Abstract
When electrons are subject to a large external magnetic field, the conventional charge quantum Hall effect^{1,2} dictates that an electronic excitation gap is generated in the sample bulk, but metallic conduction is permitted at the boundary. Recent theoretical models suggest that certain bulk insulators with large spin–orbit interactions may also naturally support conducting topological boundary states in the quantum limit^{3,4,5}, which opens up the possibility for studying unusual quantum Halllike phenomena in zero external magnetic fields^{6}. Bulk Bi_{1x}Sb _{x} single crystals are predicted to be prime candidates^{7,8} for one such unusual Hall phase of matter known as the topological insulator^{9,10,11}. The hallmark of a topological insulator is the existence of metallic surface states that are higherdimensional analogues of the edge states that characterize a quantum spin Hall insulator^{3,4,5,6,7,8,9,10,11,12,13}. In addition to its interesting boundary states, the bulk of Bi_{1x}Sb _{x} is predicted to exhibit threedimensional Dirac particles^{14,15,16,17}, another topic of heightened current interest following the new findings in twodimensional graphene^{18,19,20} and charge quantum Hall fractionalization observed in pure bismuth^{21}. However, despite numerous transport and magnetic measurements on the Bi_{1x}Sb _{x} family since the 1960s^{17}, no direct evidence of either topological Hall states or bulk Dirac particles has been found. Here, using incidentphotonenergymodulated angleresolved photoemission spectroscopy (IPEMARPES), we report the direct observation of massive Dirac particles in the bulk of Bi_{0.9}Sb_{0.1}, locate the Kramers points at the sample’s boundary and provide a comprehensive mapping of the Dirac insulator’s gapless surface electron bands. These findings taken together suggest that the observed surface state on the boundary of the bulk insulator is a realization of the ‘topological metal’^{9,10,11}. They also suggest that this material has potential application in developing nextgeneration quantum computing devices that may incorporate ‘lightlike’ bulk carriers and spintextured surface currents.
Main
Bismuth is a semimetal with strong spin–orbit interactions. Its band structure is believed to feature an indirect negative gap between the valence band maximum at the Tpoint of the bulk Brillouin zone and the conduction band minima at three equivalent Lpoints^{17,22} (here we generally refer to these as a single point, L). The valence and conduction bands at L are derived from antisymmetric (L_{a}) and symmetric (L_{s}) ptype orbitals, respectively, and the effective lowenergy hamiltonian at this point is described by the (3+1)dimensional relativistic Dirac equation^{14,15,16}. The resulting dispersion relation, , is highly linear owing to the combination of an unusually large band velocity v and a small gap Δ (such that Δ/v ≈ 5 × 10^{3} Å^{1}) and has been used to explain various peculiar properties of bismuth^{14,15,16}. Substituting bismuth with antimony is believed to change the critical energies of the band structure as follows (see Fig. 1). At an Sb concentration of x ≈ 4%, the gap Δ between L_{a} and L_{s} closes and a massless, threedimensional (3D) Dirac point is realized. As x is further increased this gap reopens with inverted symmetry ordering, which leads to a change in sign of Δ at each of the three equivalent Lpoints in the Brillouin zone. For concentrations greater than x ≈ 7% there is no overlap between the valence band at T and the conduction band at L, and the material becomes an invertedband insulator. Once the band at T drops below the valence band at L, at x ≈ 8%, the system evolves into a directgap insulator whose lowenergy physics is dominated by the spin–orbitcoupled Dirac particles at L^{7,17}.
Recently, semiconductors with inverted bandgaps have been proposed to manifest the twodimensional (2D) quantum spin Hall phase, which is predicted to be characterized by the presence of metallic onedimensional edge states^{3,4,5,12}. Although a bandinversion mechanism and edge states have been invoked to interpret the transport results in 2D mercury telluride semiconductor quantum wells^{13}, no onedimensional edge states are directly imaged, so their topological character is yet to be uniquely determined. Recent theoretical treatments^{7,8} have focused on the strongly spin–orbitcoupled, bandinverted Bi_{1x}Sb _{x} series as a possible 3D bulk realization of the quantum spin Hall phase in which the onedimensional edge states are expected to take the form of 2D surface states^{7,8,9} that may be directly imaged and spectroscopically studied, making it feasible to identify their topological order parameter character.
Highmomentumresolution angleresolved photoemission spectroscopy performed with varying incident photon energy (IPEMARPES) allows for the measurement of electronic band dispersion along various momentum space (kspace) trajectories in the 3D bulk Brillouin zone. ARPES spectra taken along two orthogonal cuts through the Lpoint of the bulk Brillouin zone of Bi_{0.9}Sb_{0.1} are shown in Fig. 1a, c. A shaped dispersion whose tip lies less than 50 meV below the Fermi energy E_{F} can be seen along both directions. Additional features originating from surface states that do not disperse with incident photon energy are also seen. Owing to the finite intensity between the bulk and surface states, the exact binding energy E_{B} where the tip of the shaped dispersion lies is unresolved. The linearity of the bulk shaped bands is observed by locating the peak positions at higher E_{B} in the momentum distribution curves and the energy at which these peaks merge is obtained by extrapolating linear fits to the momentum distribution curves. Therefore, 50 meV represents a lower bound on the energy gap Δ between L_{a} and L_{s}. The magnitudes of the extracted band velocities along the k_{x} and k_{y} directions are 7.9 ± 0.5 × 10^{4} m s^{1} and 10.0 ± 0.5 × 10^{5} m s^{1}, respectively, which are similar to the tight binding values of 7.6 × 10^{4} m s^{1} and 9.1 × 10^{5} m s^{1} calculated for the L_{a} band of bismuth^{22}. Our data are consistent with the extremely small effective mass of 0.002m_{e} (where m_{e} is the electron mass) observed in magnetoreflection measurements on samples with x = 11% (ref. 23). The Dirac point in graphene, coincidentally, has a band velocity (v_{F} ≈ 10^{6} m s^{1})^{18} comparable to what we observe for Bi_{0.9}Sb_{0.1}, but its spin–orbit coupling is several orders of magnitude weaker, and the only known method of inducing a gap in the Dirac spectrum of graphene is by coupling to an external chemical substrate^{20}. The Bi_{1x}Sb _{x} series thus provides a rare opportunity to study relativistic Dirac hamiltonian physics in a 3D condensed matter system where the intrinsic (rest) mass gap can easily be tuned.
Studying the band dispersion perpendicular to the sample surface provides a way to differentiate bulk states from surface states in a 3D material. To visualize the nearE_{F} dispersion along the 3D L–X cut (X is a point that is displaced from L by a k_{z} distance of 3π/c, where c is the lattice constant), in Fig. 2a we plot energy distribution curves (EDCs) taken such that electrons at E_{F} have fixed inplane momentum (k_{x} , k_{y} ) = (L_{x} , L_{y} ) = (0.8 Å^{1}_{,} 0.0 Å^{1}), as a function of photon energy hν. There are three prominent features in the EDCs: a nondispersing, k_{z} independent peak centred just below E_{F}, at about 0.02 eV; a broad, nondispersing hump centred near 0.3 eV; and a strongly dispersing hump that coincides with the latter near hν = 29 eV. To understand which bands these features originate from, we show ARPES intensity maps along an inplane cut (parallel to the k_{y} direction) taken using hν values of 22 eV, 29 eV and 35 eV, which correspond to approximate k_{z} values of L_{z}  0.3 Å^{1}, L_{z} , and L_{z} + 0.3 Å^{1}, respectively (Fig. 2b). At hν = 29 eV, the lowenergy ARPES spectral weight reveals a clear shaped band close to E_{F}. As the photon energy is either increased or decreased from 29 eV, this intensity shifts to higher binding energies as the spectral weight evolves from the shaped band into a ‘U’shaped band. Therefore, the dispersive peak in Fig. 2a comes from the bulk valence band, and for hν = 29 eV the highsymmetry point L = (0.8, 0, 2.9) appears in the third bulk Brillouin zone. In the maps of Fig. 2b with respective hν values of 22 eV and 35 eV, overall weak features near E_{F} that vary in intensity remain even as the bulk valence band moves far below E_{F}. The survival of these weak features over a large photon energy range (17–55 eV) supports their surface origin. The nondispersing feature centred near 0.3 eV in Fig. 2a comes from the higherbindingenergy (valence band) part of the full spectrum of surface states, and the weak nondispersing peak at 0.02 eV reflects the lowenergy part of the surface states that cross E_{F} away from the point and forms the surface Fermi surface (Fig. 2c).
Having established the existence of an energy gap in the bulk state of Bi_{0.9}Sb_{0.1} (Figs 1 and 2) and observed linearly dispersive bulk bands uniquely consistent with strong spin–orbit coupling model calculations^{14,15,16,22} (see Supplementary Information for full comparison with the theoretical calculation), we now discuss the topological character of its surface states, which we found to be gapless (Fig. 2c). In general, the states at the surface of spin–orbitcoupled compounds are allowed to be spinsplit owing to the loss of space inversion symmetry (E(k,↑) = E(k,↑)). However, as required by Kramers’ theorem, this splitting must go to zero at the four timereversalinvariant momenta in the 2D surface Brillouin zone. As discussed in refs 7 and 9, along a path connecting two timereversalinvariant momenta in the same Brillouin zone, the Fermi energy inside the bulk gap will intersect these singly degenerate surface states either an even or an odd number of times. When there are an even number of surface state crossings, the surface states are topologically trivial because weak disorder (as may arise through alloying) or correlations can remove pairs of such crossings by pushing the surface bands entirely above or below E_{F}. When there are an odd number of crossings, however, at least one surface state must remain gapless, which makes it nontrivial^{7,8,9}. The existence of such topologically nontrivial surface states can be theoretically predicted on the basis of the bulk band structure only, using the Z_{2}invariant that is related to the quantum Hall Chern number^{24}. Materials with band structures with Z_{2} = +1 are ordinary Bloch band insulators that are topologically equivalent to the filledshell atomic insulator, and are predicted to exhibit an even number (including zero) of surface state crossings. Materials with bulk band structures with Z_{2} = 1, on the other hand, which are expected to exist in rare systems with strong spin–orbit coupling acting as an internal quantizing magnetic field on the electron system^{6} and inverted bands at an odd number of highsymmetry points in their bulk 3D Brillouin zones, are predicted to exhibit an odd number of surface state crossings, precluding their adiabatic continuation to the atomic insulator^{3,7,8,9,10,11,12,13}. Such ‘topological metals’^{9,10,11} cannot be realized in a purely 2D electron gas system.
In our experimental case, namely the (111) surface of Bi_{0.9}Sb_{0.1}, the four timereversalinvariant momenta are located at and three points that are rotated by 60° relative to one another. Owing to the threefold crystal symmetry (A7 bulk structure) and the observed mirror symmetry of the surface Fermi surface across k_{x} = 0 (Fig. 2), these three points are equivalent (and we henceforth refer to them as a single point, ). The mirror symmetry (E(k_{y} ) = E(k_{y} )) is also expected, from timereversal invariance exhibited by the system. The complete details of the surface state dispersion observed in our experiments along a path connecting and are shown in Fig. 3a; finding this information is made possible by our experimental separation of surface states from bulk states. As for bismuth, two surface bands emerge from the bulk band continuum near to form a central electron pocket and an adjacent hole lobe^{25,26,27}. It has been established that these two bands result from the spinsplitting of a surface state and are thus singly degenerate^{27,28}.
On the other hand, the surface band that crosses E_{F} at k_{x} ≈ 0.5 Å^{1}, and forms the narrow electron pocket around , is clearly doubly degenerate, as far as we can determine within our experimental resolution. This is indicated by its splitting below E_{F} between k_{x} ≈ 0.55 Å^{1} and , as well as the fact that this splitting goes to zero at in accordance with Kramers’ theorem. In semimetallic singlecrystal bismuth, only a single surface band is observed to form the electron pocket around (refs 29 and 30). Moreover, this surface state overlaps, and hence becomes degenerate with, the bulk conduction band at L (L projects to the surface point ) owing to the semimetallic character of bismuth (Fig. 3b). In Bi_{0.9}Sb_{0.1}, on the other hand, the states near fall completely inside the bulk energy gap, preserving their purely surface character at (Fig. 3a). The surface Kramers doublet point can thus be defined in the bulk insulator (unlike in bismuth^{25,26,27,28,29,30}) and is experimentally located in Bi_{0.9}Sb_{0.1} samples to lie approximately 15 ± 5 meV below E_{F} at (Fig. 3a). For the precise location of this Kramers point, it is important to demonstrate that our alignment is strictly along the – line. To do so, we contrast highresolution ARPES measurements taken along the – line with those that are slightly offset from it (Fig. 3e). Figures 3f–i show that with k_{y} offset from the Kramers point at by less than 0.02 Å^{1}, the degeneracy is lifted and only one band crosses E_{F} to form part of the bowshaped electron distribution (Fig. 3d). Our finding of five surface state crossings (an odd rather than an even number) between and (Fig. 3a), confirmed by our observation of the Kramers degenerate point at the timereversalinvariant momentum, indicates that these gapless surface states are topologically nontrivial. This corroborates our bulk electronic structure result that Bi_{0.9}Sb_{0.1} is in the insulating bandinverted (Z_{2} = 1) regime (Fig. 3c), which contains an odd number of bulk (gapped) Dirac points, and is topologically analogous to an integerquantumspin Hall insulator.
Our experimental results taken collectively strongly suggest that Bi_{0.9}Sb_{0.1} is quite distinct from graphene^{18,19} and represents a novel state of quantum matter: a strongly spin–orbitcoupled insulator with an odd number of Dirac points and a negative Z_{2} topological Hall phase. Our work points to future possibilities for further spectroscopic investigations of topological orders in quantum matter.
Methods Summary
Highresolution IPEMARPES data have been taken at beamlines 12.0.1 and 10.0.1 of the Advanced Light Source at the Lawrence Berkeley National Laboratory, as well as at the PGM beamline of the Synchrotron Radiation Center in Wisconsin, with photon energies ranging from 17 eV to 55 eV and energy resolutions ranging from 9 meV to 40 meV, and momentum resolution better than 1.5% of the surface Brillouin zone. Data were taken on highquality bulk singlecrystal Bi_{1x}Sb _{x} at a temperature of 15 K and chamber pressures less than 8 × 10^{11} torr. Throughout this paper, the bulk bands presented are from those measured in the third bulk Brillouin zone, to ensure a high degree of signaltonoise contrast, and the k_{z} values are estimated using the standard freeelectron finalstate approximation.
Online Methods
Growth method for highquality single crystals
The Bi_{1x}Sb _{x} singlecrystal samples (0 ≤ x ≤ 0.17) used for ARPES experiments were each cleaved from a boule grown from a stoichiometric mixture of highpurity elements. The boule was cooled from 650 °C to 270 °C over a period of five days and was annealed for seven days at 270 °C. The samples naturally cleaved along the (111) plane, which resulted in shiny flat silver surfaces. Xray diffraction measurements were used to check that the samples were single phase, and confirmed that the Bi_{0.9}Sb_{0.1} single crystals presented in this paper have a rhombohedral A7 crystal structure (point group Rm), with roomtemperature (300 K) lattice parameters a = 4.51 Å and c = 11.78 Å indexed using a rhombohedral unit cell. The Xray diffraction patterns of the cleaved crystals exhibit only the (333), (666), and (999) peaks, showing that the cleaved surface is oriented along the trigonal (111) axis. Roomtemperature data were recorded on a Bruker D8 diffractometer using Cu Kα radiation (λ = 1.54 Å) and a diffractedbeam monochromator. The inplane crystal orientation was determined by Laue Xray diffraction. During the angleresolved photoemission spectroscopy (ARPES) measurements a fine alignment was achieved by carefully studying the band dispersions and Fermi surface symmetry as an internal check for crystal orientation.
Transport measurements
Temperaturedependent resistivity measurements were carried out on singlecrystal samples in a Quantum Design PPMS9 instrument, using a standard fourprobe technique on approximately 4 × 1 × 1mm^{3}, rectangular samples with the current in the basal plane, which was perpendicular to the trigonal axis. The four contacts were made by using roomtemperature silver paste. The data for samples with concentrations ranging from x = 0 to x = 0.17 showed a systematic change from semimetallic to insulatinglike behaviour with increasing x, in agreement with previously published works^{15}, which was used as a further check of the antimony concentrations. Conventional magnetic and transport measurements^{7,17,31} such as these cannot separately measure the contributions of the surface and bulk states to the total signal. ARPES, on the other hand, is a momentumselective technique^{32}, which allows for a separation of 2D (surface) from 3D (bulk) dispersive energy bands. This capability is especially important for Bi_{1x}Sb _{x} because the Dirac point lies at a single point in the 3D Brillouin zone, unlike for 2D graphene, where the Dirac points can be studied at any arbitrary perpendicular momentum along a line^{33,34}.
Systematic methods for separating bulk from surface electronic states
ARPES is a photonin, electronout technique^{32}. Photoelectrons ejected from a sample by a monochromatic beam of radiation are collected by a detector capable of measuring its kinetic energy E_{kin}. By varying the detector angles, θ and ϕ, relative to the sample surface normal, the momentum of the photoelectrons, K, can also be determined (as illustrated in Supplementary Fig. 1a). By employing the commonly used freeelectron final state approximation, we can fully convert from the measured kinetic energy and momentum values of the photoelectron to the binding energy, E_{B}, and Bloch momentum values k of its initial state inside the crystal, via
where we have set ϕ = 0, W is the work function, m_{e} is the electron mass and V_{0} is an experimentally determined parameter, which is approximately 10 eV for bismuth^{35,36}. Features in the ARPES spectra originating from bulk initial states (dispersive along the k_{z} direction) were distinguished from those originating from surface initial states (nondispersive along the k_{z} direction) by studying their dependence on incident photon energy, hν, and converting this to dependence on k_{z} via the displayed equations. ARPES data were collected at beamlines 12.0.1 and 10.0.1 of the Advanced Light Source at the Lawrence Berkeley National Laboratory, as well as at the PGM beamline of the Synchrotron Radiation Center in Wisconsin, with incident photon energies ranging from 17 eV to 55 eV, energy resolutions ranging from 9 meV to 40 meV and momentum resolution better than 1.5% of the surface Brillouin zone, using Scienta electron analysers. The combination of high spatial resolution and high crystalline quality enabled us to probe only the highly ordered and cleanest regions of our samples. Singlecrystal Bi_{1x}Sb _{x} samples were cleaved in situ at a temperature of 15 K and chamber pressures less than 8 × 10^{11} torr, and high surface quality was checked throughout the measurement process by monitoring the EDC linewidths of the surface state. To measure the nearE_{F} dispersion of an electronic band along a direction normal to the sample surface, such as the direction from X (2π/√3a, 0, 8π/c) to L (2π/√3a, 0, 11π/c) shown in Fig. 2a, EDCs were taken at several incident photon energies. The kinetic energy of the photoelectron at E_{F} is different for each value of hν, so the angle θ was first adjusted and then held fixed for each hν so as to keep k_{x} constant at 2π/√3a = 0.8 Å^{1} for electrons emitted near E_{F}. To ensure that the inplane momentum remained constant at (the L–X line projects onto ) for each EDC, a complete nearE_{F} intensity map was generated for each photon energy to precisely locate the point (see Supplementary Fig. 1d). We note that because the bulk crystal has only threefold rotational symmetry about the k_{z} axis, the reciprocal lattice does not have mirror symmetry about the k_{x} = 0 plane. Therefore, scans taken at +θ and –θ for the same photon energy probe different points in the bulk 3D Brillouin zone; this is responsible for the absence of the bulk shaped band in Fig. 3f.
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Acknowledgements
We thank P. W. Anderson, B. A. Bernevig, L. Balents, E. Demler, A. Fedorov, F. D. M. Haldane, D. A. Huse, C. L. Kane, R. B. Laughlin, J. E. Moore, N. P. Ong, A. N. Pasupathy, D. C. Tsui and S.C. Zhang for discussions. The synchrotron experiments are supported by the DOEBES and materials synthesis is supported by the NSFMRSEC at Princeton Center for Complex Materials.
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Supplementary information
This file contains Supplementary Methods, Supplementary Notes, Supplementary Figures S1S4 with Legends and additional references.
The Supplementary Information (SI) describes our method of comparing experimentally measured deeper lying bands of Bi0.9Sb0.1 with theoretical calculations of bulk Bi as further evidence of their bulk origin, and as an alternate way of extracting the kz values from ARPES measurements. The SI also provides further theoretical justification that spinorbit coupling is essential to account for the 3D bulk Dirac point, and provides further experimental evidence for an odd number of surface state Fermi crossings. (PDF 503 kb)
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Hsieh, D., Qian, D., Wray, L. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008) doi:10.1038/nature06843
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