers. A rigid shift of E F is included to match the lowest energyexcitations in the ARPES data with calculations, a consequence of the system being somewhatelectron-doped. The strongest signals are observed from the surface states (see text). -0.15 0.00 0.15 -0.3 0.0 0.3-0.4-0.20.0 -0.8 0.0 0.82.42.83.2 (b) E ( e V ) B -0.15 0.00 0.15-0.4-0.3-0.2-0.10.00.1 k y (Å ) -1 k x (Å ) -1 E ( e V ) B Γ MM Γ KK(a)
SS RS SS RS
Γ Z Γ -0.3 0.0 0.3 -0.3 0.0 0.3
25 eV
21 eV k y (Å ) LFZΓΓM M k y (Å ) -1 k z ( Å ) - F
25 eV 31 eV (c) (d) Γ FIG. 3: k z dependence of low-lying excitations near ¯Γ: High resolution surface band dispersiondata near the ¯Γ point along the (a) ¯Γ- ¯ M and (b) ¯Γ- ¯ K directions measured at 22eV photon energyare presented. The ”V” shaped pure surface state (SS) band is nearly isotropic along both axes,forming a cone in the k x − k y space. (c) Results from the energy dependence study is presentedalong ¯Γ- ¯ M going from Γ at 21 eV to 0.41 Γ − Z at 31 eV in 1eV intervals. Solid arcs indicate pointsin the bulk 3D BZ seen by the detector over a θ range of ± ◦ . (d) The energy dispersion data atthree selected energies show no k z dependence for the outer SS band as well as the inner resonantstate (RS) feature. At each energy the corresponding energy distribution curve is presented (top).The calculated bandwidth of the lowest 3D bulk conduction band (inset) is about 0.3 eV which issignificantly larger than the instrumental resolution. k z value, after projected ontothe 2D surface BZ. A good agreement is seen by shifting the E F of the calculated result to20 meV above the bottom of the lowest conduction band - a consequence of doping the semi-conductor matrix with electrons. While this observation suggests that the bulk is slightlyelectron doped (n-type) as observed in many transport measurements, it is an advantagefor the fact that we can have spectroscopic (ARPES) look at the large part of the completeband-structure of the surface states which was not possible in fully insulating BiSb [3]. Acomplete spectroscopic view is of significant advantage for we would like to study the con-nectivity of the surface and bulk states at all energies between the valence and conductionbands if possible to determine the unique and specific class of topological order of the parentmaterials [4].In the measured spectra, the strongest quasiparticle signals are typically observed nearnormal electron emission, at around 1.5-2.0 eV. This band corresponds to a pure surfaceband lying outside the bulk projection of states. Strong quasiparticle signals near ¯Γ are ofsurface origin, based on their direct correspondence with the band calculation. Our energydependent study of the bands suggest insignificant k z dependence of these state, confirmingthe 2D (surface) nature of the bands. Near E F , one finds an additional 150 meV electronpocket-like feature inside the spin-split surface band. This feature is a hybrid resonant stateformed by the superposition of the surface and the bulk.To investigate the resonant state (RS) feature centered at ¯Γ, high momentum resolutiondata is presented in Fig. 3. The extracted band velocity of the outer pure SS in both¯Γ- ¯ K and ¯Γ- ¯ M directions are approximately 5 × m/s, close to the calculated values. TheSS crosses E F at 0.09 ˚ A − in the ¯Γ- ¯ M direction and 0.11 ˚ A − in the ¯Γ- ¯ K direction. Thedependence of the bands at three selected energies and their respective energy distributioncurves (EDCs) are presented in panel (d). By changing the energy of the incident photon,one can move to different k z values in the 3D bulk BZ (Fig. 3(c)). The inner potential V used is approximately 10 eV, given by the muffin-tin zero of the calculation. The valueis obtained by averaging the atomic potential over the interstitial region, using a muffin-tinradius of 2.5 bohr. Moving the photon energy at normal emission from 21eV, correspondingto Γ in the bulk 3D BZ, to 31eV, corresponding to 0.41 Γ − Z (or -0.59 Γ − Z after mapping6 -0.08 0.00 0.08-0.12-0.060.000.060.12 (b)(a) k y ( Å ) - k x (Å ) -1 Γ RSSS MΓ (c) k x (Å ) -1 MM ΓK
Gold k y ( Å ) - Bi Se
Bi Se
FIG. 4: The surface Fermi surface : (a) the surface FS is a small pocket around Γ. (b) Highmomentum resolution data around ¯Γ show a single ring formed by the pure surface state band.In the middle of the ring is a filled-in disk-shaped spectral intensity reflecting resonant states (seetext). The pure surface Fermi surface of BiSe is different from that of spin-orbit pair Fermi surfaceobserved on metallic gold Au(111). (c) The Au(111) surface FS, which has two rings (each non-degenerate) surrounding the ¯Γ TRIM. An electron circling the gold FS can only carry a Berry’sphase of 0, characteristic of a trivial topological metal, Z ==+1. The single surface Fermi surfaceobserved in Bi Se reflects its non-trivial topological character Z =-1. into the first BZ), one finds no k z dispersion of the bands. A variation of the quasiparticleintensity is however observed with changing photon energy, due to matrix element effects.The calculated Γ − Z dispersion of the lowest conduction band is presented in the inset. Theresulting bandwidth is approximately 300 meV. Therefore, if the features above 0.15eV arepurely due to the bulk, one should observe a clear dispersion as k z moves away from the Γpoint. The two closest bulk bands - the highest bulk valence band and the next conductionband, are at least 500 meV away with bandwidths of approximately 200 meV, so they arenot to affect the quasiparticle signal near E F . This finding confirms that the outer ”V”band as well as the inner electron pocket-like feature are due to the surface.The surface FS is then presented in Figure 4. Panel (a) presents the ARPES dataover the entire 2D (111) surface BZ. No E F crossings are observed other than the featurescentered at ¯Γ. The three TRIMs located at ¯ M are not enclosed by any Fermi arcs. Thisobservation is in contrast to the surface FS of Bi − x Sb x [3], which contains a electron-holepocket features around each of the ¯ M points. The region around ¯Γ is zoomed in with7ner momentum resolution data (panel (b)). One finds a ring-like feature formed by theouter ”V” SS surrounding the filled-in RS disk. This ring is single degenerate thus spin-polarized as expected from the correspondence bewteen the LDA band calculation. Anelectron encircling the surface FS that encloses a TRIM collects a Berry’s phase of π mod2 π , picking up a value of π from each of the Fermi arcs that enclose the TRIM. At thefirst sight the surface FS in Bi Se may look like the well-known surface FS in bulk metallicstrong spin-orbit coupled gold (Au). However, the SS in gold(111) is split into two parabolicbands each singly degenerate, which are shifted in momentum-space from each other andboth enclose the Γ-point [24, 25]. The resulting surface FS topology contains two concentricrings around ¯Γ, instead of one as observed in our data for Bi Se . Therefore, the Fermi arcsin gold encloses the TRIM an even number of times, with two E F crossings between a pair¯Γ and ¯ M points. Because each of the rings is singly degenerate (necessarily spin polarized),an electron encircling the FS can only carry a trivial Berry’s phase of 0 mod 2 π , by pickingup one π from each of the rings. This makes gold surface states topologically trivial.To determine the topological nature of Bi Se , one needs to determine if the RS con-tributes an odd or even number of crossings between two TRIMs. In order to determinethe nature of the RS states and its connection to the conduction band structure we needto doped the surface of Bi Se with atoms that donates electrons on the surface. This canbe done by depositing alkali atoms on the cleaved surface under UHV conditions. For thisparticular experiment, we have chosen to deposit iron on the surface. Figure 5(a) presentsARPES data at 29eV along ¯Γ − ¯ M after Fe atoms of less than 1% of a monolayer is depositedonto the sample. The system becomes strongly electron doped, with the bottom of the ”V”band shifted by more than 200meV. In addition to a shift in the chemical potential, theouter ”V” band also becomes less dispersive, with a decreases in the band velocity near E F .More interestingly, the inner RS appears to be the bottom of a pair of spin-split parabolicbands shifted from each other in k-space. These bands are reminiscent of those observedin the SS of Au(111) [24, 25] and Sb(111) [27] at ¯Γ. The result shows that the RS featurecrosses E F twice, bringing the total number of crossings between a ¯Γ- ¯ M pair to three.Since iron is magnetic its use in doping the surface also serves a separate but importantpurpose which has to do with the possiblity of observing the system’s response to the break-ing of time reversal invariance on the surface. With magnetic irons on the surface Kramer’sdegeneracy is broken at the TRIMs and a gap is expected to open up. A small gap or8 ulk Cond.Bulk ValenceΓM MSSRS SS RS RS (a) Sb (c) Γ M
Bi Se E F Γ M -0.2 -0.1 0.0 0.1 0.2-0.8-0.6-0.4-0.20.0 E ( e V ) B k y (Å ) -1 (b) (d)(e) SSRS
FIG. 5: The iron deposited surface: band structure in a broken time-reversal setting and thedetermination of contribution of the RS feature to the Berry’s phase counting: (a) a pair of spin-split bands are extended from the RS feature after the chemical potential is shifted with doping. Asmall gap opens at ¯Γ, due to the loss of time reversal symmetry. The corresponding EDC is shownin (b). Similar to (c) n-Bi Se , (d) the band structure of the strong topological metal Sb(111)contains RS at some TRIMs. The pair of spin-split RS bands in Bi Se resembles the SS in Sb. Inboth cases (e) a ”partner-switching” behavior is observed, a critical characteristics in generatingan odd number of surface state Fermi crossings. strong spectral weight suppression is indeed observed at ¯Γ (Fig. 5(a)). The outer ”V” bandbecomes more parabolic and is detached from the lower ” ∧ ” band. The signal intensity at ¯Γbecomes very weak, whereas in the undoped case, the signal is strongest at the Dirac point.For this reason, an exact measurement of the gap size is difficult. Spectral suppression ora weak gap opening at the Kramers’ point suggest that magnetic field and disorder cancut the k-space surface band thread that connects the bulk valence and conduction bandsdriving the topological insulator into a trivial band insulator as expected in Z theory ofthese materials [26]Therefore, the FS of the Bi Se (111) SS only encloses the ¯Γ point an odd number oftimes, giving it a ν = 1 and Z =-1 band topology similar to Bi − x Sb x . However, thesurface transport of pure undoped Bi Se would then be dominated by topological effectssince there is only one surface band that is also topological. Our calculation suggests thatthe topological character should be preserved in the undoped compound, which is insulatingwith metallic boundary states. One might be able to obtain the pure undoped compound9y hole-doping the n-type Bi Se thereby shifting the chemical potential downwards. Sucha chemical alteration would also remove the resonant state discussed previously, leaving asingle ring in the surface FS which carries a π Berry’s phase. Treated along these lines ourdata in Fig.-3 and 5 suggest that the undoped material should have a band gap of about 0.3eV which is sufficient to keep it insulating at room temperature.In conclusion, we have calculated the band structure of Bi Se (111) surface and found thatspin-orbit coupling induces a single non-degenerate band crossing at E F . Our experimentaldata agrees with our surface band calculations as far as the band topology is concerned whichdemonstrates that the stoichiometric compound belongs to the Z =-1 topological class. Wehave also shown that since that Fermi surface contains only one surface state, the transportproperties of undoped Bi Se would be dominated by topological effects and since the bandgap is large (Fig.-3 and 5) the material can be considered as a room temperature topologicalinsulator without any alloying disorder. [1] C.L. Kane and E.J. Mele, Phys. Rev. Lett. , 226801 (2005); B.A. Bernevig and S-C. Zhang,Phys. Rev. Lett. , 106802 (2006).[2] M. Konig et al. , Science , 766 (2007).[3] D. Hsieh et al. , Nature , 970 (2008).[4] L. Fu and C.L. Kane, Phys. Rev. B , 045302 (2007).[5] S.-C. Zhang, Physics , 6 (2008).[6] K.S. Novoselov et al. , Nature , 197-200 (2005); Y. Zhang et al. , Nature , 201-204(2005).[7] X.-L. Qi et al. , Nature Phys. , 273 (2008).[8] Y. Ran et al. , arxiv.org/abs/0801.0627 (2008).[9] J.E. Moore et al. , arxiv.org/abs/0804.4527 (2008).[10] L. Fu and C.L. Kane, Phys. Rev. Lett. , 096407 (2008).[11] S. Murakami, Phys. Rev. Lett. , 236805 (2006).[12] F.J. DiSalvo, Science , 703 (1999).[13] R.W.G. Wyckoff, Crystal Structures (Krieger, Malabar, 1986).[14] B. Schroeder et al. , Phys. Stat. Sol. B , 561 (1973).
15] J. Black et al. , J. Phys. Chem. Solids , 240 (1957).[16] P. Larson et al. , Phys. Rev. B , 085108 (2002).[17] S.K. Mishra et al. , J. Phys.: Condens. Matter , 461 (1997).[18] V.A. Greanya et al. , J. App. Phys. , 6658 (2002).[19] G.R. Hyde et al. , J. Phys. Chem. Solids , 1719 (1974).[20] H. Kohler and A. Fabricius, Phys. Status Solidi B , 487 (1975).[21] P. Blaha et al. , computer code WIEN2K (Vienna University of Technology, Vienna, 2001).[22] J.P. Perdew et al. , Phys. Rev. Lett. , 1996 (1996).[23] C. Heske et al. , Phys. Rev. B , 4680 (1999).[24] S. LaShell et al. , Phys. Rev. Lett. , 3419 (1996).[25] M. Hoesch et al. , Phys. Rev. B , 241401 (2004).[26] A.P. Schnyder et al. , Phys. Rev. B , 195125 (2008).[27] K. Sugawara et al. , Phys. Rev. Lett. , 046411 (2006).[28] D. Hsieh, M.Z. Hasan et al. , submitted., submitted.