Abstract
Recent experiments and theories have suggested that strong spinâorbit coupling effects in certain band insulators can give rise to a new phase of quantum matter, the so-called topological insulator, which can show macroscopic quantum-entanglement effects1,2,3,4,5,6,7. Such systems feature two-dimensional surface states whose electrodynamic properties are described not by the conventional Maxwell equations but rather by an attached axion field, originally proposed to describe interacting quarks8,9,10,11,12,13,14,15. It has been proposed that a topological insulator2 with a single Dirac cone interfaced with a superconductor can form the most elementary unit for performing fault-tolerant quantum computation14. Here we present an angle-resolved photoemission spectroscopy study that reveals the first observation of such a topological state of matter featuring a single surface Dirac cone realized in the naturally occurring Bi2Se3 class of materials. Our results, supported by our theoretical calculations, demonstrate that undoped Bi2Se3 can serve as the parent matrix compound for the long-sought topological device where in-plane carrier transport would have a purely quantum topological origin. Our study further suggests that the undoped compound reached via n-to-p doping should show topological transport phenomena even at room temperature.
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It has been experimentally shown that spinâorbit coupling can lead to new phases of quantum matter with highly non-trivial collective quantum effects4,5,6. Two such phases are the quantum spin Hall insulator4 and the strong topological insulator 5,6,7, both realized in the vicinity of a Dirac point but yet quite distinct from graphene16. The strong-topological-insulator phase contains surface states (SSs) with novel electromagnetic properties7,8,9,10,11,12,13,14,15. It is currently believed that the Bi1âxSbx insulating alloys realize the only known topological-insulator phase in the vicinity of a three-dimensional Dirac point5, which can in principle be used to study topological electromagnetic and interface superconducting properties8,9,10,14. However, a particular challenge for the topological-insulator Bi1âxSbx system is that the bulk gap is small and the material contains alloying disorder, which makes it difficult to gate for the manipulation and control of charge carriers to realize a device. The topological insulator Bi1âxSbx features five surface bands, of which only one carries the topological quantum number6. Therefore, there is an extensive world-wide search for topological phases in stoichiometric materials with no alloying disorder, with a larger gap and with fewer yet still odd-numbered SSs that may work as a matrix material to observe a variety of topological quantum phenomena.
The topological-insulator character of BiSb5,6 led us to investigate the alternative Bi-based compounds Bi2X3 (X=Se, Te). The undoped Bi2Se3 is a semiconductor that belongs to the class of thermoelectric materials Bi2X3 with a rhombohedral crystal structure (space group ; refs 17, 18). The unit cell contains five atoms, with quintuple layers ordered in the Se(1)âBiâSe(2)âBiâSe(1) sequence. Electrical measurements report that, although the bulk of the material is a moderately large-gap semiconductor, its charge transport properties can vary significantly depending on the sample preparation conditions19, with a strong tendency to be n-type20,21 owing to atomic vacancies or excess selenium. An intrinsic bandgap of approximately 0.35âeV is typically measured in experiments22,23, whereas theoretical calculations estimate the gap to be in the range of 0.24â0.3âeV (refs 20, 24).
It has been shown that spinâorbit coupling can lead to topological effects in materials that determine their spin Hall transport behaviours4,5,6,7. Topological quantum properties are directly probed from the nature of the electronic states on the surface by studying the way surface bands connect the materialâs bulk valence and conduction bands in momentum space5,6,7. The surface electron behaviour is intimately tied to the number of bulk band inversions that exist in the band structure of a material7. The origin of topological Z2 order in Bi1âxSbx is bulk-band inversions at three equivalent L-points5,7, whereas in Bi2Se3 only one band is expected to be inverted, making it similar to the case in the two-dimensional quantum spin Hall insulator phase. Therefore, a much simpler surface spectrum is naturally expected in Bi2Se3. All previous experimental studies of Bi2Se3 have focused on the materialâs bulk properties; nothing is known about its SSs. It is this key experimental information that we provide here that, for the first time, enables us to determine its topological quantum class.
The bulk crystal symmetry of Bi2Se3 fixes a hexagonal Brillouin zone (BZ) for its (111) surface (Fig. 1d) on which and are the time-reversal invariant momenta (TRIMs) or the surface Kramers points. We carried out high-momentum-resolution angle-resolved photoemission spectroscopy (ARPES) measurements on the (111) plane of naturally grown Bi2Se3 (see the Methods section). The electronic spectral weight distributions observed near the point are presented in Fig. 1aâc. Within a narrow binding-energy window, a clear V-shaped band pair is observed to approach the Fermi level (EF). Its dispersion or intensity had no measurable time dependence within the duration of the experiment. The âVâ bands cross EF at 0.09âà â1 along and at 0.10âà â1 along , and have nearly equal band velocities, approximately 5Ã105âmâsâ1, along the two directions. A continuum-like manifold of statesâa filled U-shaped featureâis observed inside the V-shaped band pair. All of these experimentally observed features can be identified, to first order, by a direct one-to-one comparison with the LDA band calculations. Figure 1f shows the theoretically calculated (see the Methods section) (111)-surface electronic structure of bulk Bi2Se3 along the â-space cut. The calculated band structure with and without SOC are overlaid together for comparison. The bulk band projection continuum on the (111) surface is represented by the shaded areas, blue with SOC and green without SOC. In the bulk, time-reversal symmetry demands E(k,â)=E(âk,â) whereas space inversion symmetry demands E(k,â)=E(âk,â). Therefore, all the bulk bands are doubly degenerate. However, because space inversion symmetry is broken at the terminated surface in the experiment, SSs are generally spin-split on the surface by spinâorbit interactions except at particular high-symmetry pointsâthe Kramers points on the surface BZ. In our calculations, the SSs (red dotted lines) are doubly degenerate only at (Fig. 1f). This is generally true for all known spinâorbit-coupled material surfaces such as gold25,26 or Bi1âxSbx (ref. 5). In Bi2Se3, the SSs emerge from the bulk continuum, cross each other at , pass through the Fermi level (EF) and eventually merge with the bulk conduction-band continuum, ensuring that at least one continuous band-thread traverses the bulk bandgap between a pair of Kramers points. Our calculated result shows that no surface band crosses the Fermi level if SOC is not included in the calculation, and only with the inclusion of the realistic values of SOC (based on atomic Bi) does the calculated spectrum show singly degenerate gapless surface bands that are guaranteed to cross the Fermi level. The calculated band topology with realistic SOC leads to a single ring-like surface FS, which is singly degenerate so long as the chemical potential is inside the bulk bandgap. This topology is consistent with the Z2=â1 class in the FuâKaneâMele classification scheme7.
A global agreement between the experimental band structure (Fig. 1aâc) and our theoretical calculation (Fig. 1f) is obtained by considering a rigid shift of the chemical potential by about 200âmeV with respect to our calculated band structure (Fig. 1f) of the formula compound Bi2Se3. The experimental sign of this rigid shift (the raised chemical potential) corresponds to an electron doping of the Bi2Se3 insulating formula matrix (see Supplementary Information). This is consistent with the fact that naturally grown Bi2Se3 semiconductor used in our experiment is n-type, as independently confirmed by our transport measurements. The natural doping of this material, in fact, comes as an advantage in determining the topological class of the corresponding undoped insulator matrix, because we would like to image the SSs not only below the Fermi level but also above it, to examine the way surface bands connect to the bulk conduction band across the gap. A unique determination of the surface band topology of purely insulating Bi1âxSbx (refs 5, 6) was clarified only on doping with a foreign element, Te. In our experimental data on Bi2Se3, we observe a V-shaped pure SS band to be dispersing towards EF, which is in good agreement with our calculations. More remarkably, the experimental band velocities are also close to our calculated values. By comparison with calculations combined with a general set of arguments presented above, this V-shaped band is singly degenerate. Inside this âVâ band, an electron-pocket-like U-shaped continuum is observed to be present near the Fermi level. This filled U-shaped broad feature is in close correspondence to the bottom part of the calculated conduction-band continuum (Fig. 1f). Considering the n-type character of the naturally occurring Bi2Se3 and by correspondence to our band calculation, we assign the broad feature to correspond roughly to the bottom of the conduction band.
To systematically investigate the nature of all the band features imaged in our data, we have carried out a detailed photon-energy-dependence study, of which selected data sets are presented in Fig. 2a,b. A modulation of incident photon energy enables us to probe the kz dependence of the bands sampled in an ARPES study (Fig. 2c), allowing for a way to distinguish surface from bulk contributions to a particular photoemission signal5. Our photon-energy study did not indicate a strong kz dispersion of the lowest-lying energy bands on the âUâ, although the full continuum does have some dispersion (Fig. 2). Some variation of the quasiparticle intensity near EF is, however, observed owing to the variation of the electronâphoton matrix element. In light of the kz-dependence study (Fig. 2b), if the features above â0.15âeV were purely due to the bulk, we would expect to observe dispersion as kz moved away from the Î-point. The lack of strong dispersion yet close one-to-one correspondence to the calculated bulk band structure suggests that the inner electron pocket continuum features are probably a mixture of surface-projected conduction-band states, which also includes some band-bending effects near the surface and the full continuum of bulk conduction-band states sampled from a few layers beneath the surface. Similar behaviour is also observed in the ARPES study of other semiconductors27. In our kz-dependent study of the bands (Fig. 2b) we also observe two bands dispersing in kz that have energies below â0.3âeV (blue dotted bands), reflecting the bulk valence bands, in addition to two other non-dispersive features associated with the two sides of the pure SS Dirac bands. The red curve is measured right at the Î-point, which suggests that the Dirac point lies inside the bulk bandgap. Taking the bottom of the âUâ band as the bulk conduction-band minimum, we estimate that a bandgap of about 0.3âeV is realized in the bulk of the undoped material. Our ARPES estimated bandgap is in good agreement with the value deduced from bulk physical measurements23 and from other calculations that report the bulk band structure20,24. This suggests that the magnitude of band bending near the surface is not larger than 0.05 eV. We note that in purely insulating Bi2Se3 the Fermi level should lie deep inside the bandgap and only pure surface bands will contribute to surface conduction. Therefore, in determining the topological character of the insulating Bi2Se3 matrix the âUâ feature is not relevant.
We therefore focus on the pure SS part. The complete surface FS map is presented in Fig. 3. Figure 3a presents electron distribution data over the entire two-dimensional (111) surface BZ. All the observed features are centred around . None of the three TRIMs located at are enclosed by any FS, in contrast to what is observed in Bi1âxSbx (ref. 5). The detailed spectral behaviour around is shown in Fig. 3b, which was obtained with high momentum resolution. A ring-like feature formed by the outer âVâ pure SS band (a horizontal cross-section of the upper Dirac cone in Fig. 1) surrounds the conduction-band continuum centred at . This ring is singly degenerate from its one-to-one correspondence to band calculation. An electron encircling the surface FS that encloses a TRIM or a Kramers point obtains a geometrical quantum phase (Berry phase) of Ï mod 2Ï in its wavefunction7. Therefore, if the chemical potential (Fermi level) lies inside the bandgap, as it should in purely insulating Bi2Se3, its surface must carry a global Ï mod 2Ï Berry phase. In most spinâorbit materials, such as gold (Au[111]), it is known that the surface FS consists of two spinâorbit-split rings generated by two singly degenerate parabolic (not Dirac-like) bands that are shifted in momentum space from each other, with both enclosing the -point25,26. The resulting FS topology leads to a 2Ï or 0 Berry phase because the phases from the two rings add or cancel. This makes gold-like SSs topologically trivial despite their spinâorbit origin.
Our theoretical calculation supported by our experimental data suggests that in insulating Bi2Se3 there exists a singly degenerate surface FS which encloses only one Kramers point on the surface Brillouin zone. This provides evidence that insulating Bi2Se3 belongs to the Z2=â1 topological class in the FuâKaneâMele topological classification scheme for band insulators. On the basis of our ARPES data we suggest that it should be possible to obtain the fully undoped compound by chemically hole-doping the naturally occurring Bi2Se3, thereby shifting the chemical potential to lie inside the bulk bandgap. The surface transport of Bi2Se3 prepared as such would therefore be dominated by topological effects as it possesses only one Dirac fermion that carries the non-trivial Z2 index. The existence of a large bulk bandgap (0.3âeV) within which the observed Z2 Dirac fermion state lies suggests the realistic possibility for the observation of topological effects even at room temperature in this material class. Because of the simplest possible topological surface spectrum realized in Bi2Se3, it can be considered as the âhydrogen atomâ of strong topological insulators. Its simplest topological surface spectrum would make it possible to observe and study many exotic quantum phenomena predicted in topological field theories, such as the Majorana fermions14, magnetic monopole image 9,10 or topological exciton condensates15, by transport probes.
Methods
Theoretical calculations.
The theoretical band calculations were performed with the LAPW method in slab geometry using the WIEN2K package28. The generalized gradient approximation of Perdew, Burke and Ernzerhof29 was used to describe the exchangeâcorrelation potential. SOC was included as a second variational step using scalar-relativistic eigenfunctions as basis after the initial calculation was converged to self-consistency. The surface was simulated by placing a slab of 12 quintuple layers in vacuum. A grid of 21Ã21Ã1 points was used in the calculations, equivalent to 48 k points in the irreducible BZ and 300 k points in the first BZ. To calculate the kz of the ARPES measurements (), an inner potential V0 of approximately 11.7âeV was used, given by a fit on the ARPES data at normal emission.
Experimental methods.
Single crystals of Bi2Se3 were grown by melting stoichiometric mixtures of high-purity elemental Bi and Se in a 4-mm-inner-diameter quartz tube. The sample was cooled over a period of two days, from 850 to 650ââC, and then annealed at this temperature for a week. Single crystals were obtained and could be easily cleaved from the boule. High-resolution ARPES measurements were then performed using 17â45âeV photons on beamline 12.0.1 of the Advanced Light Source at the Lawrence Berkeley National Laboratory and beamline 5â4 at the Stanford Synchrotron Radiation Laboratory. The energy and momentum resolutions were 15âmeV and 1.5% of the surface BZ respectively using a Scienta analyser. The samples were cleaved in situ between 10 and 55âK under pressures of less than 5Ã10â11âtorr, resulting in shiny flat surfaces. The surface band quasiparticle signal is stable throughout the entire measurement duration.
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Acknowledgements
We thank N. P. Ong, B.A. Bernevig, D. Haldane and D.A. Huse for discussions. The synchrotron X-ray experiments are supported by the DOE-BES (contract DE-FG02-05ER46200) and materials synthesis is supported by the NSF-MRSEC (NSF-DMR-0819860) at Princeton Center for Complex Materials at Princeton University. Theoretical work is supported by the US Department of Energy, Office of Science, Basic Energy Sciences contract DEFG02-07ER46352, and benefited from the allocation of supercomputer time at NERSC and Northeastern Universityâs Advanced Scientific Computation Center (ASCC). D.Q. was partly supported by the NNSF-China (grant No. 10874116).
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Y.X., D.Q. and D.H., carried out the experiment with the assistance of L.W. and A.P. D.G., Y.S.H. and R.J.C. provided the samples. H.L., Y.X. and A.B. carried out the theoretical calculations and the data analysis. M.Z.H. conceived the idea for the Bi2X3 topological class before any theoretical proposal and was responsible for overall project direction, planning and management.
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Xia, Y., Qian, D., Hsieh, D. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nature Phys 5, 398â402 (2009). https://doi.org/10.1038/nphys1274
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DOI: https://doi.org/10.1038/nphys1274
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