When electrons are subject to a large external magnetic field, the
conventional charge quantum Hall effect \cite{Klitzing,Tsui} 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 extreme quantum limit,
which opens up the possibility for studying unusual quantum Hall-like phenomena
in zero external magnetic field. Bulk Bi$_{1-x}$Sb$_x$ single crystals are
expected to be prime candidates for one such unusual Hall phase of matter known
as the topological insulator. The hallmark of a topological insulator is the
existence of metallic surface states that are higher dimensional analogues of
the edge states that characterize a spin Hall insulator. In addition to its
interesting boundary states, the bulk of Bi$_{1-x}$Sb$_x$ is predicted to
exhibit three-dimensional Dirac particles, another topic of heightened current
interest. Here, using incident-photon-energy-modulated (IPEM-ARPES), we report
the first 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 topological Dirac insulator's gapless
surface modes. These findings taken together suggest that the observed surface
state on the boundary of the bulk insulator is a realization of the much sought
exotic "topological metal". They also suggest that this material has potential
application in developing next-generation quantum computing devices.Comment: 16 pages, 3 Figures. Submitted to NATURE on 25th November(2007

A Bostwick

B Lenoir

BA Bernevig

CL Kane

CR Ast

D. Hsieh

D. Qian

DC Tsui

DN Sheng

FA Buot

FDM Haldane

G Jezequel

H Fukuyama

H Hochst

JE Moore

K Behnia

K von Klitzing

KS Novoselov

L Fu

L. Wray

LC Hebel

M Hengsberger

M Konig

M. Z. Hasan

P Hofmann

PA Wolff

R. J. Cava

S Hufner

S Murakami

SY Zhou

T Hirahara

Y Kopelevich

Y Liu

Y Zhang

Y. S. Hor

Y. Xia

English

arXiv

When electrons are subject to a large external magnetic field, the conventional charge quantum Hall effect 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, which opens up the possibility for studying unusual quantum Hall-like phenomena in zero external magnetic fields. Bulk Bi_(1-x)Sb_x single crystals are predicted to be prime candidates for one such unusual Hall phase of matter known as the topological insulator. The hallmark of a topological insulator is the existence of metallic surface states that are higher-dimensional analogues of the edge states that characterize a quantum spin Hall insulator. In addition to its interesting boundary states, the bulk of Bi_(1-x)Sb_x is predicted to exhibit three-dimensional Dirac particles, another topic of heightened current interest following the new findings in two-dimensional graphene and charge quantum Hall fractionalization observed in pure bismuth. However, despite numerous transport and magnetic measurements on the Bi_(1-x)Sb_x family since the 1960s, no direct evidence of either topological Hall states or bulk Dirac particles has been found. Here, using incident-photon-energy-modulated angle-resolved photoemission spectroscopy (IPEM-ARPES), 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'. They also suggest that this material has potential application in developing next-generation quantum computing devices that may incorporate 'light-like' bulk carriers and spin-textured surface currents

Hsieh, D.

Qian, D.

Wray, L.

Xia, Y.

Hor, Y. S.

Cava, R. J.

Hasan, M. Z.

Caltech Authors - Main

 
 
A Dirac point insulator with topologically non-trivial surface 
states 
D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, and M.Z. Hasan 
 
Topics: 
1. Confirming the bulk nature of electronic bands by comparison with theoretical calculations 
2. Spin-orbit coupling is responsible for the unique (Dirac) dispersion of the bulk bands near EF  
3. Matching the surface state Fermi crossings to the topology of the surface Fermi surface 
 
1. Confirming the bulk nature of electronic bands by comparison with 
theoretical calculations  
 
In an ARPES experiment (Fig.S1a), three dimensional (3D) dispersive bulk electronic 
states can be identified as those that disperse with incident photon energy, whereas 
surface states do not. As an additional check that we have indeed correctly identified the 
bulk bands of Bi0.9Sb0.1 in Figs 1 and 2, we also measured the dispersion of the deeper 
lying bands well below the Fermi level (EF) and compared them to tight binding 
theoretical calculations of the bulk bands of pure bismuth following the model of Liu and 
Allen (1995)22. A tight-binding approach is known to be valid since Bi0.9Sb0.1 is not a 
strongly correlated electron system. As Bi0.9Sb0.1 is a random alloy (Sb does not form a 
superlattice17) with a relatively small Sb concentration (~0.2 Sb atoms per rhombohedral 
unit cell), the deeper lying band structure of Bi0.9Sb0.1 is expected to follow that of pure 
Bi because the deeper lying (localized wave function) bands of Bi0.9Sb0.1 are not greatly 
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affected by the substitutional disorder, and no additional back folded bands are expected 
to arise. Since these deeper lying bands are predicted to change dramatically with kz, they 
help us to finely determine the experimentally probed kz values. Fig.S2f shows the 
ARPES second derivative image (SDI) of a cut parallel to KMK  that passes through the 
L point of the 3D Brillouin zone (BZ), and Fig.S2h shows a parallel cut that passes 
through the 0.3 LX point (Fig.S2c). The locations of these two cuts in the 3D bulk BZ 
were calculated from the kinematic relations described in the Methods section, from 
which we can construct the constant energy contours shown in Fig. S1c. By adjusting θ 
such that the in-plane momentum kx is fixed at approximately 0.8 Å-1 (the surface M  
point), at a photon energy hν=29 eV, electrons at the Fermi energy (EB=0 eV) have a kz 
that corresponds to the L point in the 3rd bulk BZ. By adjusting θ such that the in-plane 
momentum kx is fixed at approximately -0.8 Å-1, at a photon energy hν=20 eV, electrons 
at a binding energy of -2 eV have a kz near 0.3 LX .  
 
 
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Figure S1: Method of locating high symmetry bulk reciprocal lattice points of Bi0.9Sb0.1 
using incident photon energy modulated ARPES. a, Geometry of an ARPES experiment. b, 
Key parameters relevant to the calculation of the positions of the high symmetry points in the 3D 
BZ. The lattice constants refer to the rhombohedral A7 lattice structure. c, Location of L and X 
points of the bulk BZ in the kx - kz plane together with the constant energy contours that can be 
accessed by changing the angle θ. d, Near EF intensity map (hν = 55 eV) of the Fermi surface 
formed by the surface states covering an entire surface BZ, used to help locate various in-plane 
momenta, in units of the photoelectron emission angle along two orthogonal spatial directions. 
The electron pockets near M  in Fig.2c (main text) appear as lines in Fig.S1d due to relaxed k-
resolution in order to cover a large k-space in a single shot. 
 
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Figure S2: Identification of the bulk band features of Bi0.9Sb0.1. Experimental band-
structure determined by ARPES is compared to bulk tight-binding calculations of bismuth 
to further identify the deeper lying bulk bands and their symmetry origins. a, Energy 
distribution curves (EDCs) along a k-space cut given by the upper yellow line shown in 
schematic c which goes through the bulk L point in the 3rd BZ (hν=29eV). The corresponding 
ARPES intensity in the vicinity of L is shown in e.  b, EDCs along the lower yellow line of c 
which goes through the point a fraction 0.3 of the k-distance from X to L (hν=20eV), showing a 
dramatic change of the deeper lying band dispersions.  (This cut was taken at a kx value equal in 
magnitude but opposite in sign to that in a as described in the SI text). f,h, The ARPES second 
derivative images (SDI) of the raw data shown in a and b to reveal the band dispersions. The flat 
band of intensity at EF is an artifact of taking SDI. g,i, Tight binding band calculations of bismuth 
including spin-orbit coupling, using Liu and Allen model22, along the corresponding experimental 
cut directions shown in f and h. The bands (colored solid lines) labelled 3 to 6 are derived from 
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the symmetries associated with the 6p-orbitals and their dispersion is thus strongly influenced by 
spin-orbit coupling. The inter-band gap between bands 5 and 6 is barely visible on the scale of 
Fig. S2g. The circled curves mark the surface state dispersion which is present at all measured 
photon energies (no kz dispersion). There is a close match of the bulk band dispersion between 
the data and calculations, confirming the presence of strong spin-orbit coupling. d, Tight binding 
valence band (5) dispersion of bismuth in the ky-kz momentum plane showing linearity along both 
directions. The close match between data and calculation along ky suggests that the dispersion 
near EF along kz is also linear.   
 
There is a clear kz dependence of the dispersion of measured bands A, B and C, pointing 
to their bulk nature. The bulk origin of bands A, B and C is confirmed by their good 
agreement with tight binding calculations (bands 3, 4 and 5 in Figs S2g and i), which 
include a strong spin-orbit coupling constant of 1.5 eV derived from bismuth22. Band 3 
drops below -5eV at the 0.3 LX point. The slight differences between the experimentally 
measured band energies and the calculated band energies at ky = 0 Å-1 shown in Fig.S2f-i 
are due to the fact that the ARPES data were collected in a single shot, taken in constant 
θ mode. This means that electrons detected at different binding energies will have 
slightly different values of kz as described in Methods, whereas the presented tight 
binding calculations show all bands at a single kz. We checked that the magnitude of 
these band energy differences is indeed accounted for by this explanation. Even though 
the La and Ls bands in Bi0.9Sb0.1 are inverted relative to those of pure semimetallic Bi, 
calculations show that near EF, apart from an insulating gap, they are “mirror” bands in 
terms of k dispersion (see bands 5 and 6 in Fig.S2g). Such a close match to calculations, 
which also predict a linear dispersion along the kz cut near EF (Fig.S2d), provides strong 
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support that the dispersion of band C, near EF, is in fact linear along kz. Focusing on the 
Λ-shaped valence band at L, the EDCs (Fig.S2a) show a single peak out to ky ≈ ±0.15 Å-1 
demonstrating that it is composed of a single band feature. Outside this range however, 
an additional feature develops on the low binding energy side of the main peak in the 
EDCs, which shows up as two well separated bands in the SDI image (Fig.2f) and signals 
a splitting of the band into bulk representative and surface representative components 
(Fig.S2a,f). Unlike the main peak that disperses strongly with incident photon energy, 
this shoulder-like feature is present and retains the same Λ-shaped dispersion near this k-
region (open circles in Figs S2g and i) for all photon energies used, supporting its 2D 
surface character. This behaviour is quite unlike bulk band C, which attains the Λ-shaped 
dispersion only near 29 eV (see main text Fig. 2b).  
 
 
 
 
 
 
 
 
 
 
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2. Spin-orbit coupling is responsible for the unique Dirac-like dispersion 
behaviour of the bulk bands near EF : 
 
Figure S3: Spin-orbit coupling has a profound effect on the band structure of bismuth near 
L point. a, Schematic of the bulk 3D BZ and the projected BZ of the (111) surface. b,c, 
Calculated tight binding band structure of bismuth including a spin-orbit coupling strength of 1.5 
eV22 along two orthogonal cuts through the L point in the 1st bulk BZ. d,e, Tight binding band 
structure along the same two directions as b and c calculated without spin-orbit coupling. The 
inter-band gap of 13.7 meV is barely visible on the scale of b and c. 
 
According to theoretical models, a strongly spin-orbit coupled bulk band structure is 
necessary for topological surface states to exist7-11. Therefore it is important to show that 
our experimentally measured bulk band structure of Bi0.9Sb0.1 can only be accounted for 
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by calculations that explicitly include a large spin-orbit coupling term. As shown in the 
previous section, the measured bulk band dispersion of Bi0.9Sb0.1 generally follows the 
calculated bulk bands of pure Bi from a tight binding model. The dispersion of the bulk 
valence and conduction bands of pure bismuth near EF at the L point from such a tight 
binding calculation22 with a spin-orbit coupling constant of 1.5eV are shown in Fig. S3b 
and c, which show a high degree of linearity. The high degree of linearity can be 
understood from a combination of the large Fermi velocity (vF ≈ 6 eV Å along ky) and 
small inter-band (below EF) gap Δ = 13.7 meV (Fig. S3). This calculated inter-band gap 
of Bi (13.7 meV) is smaller than our measured lower limit of 50 meV (main text Fig. 1a) 
for the insulating gap of Bi0.9Sb0.1. To illustrate the importance of spin-orbit coupling in 
determining the band structure near L, we show the dispersion along ky and kz calculated 
without spin-orbit coupling (Fig. S3d and e). While the dispersion along kz is not 
drastically altered by neglecting the spin-orbit coupling, the dispersion along ky changes 
from being linear to highly parabolic. This is further evidence that our measured Dirac 
point can be accounted for only by including spin-orbit coupling. A strong spin-orbit 
coupling constant acts as an internal quantizing magnetic field for the electron system6 
which can give rise to a quantum spin Hall effect without any externally applied magnetic 
field3,4,5,12,37. Therefore, the existence or the spontaneous emergence of the surface or 
boundary states does not require an external magnetic field. 
 
 
 
 
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3. Matching the surface state Fermi crossings and the topology of the 
surface Fermi surface in bulk insulating Bi0.9Sb0.1 
 
 
Figure S4: The Kramers’ point, the gapless nature and topology of surface states in 
insulating Bi0.9Sb0.1 is revealed through high spatial and k-resolution ARPES. a, Energy 
distribution curves (EDCs) of a low resolution ARPES scan along the surface Γ - M cut of 
Bi0.9Sb0.1. b, The surface band dispersion second derivative image along Γ - M obtained by 
piecing together four high resolution ARPES scans. See main text Fig.3 for explanation of other 
features. c, EDCs of high resolution ARPES scans in the vicinity of surface Fermi crossings 1 and 
2 and crossings 3, 4 and 5 (left panels). These crossings form the surface Fermi surface shown in 
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the upper right panel of c (see also main text Fig.2). High resolution ARPES scans along cut 
directions A, B and C are further evidence for a surface Fermi surface.  
 
 
In order to count the number of singly degenerate surface state Fermi crossings24,28,38 
along the Γ - M cut of the surface BZ, high photon energy ARPES scans, which allow 
mapping of the entire k range from Γ - M to fall within the detector window at the 
expense of lower instrument resolution, were taken to preliminarily identify the k-space 
locations of the Fermi crossings (Fig. S4a). Having determined where these surface state 
Fermi crossings lie in k-space, we performed several high resolution ARPES scans, each 
covering a successive small k interval in the detector window, in order to construct a high 
resolution band mapping of the surface states from Γ to M . The second derivative 
image of the surface band dispersion shown in Fig.S4b was constructed by piecing 
together four such high resolution scans. Fig.S4c shows energy distribution curves of 
high resolution ARPES scans in the vicinity of each surface Fermi crossing, which 
together give rise to the surface Fermi surface shown. No previous work24,26-30,35,36 has 
reported the band dispersion near the L-point (thus missing the Dirac bands) or resolved 
the Kramers point near the M point, which is crucial to determine the topology of the 
surface states. For this reason there is no basis for one-to-one comparison with previous 
work, since no previous ARPES data exists in the analogous k-range. Note that surface 
band dispersions along the cuts A, B and C are highly linear. This is indirect evidence for 
the existence of the bulk Dirac point since surface states are formed when the bulk state 
wave functions are subjected to the boundary conditions at the cleaved plane.  
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37. Sheng, L., Sheng, D. N., Ting, C. S. & Haldane, F. D. M. Nondissipative spin Hall 
effect via quantized edge transport. Phys. Rev. Lett. 95, 136602 (2005). 
38. Kim, T. K. et al. Evidence against a charge density wave on Bi(111). Phys. Rev. B 72, 
085440 (2005). 
 
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A topological Dirac insulator in a quantum spin Hall phase

When electrons are subject to a large external magnetic field, the conventional charge quantum Hall effect 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, which opens up the possibility for studying unusual quantum Hall-like phenomena in zero external magnetic fields. Bulk Bi1-xSbx single crystals are predicted to be prime candidates for one such unusual Hall phase of matter known as the topological insulator. The hallmark of a topological insulator is the existence of metallic surface states that are higher-dimensional analogues of the edge states that characterize a quantum spin Hall insulator. In addition to its interesting boundary states, the bulk of Bi1-xSbx is predicted to exhibit three-dimensional Dirac particles, another topic of heightened current interest following the new findings in two-dimensional graphene and charge quantum Hall fractionalization observed in pure bismuth. However, despite numerous transport and magnetic measurements on the Bi1-xSbx family since the 1960s, no direct evidence of either topological Hall states or bulk Dirac particles has been found. Here, using incident-photon-energy-modulated angle-resolved photoemission spectroscopy (IPEM-ARPES), we report the direct observation of massive Dirac particles in the bulk of Bi0.9Sb0.1, locate the Kramers points at the sample\u27s boundary and provide a comprehensive mapping of the Dirac insulator\u27s 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 \u27topological metal\u27. They also suggest that this material has potential application in developing next-generation quantum computing devices that may incorporate \u27light-like\u27 bulk carriers and spin-textured surface currents

Hsieh, David

Qian, Dong

Wray, Lewis Andrew

Xia, Yuqi

Hor, Yew San

Cava, Robert Joseph

Hasan, Md Zahid

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

A Topological Dirac Insulator in a Quantum Spin Hall Phase

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https://authors.library.caltech.edu/49766/13/nature06843-s1.pdf

A topological Dirac insulator in a quantum spin Hall phase :
  Experimental observation of first strong topological insulator

A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator

Abstract

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Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

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