Observation of a large-gap topological-insulator class with a single Dirac cone on the surface

Abstract

Recent experiments and theories have suggested that strong spin-orbit 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 effects. 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 quarks. It has been proposed that a topological insulator with a single Dirac cone interfaced with a superconductor can form the most elementary unit for performing fault-tolerant quantum computation. 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.

Full text

arXiv:0908.3513v1 [cond-mat.mes-hall] 24 Aug 2009
Published version: Y. Xia et.al., Nature Physics 5, 398 (2009).
”Topological Insulators: Large-bandgap family found” Nat.Physics Cover (June, 2009).
”Topological Insulators: The Next Generation” Nature Physics 5, 378 (2009).
[http://dx.doi.org/10.1038/nphys1294]
Discovery (theoretical prediction and experimental observation)
of a large-gap topological-insulator class with spin-polarized
single-Dirac-cone on the surface
Y. Xia,
1, 2
D. Qian,
1, 3
D. Hsieh,
1, 2
L. Wray,
1
A. Pal,
1
H. Lin,
4
A.
Bansil,
4
D. Grauer,
5
Y. S. Hor,
5
R. J. Cava,
5
and M. Z. Hasan
1, 2
1
Department of Physics, Princeton University, Princeton, NJ 08544, USA
2
Princeton Center for Complex Materials,
Princeton University, Princeton, NJ 08544, USA
3
Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
4
Department of Physics, Northeastern University, Boston, MA 02115, USA
5
Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
(Dated: Submitted for publication in December 2008)
1

Recent theories and experiments have suggested that strong spin–orbit cou-
pling effects in certain band insulators can give rise to a new phase of quan-
tum matter, the so-called topological insulator, which can show macroscopic
entanglement effects [1, 2, 3, 4, 5, 6, 7]. Such systems feature two-dimensional
surface states whose electrodynamic properties are described not by the con-
ventional Maxwell equations but rather by an attached axion field, originally
proposed t o describe strongly interacting particles [8, 9, 10, 11, 12, 13, 14, 15].
It has been proposed that a topological insulator [2] with a single spin-textured
Dirac cone interfaced with a superconductor c an form the most elementary unit
for performing fault-tolerant quantum computation [14]. Here we present an
angle-resolved photoemission spectroscopy study and first-principle theoretical
calculation-predictions that reveal the first observation of such a topological
state of matter featuring a single-surface-Dirac-cone realized in the naturally
occurring Bi
2
Se
3
class of materials. Our results, supported by our theoreti-
cal predictions and calculations, demonstrate that undoped c ompound of this
class of materials can serve as the parent matrix compound for the long-sought
topological device where in-plane surface carrier transport would have a purely
quantum top ological origin. Our study further suggests that t he undoped com-
pound reached via n-to-p doping should show t opological transport phenomena
even at room temperature.
Note added: The counter-doping method to reach the fully undoped compound - the
bulk-insulator state, proposed in this paper, has been achieved in a recent work in Nature:
[http://dx.doi.org/10.1038/nature08234 (2009)]
2

It has been experimentally shown that spin-orbit coupling can lead to new phases of
quant um matter with highly nontrivial collective quantum effects [4, 5, 6]. Two such phases
are the quant um spin Hall insulator [4] and the strong topological insulator [5 , 6, 7] both
realized in the vicinity of a Dirac point but yet quite distinct from graphene [1 6]. The
strong-topological-insulator phase contains surface states (SSs) with novel electromagnetic
properties [7, 8, 9, 10, 11, 12, 13, 14, 15]. It is currently believed that the Bi
1−x
Sb
x
insu-
lating alloys realize the only known topological-insulator phase in the vicinity of a three-
dimensional Dirac point [5], which can in principle be used to study topological electromag-
netic and interface superconducting properties [8, 9, 10, 14]. However, a particular challenge
for the topological-insulator Bi
1−x
Sb
x
system is t hat 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 Bi
1−x
Sb
x
features five surface
bands, of which o nly one carries the topological quantum number [6]. 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 topolog ical quantum phenomena.
The topological-insulator character o f Bi
1−x
Sb
x
[5, 6] led us to investigate the alternative
Bi-based compounds Bi
2
X
3
(X=Se, Te). The undoped Bi
2
Se
3
is a semiconductor that be-
longs to the class of thermoelectric materials Bi
2
X
3
with a r hombohedral crystal structure
(space group D
5
3d
(R
¯
3m); refs 17,18). The unit cell conta ins five atoms, with quintuple lay-
ers ordered in the Se(1)-Bi-Se(2)-Bi-Se(1) sequence. Electrical measurements report that,
although the bulk of the material is a moderately large-gap semiconductor, its charge trans-
port properties can vary significantly depending on the sample preparation conditions [19],
with a strong tendency to be n-type [20, 21] owing to atomic vacancies or excess selenium.
An intrinsic bandgap of approximately 0.35 eV is typically measured in experiments [22, 23],
whereas theoretical calculations estimate the gap to be in the range of 0.24- 0.3 eV (refs 20,
24).
It has been shown that spin-orbit coupling can lead to topological effects in materials
that determine their spin Hall transport behaviors [4, 5, 6, 7]. Topological quantum proper-
ties are directly probed from the nature of the electronic states on the surface by studying
the way surface bands connect the material’s bulk valence and conduction bands in mo-
mentum space [5, 6, 7]. The surface electron behavior is intimately tied to the number
3

-0.15 0.00 0.15
-0.15 0.00 0.15
-0.4
-0.3
-0.2
-0.1
0.0
0.1
k
x
k
y
k
z
Γ
M
K
L
F
Z
Γ
(111)
M
2
M
1
M
3
Γ
k
x
k
y
(a)
(e)
(d)
nosoc
soc
nosoc
soc
Bulk
Surface
k
y
(Å )
-1
k
x
(Å )
-1
E (eV)
B
Γ M
M
Γ
K
K
SS
SS
(b)
(c)
-0.12 0.00 0.12
k
y
(Å )
-1
Intensity(arb.units)
Low
High
(f)
K
E
F
TheoreticalCalculations
FIG. 1: Theoretical calculations and experimental results : Strong spin-orbit interac-
tion gives rise to a single-surface-state-Dirac-cone on the surface of the topological
insulator.
-0.4 0.0
-0.2 0.0 0.2
-0.2 0.0 0.2
-0.2 0.0 0.2
-0.6
-0.4
-0.2
0.0
-0.8 0.0 0.8
2.4
2.8
3.2
E (eV)
B
21eV
k
y
(Å )
-1
L
F
Z
Γ
Γ
M
M
21
22
19
31
eV
k
y
(Å )
-1
k
z
(Å )
-1
(c)
(a)
Γ
15
19eV
31eV
E (eV)
B
(b)
19
18
17
16
15
20
21
22
25
28
31
hν
(eV)
Low
High
FIG. 2: Transverse-momentum k
z
dependence of topological Dirac bands near
¯
Γ.
of bulk band inversions that exist in the band structure of a material [7]. The origin of
topological Z
2
order in Bi
1−x
Sb
x
is bulk-band inversions at t hree equivalent L-points [5, 7]
whereas in Bi
2
Se
3
only o ne band is expected to be inverted, making it similar t o the case in
the two-dimensional quantum spin Hall insulator phase. Therefore, a much simpler surface
4

-0.2 0.0 0.2
-0.2
0.0
0.2
-1 0 1
-1
0
1
(b)
(a)
k
y
(Å )
-1
k
x
(Å )
-1
Γ
(c)
M
M
Γ
K
Gold
Bi Se
2 3
Bi Se
2 3
BulkValence
Γ
M
M
SS
(e)
Bi Se
2 3
BulkCond.
BulkValence
Γ
M
M
SS
(d)
E
F
Gold
SS
M
Γ
k
x
(Å )
-1
k
y
(Å )
-1
Bi Se
2 3
0
-0.4
-0.2
E (eV)
B
(d)
(e)
0.1L
(g)
(h)
0L
0.5L
-0.1
0.0
0.1
k (Å
-1
)
(f)
FIG. 3: The topology of the surface Dirac cone Fermi surface and the associated Berry’s
phase.
spectrum is naturally expected in Bi
2
Se
3
. All previous experimental studies of Bi
2
Se
3
have
focused on the material’s bulk properties; nothing is known about its SSs. It is this key ex-
perimental info rmatio n that we provide here that, for the first time, enables us to determine
its topological quantum class.
The bulk crystal symmetry of Bi
2
Se
3
fixes a hexagonal Brillouin zone (BZ) for its (111)
surface (Fig. 1d) on which
¯
M and
¯
Γ are t he 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
Bi
2
Se
3
(see the Methods section). The electronic spectral weight distributions observed
near the
¯
Γ point are presented in Fig. 1a-c. Within a narrow binding-energy window, a
clear V-shaped band pair is o bserved to approach the Fermi level (E
F
). Its dispersion or
intensity had no measurable time dependence within the duration of the experiment. The
‘V’ bands cross E
F
at 0.09
˚
A
−1
along
¯
Γ-
¯
M and at 0.10
˚
A
−1
along
¯
Γ-
¯
K, and have nearly equal
5

band velocities, approximately 5 × 10
5
ms
−1
, along the two directions. A continuum-like
manifold of states - a filled U-shaped feature - is 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 Bi
2
Se
3
along
the
¯
K −
¯
Γ −
¯
M k-space cut. The calculated band structure with a nd 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, ↑) = E(−
~
k, ↓) whereas space inversion symmetry
demands E(
~
k, ↑) = E(−
~
k, ↑). 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 o n the surface by spin-orbit interactions except at particular high-
symmetry points-the Kramers points on the surface BZ. In our calculations, the SSs (red
dotted lines) are doubly degenerate only at
¯
Γ (Fig. 1 f). This is generally true for all known
spin-orbit-coupled material surfaces such as gold [25, 26] or Bi
1−x
Sb
x
(ref. 5). In Bi
2
Se
3
,
the SSs emerge from the bulk continuum, cross each other at
¯
Γ, pa ss through the Fermi
level (E
F
) 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 Z
2
= −1
class in the Fu-Kane-Mele classification scheme [7].
A global agreement between the experiment al band structure (Fig. 1a-c) and our the-
oretical calculation (Fig. 1f) is obtained by considering a rigid shift of t he chemical po-
tential by about 20 0 meV with respect to our calculated band structure (Fig. 1f) of the
formula compound Bi
2
Se
3
. The experimental sign of this rigid shift (the raised chemical
potential) corresponds to an electron doping of the Bi
2
Se
3
insulating f ormula matrix (see
Supplementary Information). This is consistent with the fact that naturally g r own Bi
2
Se
3
semiconductor used in our experiment is n-type, as independently confirmed by our trans-
6

port measurements. The natural do ping 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 exam-
ine the way surface bands connect to the bulk conduction band across the gap. A unique
determination of the surface band topology of purely insulating Bi
1−x
Sb
x
(refs 5, 6) was
clarified only on doping with a foreign element, Te. In our experiment al data o n Bi
2
Se
3
,
we observe a V-shaped pure SS band to be dispersing towards E
F
, which is in good agree-
ment 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 ‘V’
band, an electron-pocket-like U-shaped continuum is observed to b e 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 Bi
2
Se
3
and by correspondence to our band calculation, we assign the
broad feature to correspond roughly to the bott om of the conduction band.
To systematically investigate the nature of all the band f eatures 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 o f incident photon energy enables us to
probe the k
z
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 signal
[5]. Our photon-energy study did not indicate a strong k
z
dispersion of the lowestlying
energy bands on the ‘U’, although the full cont inuum does have some dispersion (Fig. 2).
Some variation of the quasiparticle intensity near E
F
is, however, observed owing to the
variation of the electron-photon matrix element. In light of the k
z
-dependence study (Fig.
2b), if the features above -0.15 eV were purely due to the bulk, we would expect to observe
dispersion a s k
z
moved away from the Γ-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 behavior is also observed in the ARPES study of other semiconductors [27]. In
our k
z
-dependent study of the bands (Fig. 2b) we also observe two bands dispersing in k
z
7

that have energies below -0.3 eV (blue dotted bands), reflecting the bulk valence bands,
in addition to two other non-dispersive features associated with the two sides of t he pure
SS Dirac bands. The red curve is measured right at the Γ-point, which suggests that the
Dirac point lies inside the bulk bandgap. Taking the bottom of the ‘U’ band as the bulk
conduction-band minimum, we estimate that a bandgap of about 0.3 eV is realized in the
bulk of the undoped material. Our ARPES estimated bandgap is in good agreement with
the value deduced from bulk physical measurements [23] and from other calculations that
report the bulk band structure [20, 24]. This suggests that the magnitude of band bending
near the surface is not la rger than 0.05 eV. We note that in purely insulating Bi
2
Se
3
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
Bi
2
Se
3
matrix the ‘U’ 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 centered around
¯
Γ. None of the three TRIMs
located at
¯
M are enclosed by any FS, in contrast to what is observed in Bi
1−x
Sb
x
(ref.
5). The detailed spectral behavior around
¯
Γ is shown in Fig. 3b, which was obtained with
high momentum resolution. A r ing-like feature formed by the outer ‘V’ pure SS band (a
horizontal cross-section of the upper Dirac cone in Fig . 1) surrounds the conduction-band
continuum centered at
¯
Γ. 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 π mod 2π in its wavefunction
[7]. Therefore, if the chemical potential (Fermi level) lies inside the bandgap, as it should
in purely insulating Bi
2
Se
3
, its surface must carry a global π mod 2π Berry phase. In
most spin-orbit materials, such as gold (Au[111]), it is known that the surface FS consists
of two spin-orbit-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
¯
Γ-point
[25, 26]. The resulting FS topology leads to a 2π or 0 Berry phase because the phases from
the two rings add or cancel. This makes gold-like SSs to polo gically trivial despite their
spin-orbit origin.
Our theoretical calculation supported by our experimental data suggests that in insulating
Bi
2
Se
3
there exists a singly degenerate surface FS which encloses only one Kramers point
8

on the surface Brillouin zone. This provides evidence t hat insulating Bi
2
Se
3
belongs to
the Z
2
= −1 topological class in the Fu-Kane-Mele topolog ical classification scheme fo r
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
Bi
2
Se
3
, thereby shifting the chemical potential to lie inside the bulk bandgap. The surface
transport of Bi
2
Se
3
prepared as such would therefore be dominated by topological effects as
it po ssesses only one Dirac fermion that carries the non-trivial Z
2
index. The existence of a
large bulk bandgap (0.3 eV) within which the observed Z
2
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 Bi
2
Se
3
, it can be considered as the ‘hydrogen atom’ 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
fermions [14], magnetic monopole image [9, 10] or topolog ical exciton condensates [15], by
transport probes.
NOTE AD DED: The counter-doping method to reach the fully undoped
compound - the bulk-insulator state, proposed in this paper, and the spin-
texture measurements have been achieved in a recent work in Nature:
[http://dx.doi.org/10.1038/nature08234 (2009)]
Methods
Theoretical calculations. The theoretical band calculations were performed with the
LAPW method in slab geometry using the WIEN2K package [28]. The g eneralized gradient
approximation o f Perdew, Burke and Ernzerhof [29] was used to describe the exchange-
correlation p otential. SOC was included as a second variational step using scalar-relativistic
eigenfunctions as basis after the initial calculation was converged t o self-consistency. The
surface was simulated by placing a slab of 12 quintuple layers in vacuum. A grid of
21×21×1 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 k
z
of the ARPES measurements
(k
z
= (1/hbar)
p
2m(E
kin
cos
2
θ + V
0
)), an inner potential V
0
of approximately 11.7 eV was
used, given by a fit on the ARPES data at normal emission.
9

Experimental methods. Single crystals of Bi
2
Se
3
were grown by melting stoichio-
metric 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 650
◦
C, 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 17-45
eV photons on beamline 12.0.1 of the Advanced Light Source at the Lawrence Berkeley Na-
tional Laboratory and beamline 5-4 at the Stanford Synchrotron Radiation Laboratory. The
energy and momentum resolutions were 15 meV and 1.5% of t he surface BZ respectively
using a Scient a analyser. The samples were cleaved in situ between 10 and 55 K under
pressures of less than 5×10
−11
torr, resulting in shiny flat surfaces. The surface band quasi-
particle signal is stable throughout the entire measurement duration. The counter- doping
method to r each the fully undoped compound - the bulk-insulator state, proposed in this
paper, and the spin-texture measurements have been achieved in a recent work in Nature:
[http://dx.doi.org/10.1038/nature08234 (2009)]
[1] Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett.
98, 106803 (2007).
[2] Moore, J. E. & Balents, L. Topological invariants of time-reversal-invariant band structures.
Phys. Rev. B 75, 121306(R) (2007).
[3] Zhang, S.-C. Topological states of quantu m matter. Physics 1, 6 (2008).
[4] Konig, M. et al. Qu antum spin Hall insulator state in HgTe quantum wells. Science 318,
766-770 (2007).
[5] Hsieh, D. et al. A topological Dirac insulator in a quantum sp in Hall phase. Nature 452,
970-974 (2008).
[6] Hsieh, D. et al. Observation of unconventional quantum spin textures in topological insulators.
Science 323, 919-922 (2009).
[7] Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302
(2007).
[8] Wilczek, F. Two applications of axion electrodynamics. Phys. Rev. Lett. 58, 1799-1802 (1987).
[9] Franz, M. High-energy physics in a new guise. Physics 1, 36 (2008).
10

[10] Qi, X.-L. et al., Inducing a magnetic monopole with topological surface states. Science 323,
1184-1187 (2009).
[11] Qi, X.-L. et al. Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78,
195424 (2008).
[12] Ran, Y., Zhang, Y. & Vishwanath, A. One-dimensional topologically protected modes in
topological insulators with lattice dislocations. Nature Phys. doi:10.1038/nphys1220 (2009).
[13] Moore, J. E ., R an, Y. & Wen, X.-G. Topological surf ace states in three-dimensional magnetic
insulators. Phys. Rev. Lett. 101, 186805 (2008).
[14] Fu, L. & Kane, C. L. Superconducting p roximity effect and Majorana fermions at the sur face
of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).
[15] Seradjeh, B., Moore, J. E. & Franz, M. Exciton condensation and charge fractionalization in
a topological insulator film. Preprint at http://arxiv.org/abs/0902.1147v1 (2009).
[16] Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183-191 (2007).
[17] DiSalvo, F. J. Thermoelectric cooling and power generation. Science 285, 703-706 (1999).
[18] Wyckoff, R. W. G. Crystal Structures (Krieger, 1986).
[19] Hyde, G. R. et al. Electronic properties of Bi
2
Se
3
crystals. J. Phys. Chem. Solids 35, 1719-1728
(1974).
[20] Larson, P. et al. Electronic structure of Bi
2
X
3
(X=S, Se, T) compounds: Comparison of
theoretical calculations with photoemission studies. Phys. Rev. B 65, 085108 (2002).
[21] Greanya, V. A. et al. Determination of the valence band dispersions f or Bi
2
Se
3
using angle
resolved photoemission. J. Appl. Phys. 92, 6658-6661 (2002).
[22] Mooser, E. & Pearson, W. B. New semiconducting compoun ds. Phys. Rev. 101, 492-493
(1956).
[23] Black, J. et al. Electrical and optical properties of some M
V −B
2
N
V I−B
3
semiconductors. J.
Phys. Chem. Solids 2, 240-251 (1957).
[24] Mishra, S. K., Satpathy, S. & Jepsen, O. Electronic structure and thermoelectric properties
of bismuth telluride and bismuth selenide. J. Phys. Condens. Matter 9, 461-479 (1997).
[25] LaShell, S., McDougal, B. A. & Jensen, E. Spin splitting of an Au(111) surface state band ob-
served with angle resolved photoelectron spectroscopy. Phys. Rev. Lett. 77, 3419-3422 (1996).
[26] Hoesch, M. et al. Spin structure of the Shockley surface state on Au(111). Phys. Rev. B 69,
241401 (2004).
11

[27] Hufner, S. Photoelectron Spectroscopy (Springer, 1995).
[28] Blaha, P. et al. Computer Code WIEN2K (Vienna Univ. Technology, 2001).
[29] Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation made simple.
Phys. Rev. Lett. 77, 3865-3868 (1996).
12

-0.15 0.00 0.15
-0.15 0.00 0.15
-0.4
-0.3
-0.2
-0.1
0.0
0.1
k
x
k
y
k
z
Γ
M
K
L
F
Z
Γ
(111)
M
2
M
1
M
3
Γ
k
x
k
y
(a)
(e)
(d)
nosoc
soc
nosoc
soc
Bulk
Surface
k
y
(Å )
-1
k
x
(Å )
-1
E (eV)
B
Γ M
M
Γ
K
K
SS
SS
(b)
(c)
-0.12 0.00 0.12
k
y
(Å )
-1
Intensity(arb.units)
Low
High
(f)
K
E
F
TheoreticalCalculations
FIG. 4: [Enlarged version of Fig-1] Theoretical calculations and experimental results : Strong
spin-orbit interaction gives rise to a single-surface-state-Dirac-cone on the surface of the topological
insulator.
13

-0.4 0.0
-0.2 0.0 0.2
-0.2 0.0 0.2
-0.2 0.0 0.2
-0.6
-0.4
-0.2
0.0
-0.8 0.0 0.8
2.4
2.8
3.2
E (eV)
B
21eV
k
y
(Å )
-1
L
F
Z
Γ
Γ
M
M
21
22
19
31
eV
k
y
(Å )
-1
k
z
(Å )
-1
(c)
(a)
Γ
15
19eV
31eV
E (eV)
B
(b)
19
18
17
16
15
20
21
22
25
28
31
hν
(eV)
Low
High
FIG. 5: [Enlarged version of Fig-2] Transverse-momentum k
z
dependence of topological Dirac
bands near
¯
Γ.
14

-0.2 0.0 0.2
-0.2
0.0
0.2
-1 0 1
-1
0
1
(b)
(a)
k
y
(Å )
-1
k
x
(Å )
-1
Γ
(c)
M
M
Γ
K
Gold
Bi Se
2 3
Bi Se
2 3
BulkValence
Γ
M
M
SS
(e)
Bi Se
2 3
BulkCond.
BulkValence
Γ
M
M
SS
(d)
E
F
Gold
SS
M
Γ
k
x
(Å )
-1
k
y
(Å )
-1
Bi Se
2 3
0
-0.4
-0.2
E (eV)
B
(d)
(e)
0.1L
(g)
(h)
0L
0.5L
-0.1
0.0
0.1
k (Å
-1
)
(f)
FIG. 6: [Enlarged version of Fig-3] The Z
2
topology of the surface Dirac cone Fermi su rface and
the associated Berry’s phase.
15

Fig. 1 (Caption). Strong spin-orbit int er act ion gives rise to a single SS Dirac
cone. Theory (see the Methods section for calculation methods) versus experiments. a,b,
High-resolution ARPES measurements of surface electronic band dispersion on Bi
2
Se
3
(111).
Electron dispersion data measured with an incident photon energy of 22 eV near the
¯
Γ-point
along the
¯
Γ −
¯
M (a) and
¯
Γ −
¯
K (b) momentum-space cuts. c, The momentum distribution
curves corresponding to a suggest that two surface bands converge into a single Dirac point
at
¯
Γ. The V-shaped pure SS band pair observed in a-c is nearly isotropic in the momentum
plane, forming a Dirac cone in the energy-k
x
− k
y
space (where k
x
and k
y
are in the
¯
Γ −
¯
K
and
¯
Γ −
¯
M directions, respectively). The U-shaped broad continuum feature inside the V-
shaped SS corresponds roughly to the bottom of the conduction band (see the text). d, A
schematic diagram of the full bulk three-dimensional BZ of Bi
2
Se
3
and the two-dimensional
BZ of the projected (111) surface. e, The surface Fermi surface (FS) of the two-dimensional
SSs along the
¯
K −
¯
Γ −
¯
M momentum-space cut is a single ring centered at
¯
Γ if the chemical
potential is inside the bulk bandgap. The band resp onsible for this ring is singly degenerate
in theory. The TRIMs on the (111) surface BZ are located at
¯
Γ and the three
¯
M points. The
TRIMs are marked by the red dots. In the presence of strong spinorbit coupling (SOC),
the surface band crosses the Fermi level only once between two TRIMs, namely
¯
Γ and
¯
M; this ensures the existence of a π Berry phase on the surface. f, The corresponding
local density approximation (LDA) band structure (see the Methods section). Bulk band
projections are represented by the shaded areas. The band-structure to polo gy calculated in
the presence of SOC is presented in blue and that without SOC is in green. No pure surface
band is observed to lie within the insulating gap in the absence of SOC (black lines) in the
theoretical calculation. One pure gapless surface band is observed between
¯
Γ and
¯
M when
SOC is included (red dotted lines).
16

Fig. 2 (Caption). Transverse-momentum k
z
dependence of topological Dirac
bands near
¯
Γ. a, The energy dispersion data along t he
¯
Γ-
¯
M cut, measured with the
photon energy of 21 eV (corresponding to 0.3 k-space length along Γ -Z k k
z
), 19 eV (Γ)
and 31 eV (-0.4 k-space length along Γ-Z of the bulk three-dimensional Γ BZ) are shown.
Although the bands below -0.4 eV binding energy show strong k
z
dependence, the linearly
dispersive Dirac-like bands and the U-shaped broad feature show weaker k
z
dispersion. The
Dirac point is observed to lie inside the bulk bandgap. A careful look at the individual
curves reveals some k
z
dependence of the U-shaped continuum (see b for details). b, The
energy distribution curves obtained from the normal-emission spectra measured using 15-31
eV photon energies reveal two dispersive bulk bands below -0.3 eV (blue dotted lines). This
is in addition to the two non-dispersive peaks from the Dirac-cone bands inside the gap.
The Dirac band intensity is strongly modulated by the photon energy changes due to the
matrix-element effects (which is also observed in BiSb; ref. 5). c, A k-space map of locations
in the bulk three-dimensional BZ scanned by the detector a t different photon energies over
a theta (θ) range of ±30
◦
. This map (k
z
, k
y
, E
photon
) was used to explore the k
z
dependence
of the observed bands.
17

Fig. 3 (Caption). The topology of the surface Dirac cone Fermi surface and
the associated Berry’s phase. a, The observed surface FS of Bi
2
Se
3
consists of a small
electron pocket around the center of the BZ,
¯
Γ. b, High-momentum-resolution data around
¯
Γ reveal a single ring formed by the pure SS V-shaped Dirac band. For the naturally
occurring Bi
2
Se
3
, the spectral intensity in the middle of the ring is due to the presence of
the ’U’ feature, which roughly images the bottom of the conduction-band continuum (see
the text). The observed topology of the pure surface FS of Bi
2
Se
3
is different from that of
most other spin-orbit materials such as gold (Au(111)). c, The Au(111) surface FS features
two rings (each non-degenerate) surrounding the
¯
Γ point. An electron encircling the gold
FS carries a Berry phase of zero, characteristic of a trivial band insulator or metal, and can
be classified by Z
2
= +1 (ref. 7). The single surface FS observed in Bi
2
Se
3
is topologically
distinct from that o f gold. The single non-degenerate surface FS enclosing a Kramers point
(
¯
Γ) constitutes the key signature of a topological-insulator phase characterized by Z
2
= −1.
d, e, Schematic SS topologies in gold and Bi
2
Se
3
for direct comparison. In gold, the chemical
potential can be continuously tuned to be placed inside the interband gap, making the SSs
fully gapped. In Bi
2
Se
3
, however, the Dirac structure of the SSs required by the Kramers
degeneracy and time-reversal invariance ensures that they r emain g apless independent of the
location of the chemical potential within the bulk gap. If the chemical potential is placed
inside the gap, as it would naturally lie in the purely insulating undoped Bi
2
Se
3
, the surface
transport would be dominated by the single Dirac fermion, which is of purely topological
origin.
18