Observation of topologically protected Dirac spin-textures and \pi Berry's phase in pure Antimony (Sb) and topological insulator BiSb

Abstract

A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry based classification of condensed matter. Exotic spin transport phenomena such as the dissipationless quantum spin Hall effect have been speculated to originate from a novel topological order whose identification requires a spin sensitive measurement. Using Spin-resolved-ARPES, we probe the spin degrees of freedom and demonstrate that topological quantum numbers are uniquely determined from spin-texture Berry Phase imaging measurements. Applying this method to pure antimony (Sb) and Bi-Sb, we identify the origin of its novel Topological Order and the negative value of the mirror Chern number. These results taken together constitute the first observation of surface electrons collectively carrying a topological Berry's phase and definite mirror Chern chirality in pure Antimony (Sb) which are the key electronic properties for realizing topological quantum computing via the interface Majorana fermion framework. This paper contains the details of the above mentioned previously reported (Science \textbf{323}, 919 (2009)) results. Comment: 6 Figures, 10 Pages, RevTex Format, Detailed version of Hsieh et.al., SCIENCE 323, 919 (2009)

Full text

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arXiv:0909.5509v1 [cond-mat.mes-hall] 30 Sep 2009
Detailed version of SCIENCE 323, 919 (2009).
Observation of topologically protected Dirac spin-textures and π Berry’s phase in
pure Antimony (Sb) and topological insulator BiSb
http://dx.doi.org/10.1126/science.1167733
D. Hsieh,1Y. Xia,1L. Wray,1D. Qian,1A. Pal,1J. H. Dil,2, 3F. Meier,2,3J.
Osterwalder,3G. Bihlmayer,4C. L. Kane,5Y. S. Hor,6R. J. Cava,6and M. Z. Hasan1,7
1Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544, USA
2Swiss Light Source, Paul Scherrer Institute, CH-5232, Villigen, Switzerland
3Physik-Institut, Universit¨ at Z¨ urich-Irchel, 8057 Z¨ urich, Switzerland
4Institut f¨ ur Festk¨ orperforschung, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany
5Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
6Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
7Princeton Center for Complex Materials, Princeton University, Princeton, NJ 08544, USA∗
(Dated: September 30, 2009)
A topologically ordered material is characterized by a rare quantum organization of electrons that
evades the conventional spontaneously broken symmetry based classification of condensed matter.
Exotic spin transport phenomena such as the dissipationless quantum spin Hall effect have been
speculated to originate from a novel topological order whose identification requires a spin sensitive
measurement. Using spin-resolved ARPES, we probe the spin degrees of freedom and demonstrate
that topological quantum numbers are uniquely determined from spin texture imaging measure-
ments. Applying this method to pure antimony (Sb) and Bi1−xSbx, we identify the origin of its
novel order and unusual chiral topological properties. These results taken together constitute the
observation of surface electrons collectively carrying a geometrical quantum (Berry’s) phase and
definite chirality in pure Antimony, Sb, and topological insulator BiSb, which are the key electronic
properties for realizing topological quantum computing via the Majorana fermion framework. This
paper contains the details of our previously reported (first reported in Science 323, 919 (2009))
observation of a negative mirror Chern quantum number for pure Sb.
PACS numbers:
Ordered phases of matter such as a superfluid or a
ferromagnet are usually associated with the breaking of
a symmetry and are characterized by a local order pa-
rameter [1], and the typical experimental probes of these
systems are sensitive to order parameters. The discovery
of the quantum Hall effects in the 1980s revealed a new
and rare type of order that is derived from an organized
collective quantum motion of electrons [2-4]. These so-
called “topologically ordered phases” do not exhibit any
symmetry breaking and are characterized by a topologi-
cal number [5] as opposed to a local order parameter. The
classic experimental probe of topological quantum num-
bers is magneto-transport, where measurements of the
quantization of Hall conductivity σxy= ne2/h (where e
is the electric charge and h is Planck’s constant) reveals
the value of the topological number n that characterizes
the quantum Hall effect state [6].
Recent theoretical and experimental studies suggest
that a new class of quantum Hall-like topological phases
can exist in spin-orbit materials without external mag-
netic fields, with interest centering on two examples, the
“quantum spin Hall insulator” [7-9] and the “strong topo-
logical insulator” [10,11]. Their topological order is be-
lieved to give rise to unconventional spin physics at the
∗Electronic address: mzhasan@Princeton.edu
sample edges or surfaces with potential applications rang-
ing from dissipationless spin currents [12] to topologi-
cal (fault-tolerant) quantum computing [13]. However,
unlike conventional quantum Hall systems, these novel
topological phases do not necessarily exhibit a quantized
charge or spin response (σxy ?= ne2/h) [14,15]. In fact,
the spin polarization is not a conserved quantity in a spin-
orbit material. Thus, their topological quantum num-
bers, the analogues of n, cannot be measured via the
classic von Klitzing-type [2] transport methods.
Here we show that spin-resolved angle-resolved photoe-
mission spectroscopy (spin-ARPES) can perform analo-
gous measurements for topological metals and insulators.
We measured all of the topological numbers for Bi1−xSbx
and provide an identification of its spin-texture, which
heretofore was unmeasured despite its surface states hav-
ing been observed [10]. The measured spin texture re-
veals the existence of a non-zero geometrical quantum
phase (Berry’s phase [16,17]) and the handedness or chi-
ral properties. More importantly, this technique enables
us to investigate aspects of the metallic regime of the
Bi1−xSbxseries, such as spin properties in pure Sb, which
are necessary to determine the microscopic origin of topo-
logical order. Our measurements on pure metallic Sb
show that its surface carries a geometrical (Berry’s) phase
and chirality property unlike the conventional spin-orbit
metals such as gold (Au), which has zero net Berry’s
phase and no net chirality [18].

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2
M
?
M
H
??
?
??
??
?
??
-0.2
-0.2
-0.1
-0.1
0.0
0.0
0.1
0.1
-0.2
-0.2
0.0
0.0
0.2
0.2
Deg
y
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-0.2
-0.2
B
BiSb?(111)?topology
D
k (Å??)
-1
MM
0.04
0.0
-0.04
k (Å??)
y
-1
0.6 0.8 1.0
0.0
-0.1
E??(eV)
B
k (Å??)
x
-1
0.6
k (Å??)
x
-
0.8
-1
1.0
E??(eV)
B
k (Å??)
x
-1
F
E
M’
SS?of?Bi??????Sb
1
23
4,5
1
2
3
E??=?-25?meV
B
Spin?resolved?Int.
G
low
high
?
0.20.4
0.6
M
k (Å??)
x
-1
k (Å??)
x
-1
?
M
0.60.0 0.2
0.4
Spin?down
Spin?up
1
2
La
Ls
T
H
L
?
M
kx
spin?down
spin?up
EF
EB
M
M
?
M
1
2
3
kx
ky
K
?
M
L
?
(111)
k
k
k
x
y
z
H
T
1-xx
A
C
S?of
=?1?topology
Surface?F
?0
Bulk?conduction?band
Bulk?valence?band
FIG. 1: Spin-texture and π Berry’s Phase in BiSb
Strong topological materials are distinguished from or-
dinary materials such as gold by a topological quantum
number, ν0= 1 or 0 respectively [14,15]. For Bi1−xSbx,
theory has shown that ν0is determined solely by the char-
acter of the bulk electronic wave functions at the L point
in the three-dimensional (3D) Brillouin zone (BZ). When
the lowest energy conduction band state is composed of
an antisymmetric combination of atomic p-type orbitals
(La) and the highest energy valence band state is com-
posed of a symmetric combination (Ls), then ν0= 1, and
vice versa for ν0= 0 [11]. Although the bonding nature
(parity) of the states at L is not revealed in a measure-
ment of the bulk band structure, the value of ν0can be
determined from the spin-textures of the surface bands
that form when the bulk is terminated. In particular, a
ν0= 1 topology requires the terminated surface to have
a Fermi surface (FS) [1] that supports a non-zero Berry’s
phase (odd as opposed to even multiple of π), which is
not realizable in an ordinary spin-orbit material.
In a general inversion symmetric spin-orbit insulator,
the bulk states are spin degenerate because of a combi-
nation of space inversion symmetry [E(?k,↑) = E(−?k,↑)]
and time reversal symmetry [E(?k,↑) = E(−?k,↓)]. Be-
cause space inversion symmetry is broken at the termi-
nated surface, the spin degeneracy of surface bands can
be lifted by the spin-orbit interaction [19-21]. However,
according to Kramers theorem [16], they must remain
spin degenerate at four special time reversal invariant
momenta (?kT =¯Γ,¯M) in the surface BZ [11], which for
the (111) surface of Bi1−xSbxare located at¯Γ and three
equivalent¯M points [see Fig.1(A)].
Depending on whether ν0equals 0 or 1, the Fermi sur-
face pockets formed by the surface bands will enclose the
four?kT an even or odd number of times respectively. If
a Fermi surface pocket does not enclose?kT (=¯Γ,¯M), it
is irrelevant for the topology [11,20]. Because the wave
function of a single electron spin acquires a geometric
phase factor of π [16] as it evolves by 360◦in momen-
tum space along a Fermi contour enclosing a?kT, an odd
number of Fermi pockets enclosing?kT in total implies a
π geometrical (Berry’s) phase [11]. In order to realize a
π Berry’s phase the surface bands must be spin-polarized
and exhibit a partner switching [11] dispersion behavior
between a pair of?kT. This means that any pair of spin-
polarized surface bands that are degenerate at¯Γ must
not re-connect at¯M, or must separately connect to the
bulk valence and conduction band in between¯Γ and¯M.
The partner switching behavior is realized in Fig. 1(C)
because the spin down band connects to and is degener-
ate with different spin up bands at¯Γ and¯M. The part-
ner switching behavior is realized in Fig. 2(A) because
the spin up and spin down bands emerging from¯Γ sepa-
rately merge into the bulk valence and conduction bands
respectively between¯Γ and¯M.
We first investigate the spin properties of the topo-
logical insulator phase. Spin-integrated ARPES [19] in-
tensity maps of the (111) surface states of insulating
Bi1−xSbxtaken at the Fermi level (EF) [Figs 1(D)&(E)]
show that a hexagonal FS encloses¯Γ, while dumbbell
shaped FS pockets that are much weaker in intensity en-
close¯M. By examining the surface band dispersion be-
low the Fermi level [Fig.1(F)] it is clear that the central
hexagonal FS is formed by a single band (Fermi crossing
1) whereas the dumbbell shaped FSs are formed by the

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M
?
M
-0.4
-0.4
-0.4
-0.2
-0.2
0
0
0.2
0.2
0.4
0.4?
-0.6
-0.6
-0.4
-0.4
-0.2
-0.2
0.0
0.0
0.2
0.2
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
Deg
1.0
1.0
0.5
0.5
-1
0.0
0.0
-0.5
-0.5
0.2
0.0
-0.2
k (Å??)
x
k (Å??)
y
-1
E??(eV)
B
C
E
D
e??RS
Sb(111)
?
M
SS
?
h?RS
?
e?RS
h?RS
?
M
high
low
BS
Band?structure?calculation
2
?1
0.0
0.4
EF
n =?-1
M
SS
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
0.0
0.0
E??(eV)
B
T
HL
?
M
?
M
K
K
=?1?topology?(Sb)
?0
-0.5
0.00.5
-1
1.0
k (Å??)
x
k (Å??)
x
-1
T
H
L
-0.2 -0.1 0.0 0.1 0.2
k (Å??)
y
-1
Sb?(111)?topology
??????????????????
E??(eV)
B
EF
?
M
kx
?
M
kx
EB
EF
EB
A
Ls
La
La
Ls
=?0?topology?(Au-like)
?0
B
FG
FIG. 2: Topological nature of the surface states in pure Sb (Antimony)
merger of two bands (Fermi crossings 4 and 5) [10].
This band dispersion resembles the partner switching
dispersion behavior characteristic of topological insula-
tors.To check this scenario and determine the topo-
logical index ν0, we have carried out spin-resolved pho-
toemission spectroscopy. Fig.1(G) shows a spin-resolved
momentum distribution curve taken along the¯Γ-¯M direc-
tion at a binding energy EB= −25 meV [Fig.1(G)]. The
data reveal a clear difference between the spin-up and
spin-down intensities of bands 1, 2 and 3, and show that
bands 1 and 2 have opposite spin whereas bands 2 and
3 have the same spin (detailed analysis discussed later
in text). The former observation confirms that bands 1
and 2 form a spin-orbit split pair, and the latter obser-
vation suggests that bands 2 and 3 (as opposed to bands
1 and 3) are connected above the Fermi level and form
one band. This is further confirmed by directly imag-
ing the bands through raising the chemical potential via
doping [see supporting online material (APPENDIX B)
[22]]. Irrelevance of bands 2 and 3 to the topology is con-
sistent with the fact that the Fermi surface pocket they
form does not enclose any?kT. Because of a dramatic
intrinsic weakening of signal intensity near crossings 4
and 5, and the small energy and momentum splittings
of bands 4 and 5 lying at the resolution limit of mod-
ern spin-resolved ARPES spectrometers, no conclusive
spin information about these two bands can be drawn
from the methods employed in obtaining the data sets
in Figs 1(G)&(H). However, whether bands 4 and 5 are
both singly or doubly degenerate does not change the fact
that an odd number of spin-polarized FSs enclose the?kT,
which provides evidence that Bi1−xSbxhas ν0 = 1 and
that its surface supports a non-trivial Berry’s phase.
We investigated the quantum origin of topological or-
der in this class of materials. It has been theoretically
speculated that the novel topological order originates
from the parities of the electrons in pure Sb and not Bi
[11,23]. It was also noted [20] that the origin of the topo-
logical effects can only be tested by measuring the spin-
texture of the Sb surface, which has not been measured.
Based on quantum oscillation and magneto-optical stud-
ies, the bulk band structure of Sb is known to evolve from
that of insulating Bi1−xSbxthrough the hole-like band at
H rising above EF and the electron-like band at L sink-
ing below EF[23]. The relative energy ordering of the La
and Lsstates in Sb again determines whether the surface
state pair emerging from¯Γ switches partners [Fig.2(A)]
or not [Fig.2(B)] between¯Γ and¯M, and in turn deter-
mines whether they support a non-zero Berry’s phase.
In a conventional spin-orbit metal such as gold, a free-
electron like surface state is split into two parabolic spin-
polarized sub-bands that are shifted in?k-space relative
to each other [18]. Two concentric spin-polarized Fermi
surfaces are created, one having an opposite sense of in-
plane spin rotation from the other, that enclose¯Γ. Such
a Fermi surface arrangement, like the schematic shown
in figure 2(B), does not support a non-zero Berry’s phase
because the?kT are enclosed an even number of times (2
for most known materials).
However, for Sb, this is not the case.
shows a spin-integrated ARPES intensity map of Sb(111)
from¯Γ to¯M. By performing a systematic incident pho-
ton energy dependence study of such spectra, previously
unavailable with He lamp sources [24], it is possible to
identify two V-shaped surface states (SS) centered at¯Γ,
a bulk state located near kx= −0.25˚ A−1and resonance
states centered about kx= 0.25˚ A−1and¯M that are hy-
brid states formed by surface and bulk states [19] (AP-
PENDIX C [22]). An examination of the ARPES inten-
sity map of the Sb(111) surface and resonance states at
EF [Fig.2(E)] reveals that the central surface FS enclos-
ing¯Γ is formed by the inner V-shaped SS only. The outer
Figure 2(C)

Page 4

4
?
?
?
?
??
?
??
??
?
?
?
?
?
?
?
?
?
?
?
0
?
?
?
?
-0.2
0.0
0.2
0.1
-0.1
0.0
-0.1
-0.2
????????????????????????????????
l2
l1
E???(eV)
B
k (Å??)
x
-1
B
A
r1
r2
-0.2
0.0
0.2
0.1
-0.1
-1
Intensity?(arb.?units)
0.3
C
D
k (Å??)
x
0.0
-0.2
-0.4
0.2
0.4
Spin?polarization
0
-1
1 0
1
-1
1
-1
0
1
F
Px
Py
Pz
Pin?plane
Intensity?(arb.?units)
E
k (Å??)
x
P
P
y ’
z ’
-0.2
0.0
0.2
0.1
-0.1
-1
E???=?-30?meV
B
r2
r1
l1
l2
e??beam
accelerating?optics
h?
Au?foil
40?kV
e
y
x
z
x’
y’
z’
Mott?spin?detector
?
sample
spin?||?y
spin?||-?y
k (Å??)
x
-0.2
0.0
0.2
0.1
-0.1
-1
E???=?-30?meV
B
0
10
20
0
2
4
Momentum?distribution?of?spin
I
I
y
y
?
?
?
2
1
r1
l1
l2
l1
r1
r2
FIG. 3: Topological Spin-Texture and π Berry’s Phase in pure Sb (Antimony)
V-shaped SS on the other hand forms part of a tear-drop
shaped FS that does not enclose¯Γ, unlike the case in
gold. This tear-drop shaped FS is formed partly by the
outer V-shaped SS and partly by the hole-like resonance
state. The electron-like resonance state FS enclosing¯M
does not affect the determination of ν0because it must
be doubly spin degenerate (APPENDIX D [22]). Such
a FS geometry [Fig.2(G)] suggests that the V-shaped SS
pair may undergo a partner switching behavior expected
in Fig.2(A). This behavior is most clearly seen in a cut
taken along the¯Γ-¯K direction since the top of the bulk
valence band is well below EF[Fig.2(F)] showing only the
inner V-shaped SS crossing EF while the outer V-shaped
SS bends back towards the bulk valence band near kx
= 0.1˚ A−1before reaching EF. The additional support
for this band dispersion behavior comes from tight bind-
ing surface calculations on Sb [Fig.2(D)], which closely
match with experimental data below EF. Our observa-
tion of a single surface band forming a FS enclosing¯Γ
suggests that pure Sb is likely described by ν0= 1, and
that its surface may support a Berry’s phase.
Confirmation of a surface π Berry’s phase rests criti-
cally on a measurement of the relative spin orientations
(up or down) of the SS bands near¯Γ so that the partner
switching is indeed realized, which cannot be done with-
out spin resolution. Spin resolution was achieved using
a Mott polarimeter that measures two orthogonal spin
components of a photoemitted electron [27,28]. These
two components are along the y′and z′directions of
the Mott coordinate frame, which lie predominantly in
and out of the sample (111) plane respectively. Each of
?
?
?
?
?
T
K
?
M
?
?
?
?
??
L
X
X
?
(111)
L
U
U
k
k
k
x
y
z
A
C
Intensity?(arb.?units)
-0.4-0.20.0 0.20.4
k??(Å??)
x
-1
E??=?0?eV
B
h =?26?eV
?
h =?14?eV
?
L
U
H
T
D
3.2
3.0
2.8
2.6
k??(Å??)
x
-1
0.0-0.2 -0.4-0.6-0.8 -1.00.2
k??(Å??)
z
-1
h =?26?eV
?
18?eV
14?eV
H
B
FIG. S2: Incident energy and Brillouin zone space in pure Sb
crystal.
these two directions represents a normal to a scattering
plane defined by the photoelectron incidence direction
on a gold foil and two electron detectors mounted on
either side (left and right) [Fig.3(A)]. Strong spin-orbit
coupling of atomic gold is known to create an asymmetry
in the scattering of a photoelectron off the gold foil that
depends on its spin component normal to the scatter-
ing plane [28]. This leads to an asymmetry between the
left intensity (IL
y′,z′) and right intensity (IR
y′,z′) given by

Page 5

5
Ay′,z′ = (IL
the spin polarization Py′,z′ = (1/Seff) × Ay′,z′ through
the Sherman function Seff= 0.085 [27,28]. Spin-resolved
momentum distribution curve data sets of the SS bands
along the −¯M-¯Γ-¯M cut at EB = −30 meV [Fig.3(B)]
are shown for maximal intensity. Figure 3(D) displays
both y′and z′polarization components along this cut,
showing clear evidence that the bands are spin polarized,
with spins pointing largely in the (111) plane. In order
to estimate the full 3D spin polarization vectors from
a two component measurement (which is not required to
prove the partner switching or the Berry’s phase), we fit a
model polarization curve to our data following the recent
demonstration in Ref-[26], which takes the polarization
directions associated with each momentum distribution
curve peak [Fig.3(C)] as input parameters, with the con-
straint that each polarization vector has length one (in
angular momentum units of ?/2). Our fitted polarization
vectors are displayed in the sample (x,y,z) coordinate
frame [Fig.3(F)], from which we derive the spin-resolved
momentum distribution curves for the spin components
parallel (I↑
PENDIX B [22]) as shown in figure 3(E). There is a clear
difference in I↑
yat each of the four momentum
distribution curve peaks indicating that the surface state
bands are spin polarized [Fig.3(E)], which is possible to
conclude even without a full 3D fitting. Each of the pairs
l2/l1 and r1/r2 have opposite spin, consistent with the
behavior of a spin split pair, and the spin polarization
of these bands are reversed on either side of¯Γ in ac-
cordance with the system being time reversal symmetric
[E(?k,↑) = E(−?k,↓)] [Fig.3(F)]. The measured spin tex-
ture of the Sb(111) surface states (Fig.3), together with
the connectivity of the surface bands (Fig.2), uniquely
determines its belonging to the ν0= 1 class. Therefore
the surface of Sb carries a non-zero (π) Berry’s phase via
the inner V-shaped band and pure Sb can be regarded
as the parent metal of the Bi1−xSbx topological insula-
tor class, in other words, the topological order originates
from the Sb wave functions.
y′,z′−IR
y′,z′)/(IL
y′,z′+IR
y′,z′), which is related to
y) and anti-parallel (I↓
y) to the y direction (AP-
yand I↓
Sb
?
M
EF
A
-0.3
-0.2
-0.1
0.0
?
M
EF
-0.1
-0.2
-0.3
0.0
-0.2 -0.10.00.10.2
??????????????????
Sb?(111)
B
C
k??(Å??)
y
-1
Scenario?1
Scenario?2
La
Ls
Ls
La
FIG. S4: Surface band topologies in pure Sb.
Our spin polarized measurement methods (Fig.1 and
3) uncover a new type of topological quantum number
L
K
?
M
L
(111)
L
k
k
k
x
y
z
A
Mirror?plane
?
?
EF
E
M
?
EF
E
spin?down
spin?up
M
bulk?valence?band
bulk?conduction?band
kx
?
M
Insulating?Bi?Sb
Pure?Sb
EF
E
EF
E
kx
kx
n =?-1
M
n =?1
M
2
1
?
?
1
2
B
C
D
E
??
??
spin?down
spin?up
M
bulk?valence?band
bulk?conduction?band
kx
FIG. S6: Z2 Topology and Berry’s Phase associated with the
spin-polarized surface states in Sb and BiSb. The value of
topological Mirror Chern number in Sb and BiSb suggest sug-
gests left-handed chirality quantum number nM = -1.
nMwhich provides information about the chirality prop-
erties. Topological band theory suggests that the bulk
electronic states in the mirror (ky= 0) plane can be clas-
sified in terms of a number nM(=±1) that describes the
handedness (either left or right handed) or chirality of
the surface spins which can be directly measured or seen
in spin-resolved experiments [20]. We now determine the
value of nMfrom our data. From figure 1, it is seen that
a single (one) surface band, which switches partners at
¯M, connects the bulk valence and conduction bands, so
|nM| = 1 (APPENDIX F [22]). The sign of nMis related
to the direction of the spin polarization ??P? of this band
[20], which is constrained by mirror symmetry to point
along ±ˆ y. Since the central electron-like FS enclosing¯Γ
intersects six mirror invariant points [see Fig.3(B)], the
sign of nM distinguishes two distinct types of handed-
ness for this spin polarized FS. Figures 1(F) and 3 show
that for both Bi1−xSbx and Sb, the surface band that
forms this electron pocket has ??P? ∝ −ˆ y along the kx
direction, suggesting a left-handed rotation sense for the
spins around this central FS thus nM = −1. Therefore,
both insulating Bi1−xSbxand pure Sb possess equivalent
chirality properties − a definite spin rotation sense (left-
handed chirality, see Fig.3(B)) and a topological Berry’s
phase.
These spin-resolved experimental measurements reveal
an intimate and straightforward connection between the
topological numbers (ν0, nM) and the physical observ-
ables. The ν0 determines whether the surface electrons
support a non-trivial Berry’s phase, and if they do, the
nMdetermines the spin handedness of the Fermi surface
that manifests this Berry’s phase. The 2D Berry’s phase
is a critical signature of topological order and is not real-
izable in isolated 2D electron systems, nor on the surfaces
of conventional spin-orbit or exchange coupled magnetic
materials. A non-zero Berry’s phase is known, theoret-
ically, to protect an electron system against the almost

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6
universal weak-localization behavior in their low temper-
ature transport [11,13] and is expected to form the key
element for fault-tolerant computation schemes [13,29],
because the Berry’s phase is a geometrical agent or mech-
anism for protection against quantum decoherence [30].
Its remarkable realization on the Bi1−xSbxsurface repre-
sents an unprecedented example of a 2D π Berry’s phase,
and opens the possibility for building realistic prototype
systems to test quantum computing modules. In general,
our results demonstrate that spin-ARPES is a powerful
probe of topological order and quantum spin Hall physics,
which opens up a new search front for topological mate-
rials for novel spin-devices and fault-tolerant quantum
computing.
Details of Materials and Methods
Spin-integrated angle-resolved photoemission spec-
troscopy (ARPES) measurements were performed with
14 to 30 eV photons on beam line 5-4 at the Stanford
Synchrotron Radiation Laboratory, and with 28 to 32
eV photons on beam line 12 at the Advanced Light
Source, both endstations being equipped with a Sci-
enta hemispherical electron analyzer (see VG Scienta
manufacturer website for instrument specifications).
Spin-resolved ARPES measurements were performed at
the SIS beam line at the Swiss Light Source using the
COPHEE spectrometer (31, p.15) with a single 40 kV
classical Mott detector and photon energies of 20 and
22 eV. The typical energy and momentum resolution
was 15 meV and 1.5% of the surface Brillouin zone
(BZ) respectively at beam line 5-4, 9 meV and 1% of
the surface BZ respectively at beam line 12, and 80
meV and 3% of the surface BZ respectively at SIS using
a pass energy of 3 eV. The undoped and Te doped
Bi1−xSbxsingle crystal samples were each cleaved from
a boule grown from a stoichiometric mixture of high
purity elements.The boule was cooled from 650 to
270◦C over a period of 5 days and was annealed for 7
days at 270◦C. Our ARPES results were reproducible
over many different sample batches.
of the Sb compositions in Bi1−xSbx to 1% precision
was achieved by bulk resistivity measurements, which
are very sensitive to Sb concentration (23), as well
as scanning electron microscopy analysis on a cleaved
surface showing lateral compositional homogeneity over
the length scale of our ARPES photon beam size. X-ray
diffraction (XRD) measurements were used to check
that the samples were single phase, and confirmed
that the single crystals presented in this paper have
rhombohedral A7 crystal structure (point group R¯3m).
The XRD patterns of the cleaved crystals exhibit only
the (333), (666), and (999) peaks showing that the
naturally cleaved surface is oriented along the trigonal
Determination
(111) axis. Room temperature data were recorded on
a Bruker D8 diffractometer using Cu Kα radiation
(λ=1.54˚ A) and a diffracted beam monochromator. The
in-plane crystal orientation was determined by Laue
x-ray diffraction prior to insertion into an ultra high
vacuum environment.Cleaving these samples in situ
between 10 K and 55 K at chamber pressures less than 5
×10−11torr resulted in shiny flat surfaces, characterized
in situ by low energy electron diffraction (LEED) to be
clean and well ordered with the same symmetry as the
bulk [Fig. S2(B)]. This is consistent with photoelectron
diffraction measurements that show no substantial
structural relaxation of the Sb(111) surface (32).
Methods of using incident photon energy
modulated ARPES to separate the bulk from
surface electronic states of pure antimony (Sb)
In this section we detail incident photon energy mod-
ulated ARPES experiments on the low lying electronic
states of single crystal Sb(111), which we employ to iso-
late the surface from bulk-like electronic bands over the
entire BZ. Figure S2(C) shows momentum distributions
curves (MDCs) of electrons emitted at EF as a function
of kx(?¯Γ-¯M) for Sb(111). The out-of-plane component
of the momentum kzwas calculated for different incident
photon energies (hν) using the free electron final state
approximation with an experimentally determined inner
potential of 14.5 eV (37, 38). There are four peaks in
the MDCs centered about¯Γ that show no dispersion
along kz and have narrow widths of ∆kx ≈ 0.03˚ A−1.
These are attributed to surface states and are similar
to those that appear in Sb(111) thin films (37).
hν is increased beyond 20 eV, a broad peak appears
at kx ≈ -0.2˚ A−1, outside the k range of the surface
states near¯Γ, and eventually splits into two peaks.
Such a strong kz dispersion, together with a broadened
linewidth (∆kx≈ 0.12˚ A−1), is indicative of bulk band
behavior, and indeed these MDC peaks trace out a Fermi
surface [Fig. S2(D)] that is similar in shape to the hole
pocket calculated for bulk Sb near H (36). Therefore
by choosing an appropriate photon energy (e.g. ≤ 20
eV), the ARPES spectrum at EF along¯Γ-¯M will have
contributions from only the surface states. The small
bulk electron pocket centered at L is not accessed using
the photon energy range we employed [Fig. S2(D)].
As
Now we describe the experimental procedure used
to distinguish pure surface states from resonant states
on Sb(111) through their spectral signatures. ARPES
spectra along¯Γ-¯M taken at three different photon ener-
gies are shown in Fig. ??. Near¯Γ there are two rather
linearly dispersive electron like bands that meet exactly
at¯Γ at a binding energy EB ∼ -0.2 eV. This behavior
is consistent with a pair of spin-split surface bands
that become degenerate at the time reversal invariant
momentum (?kT)¯Γ due to Kramers degeneracy.
surface origin of this pair of bands is established by their
The

Page 7

7
lack of dependence on hν [Fig. ??(A)-(C)]. A strongly
photon energy dispersive hole like band is clearly seen
on the negative kx side of the surface Kramers pair,
which crosses EF for hν = 24 eV and gives rise to
the bulk hole Fermi surface near H [Fig. S2(D)]. For
hν ≤ 20 eV, this band shows clear back folding near
EB ≈ -0.2 eV indicating that it has completely sunk
below EF. Further evidence for its bulk origin comes
from its close match to band calculations [Fig. S2(D)].
Interestingly, at photon energies such as 18 eV where the
bulk bands are far below EF, there remains a uniform
envelope of weak spectral intensity near EF in the
shape of the bulk hole pocket seen with hν = 24 eV
photons, which is symmetric about¯Γ.
does not change shape with hν suggesting that it is
of surface origin. Due to its weak intensity relative to
states at higher binding energy, these features cannot be
easily seen in the energy distribution curves (EDCs) in
Fig. ??(A)-(C), but can be clearly observed in the MDCs
shown in Fig. S2(C) especially on the positive kx side.
Centered about the¯M point, we also observe a crescent
shaped envelope of weak intensity that does not disperse
with kz [Fig. ??(D)-(F)], pointing to its surface origin.
Unlike the sharp surface states near¯Γ, the peaks in the
EDCs of the feature near¯M are much broader (∆E ∼80
meV) than the spectrometer resolution (15 meV).
The origin of this diffuse ARPES signal is not due to
surface structural disorder because if that were the case,
electrons at¯Γ should be even more severely scattered
from defects than those at¯M. In fact, the occurrence of
both sharp and diffuse surface states originates from a
k dependent coupling to the bulk. As seen in Fig.2(D)
of the main text, the spin-split Kramers pair near¯Γ lie
completely within the gap of the projected bulk bands
near EF attesting to their purely surface character. In
contrast, the weak diffuse hole like band centered near
kx= 0.3˚ A−1and electron like band centered near kx=
0.8˚ A−1lie completely within the projected bulk valence
and conduction bands respectively, and thus their
ARPES spectra exhibit the expected lifetime broadening
due to coupling with the underlying bulk continuum (39).
This envelope
Method of counting spin Fermi surface?kT
enclosures in pure Sb
In this section we give a detailed explanation of why
the surface Fermi contours of Sb(111) that overlap with
the projected bulk Fermi surfaces can be neglected when
determining the ν0 class of the material. Although the
Fermi surface formed by the surface resonance near¯M en-
closes the?kT¯M, we will show that this Fermi surface will
only contribute an even number of enclosures and thus
not alter the overall evenness or oddness of?kTenclosures.
Consider some time reversal symmetric perturbation that
lifts the bulk conduction Laband completely above EF
so that there is a direct excitation gap at L. Since this
perturbation preserves the energy ordering of the Laand
Lsstates, it does not change the ν0 class. At the same
time, the weakly surface bound electrons at¯M can evolve
in one of two ways. In one case, this surface band can also
be pushed up in energy by the perturbation such that it
remains completely inside the projected bulk conduction
band [Fig. S4(A)]. In this case there is no more density
of states at EFaround¯M. Alternatively the surface band
can remain below EF so as to form a pure surface state
residing in the projected bulk gap. However by Kramers
theorem, this SS must be doubly spin degenerate at¯M
and its FS must therefore enclose¯M twice [Fig. S4(B)]. In
determining ν0for semi-metallic Sb(111), one can there-
fore neglect all segments of the FS that lie within the
projected areas of the bulk FS [Fig.2(G) of main text]
because they can only contribute an even number of FS
enclosures, which does not change the modulo 2 sum of
?kT enclosures.
In order to further experimentally confirm the topo-
logically non-trivial surface band dispersion shown in
figures 2(C) and (D) of the main text, we show ARPES
intensity maps of Sb(111) along the -¯K−¯Γ−¯K direction.
Figure S4(C) shows that the inner V-shaped band that
was observed along the -¯M−¯Γ−¯M direction retains its
V-shape along the -¯K−¯Γ−¯K direction and continues to
cross the Fermi level, which is expected since it forms
the central hexagonal Fermi surface. On the other hand,
the outer V-shaped band that was observed along the
-¯M−¯Γ−¯M direction no longer crosses the Fermi level
along the -¯K−¯Γ−¯K direction, instead folding back below
the Fermi level around ky = 0.1˚ A−1and merging with
the bulk valence band [Fig. S4(C)]. This confirms that
it is the Σ1(2)band starting from¯Γ that connects to the
bulk valence (conduction) band, in agreement with the
calculations shown in figure 2(D) of the main text.
Investigation of the robustness of Sb spin states
under random field perturbations introduced by
Bi substitutional disorder
The predicted topological protection of the surface
states of Sb implies that their metallicity cannot be de-
stroyed by weak time reversal symmetric perturbations.
In order to test the robustness of the measured gapless
surface states of Sb, we introduce such a perturbation
by randomly substituting Bi into the Sb crystal matrix
(APPENDIX A). Another motivation for performing
such an experiment is that the formalism developed
by Fu and Kane (41) to calculate the Z2 topological
invariants relies on inversion symmetry being present in
the bulk crystal, which they assumed to hold true even
in the random alloy Bi1−xSbx. However, this formalism
is simply a device for simplifying the calculation and
the non-trivial ν0 = 1 topological class of Bi1−xSbx is
predicted to hold true even in the absence of inversion
symmetry in the bulk crystal (41). Therefore introducing

Page 8

8
light Bi substitutional disorder into the Sb matrix is
also a method to examine the effects of alloying disorder
and possible breakdown of bulk inversion symmetry
on the surface states of Sb(111).
spin-integrated ARPES measurements on single crystals
of the random alloy Sb0.9Bi0.1. Figure ?? shows that
both the surface band dispersion along¯Γ-¯M as well as
the surface state Fermi surface retain the same form as
that observed in Sb(111), and therefore the ‘topological
We have performed
metal’ surface state of Sb(111) fully survives the alloy
disorder.Since Bi alloying is seen to only affect the
band structure of Sb weakly, it is reasonable to assume
that the topological order is preserved between Sb and
Bi0.91Sb0.09as we observed.
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FIG. 1. Theoretical spin spectrum of a topo-
logical insulator and spin-resolved spectroscopy
results. (A) Schematic sketches of the bulk Brillouin
zone (BZ) and (111) surface BZ of the Bi1−xSbx crys-
tal series. The high symmetry points (L,H,T,Γ,¯Γ,¯M,¯K)
are identified. (B) Schematic of Fermi surface pockets
formed by the surface states (SS) of a topological insu-
lator that carries a Berry’s phase. (C) Partner switching
band structure topology: Schematic of spin-polarized SS
dispersion and connectivity between¯Γ and¯M required
to realize the FS pockets shown in panel-(B). Laand Ls
label bulk states at L that are antisymmetric and sym-
metric respectively under a parity transformation (see
text). (D) Spin-integrated ARPES intensity map of the
SS of Bi0.91Sb0.09at EF. Arrows point in the measured
direction of the spin. (E) High resolution ARPES inten-
sity map of the SS at EF that enclose the¯M1 and¯M2
points. Corresponding band dispersion (second deriva-
tive images) are shown below. The left right asymmetry
of the band dispersions are due to the slight offset of
the alignment from the¯Γ-¯M1(¯M2) direction. (F) Surface
band dispersion image along the¯Γ-¯M direction showing
five Fermi level crossings. The intensity of bands 4,5 is
scaled up for clarity (the dashed white lines are guides
to the eye). The schematic projection of the bulk va-
lence and conduction bands are shown in shaded blue and
purple areas. (G) Spin-resolved momentum distribution
curves presented at EB= −25 meV showing single spin
degeneracy of bands at 1, 2 and 3. Spin up and down cor-
respond to spin pointing along the +ˆ y and -ˆ y direction
respectively. (H) Schematic of the spin-polarized surface
FS observed in our experiments. It is consistent with a
ν0= 1 topology (compare (B) and (H)).
FIG. 2.
vealed on the (111) surface states.
of the bulk band structure (shaded areas) and surface
band structure (red and blue lines) of Sb near EF for
a (A) topologically non-trivial and (B) topological triv-
ial (gold-like) case, together with their corresponding
surface Fermi surfaces are shown. (C) Spin-integrated
ARPES spectrum of Sb(111) along the¯Γ-¯M direction.
The surface states are denoted by SS, bulk states by
BS, and the hole-like resonance states and electron-like
resonance states by h RS and e−RS respectively. (D)
Calculated surface state band structure of Sb(111) based
on the methods in [20,25]. The continuum bulk energy
bands are represented with pink shaded regions, and the
lines show the discrete bands of a 100 layer slab. The
red and blue single bands, denoted Σ1 and Σ2, are the
surface states bands with spin polarization ??P? ∝ +ˆ y
and ??P? ∝ −ˆ y respectively. (E) ARPES intensity map
of Sb(111) at EF in the kx-ky plane. The only one FS
encircling¯Γ seen in the data is formed by the inner V-
shaped SS band seen in panel-(C) and (F). The outer V-
shaped band bends back towards the bulk band best seen
Topological character of pure Sb re-
Schematic
in data in panel-(F). (F) ARPES spectrum of Sb(111)
along the¯Γ-¯K direction shows that the outer V-shaped
SS band merges with the bulk band. (G) Schematic of
the surface FS of Sb(111) showing the pockets formed by
the surface states (unfilled) and the resonant states (blue
and purple). The purely surface state Fermi pocket en-
closes only one Kramers degenerate point (?kT), namely,
¯Γ(=?kT), therefore consistent with the ν0= 1 topological
classification of Sb which is different from Au (compare
(B) and (G)). As discussed in the text, the hRS and e−RS
count trivially.
FIG. 3.
states and chirality. (A) Experimental geometry of
the spin-resolved ARPES study. At normal emission (θ
= 0◦), the sensitive y′-axis of the Mott detector is ro-
tated by 45◦from the sample¯Γ to −¯M (? −ˆ x) direction,
and the sensitive z′-axis of the Mott detector is parallel
to the sample normal (? ˆ z). (B) Spin-integrated ARPES
spectrum of Sb(111) along the −¯M-¯Γ-¯M direction. The
momentum splitting between the band minima is indi-
cated by the black bar and is approximately 0.03˚ A−1. A
schematic of the spin chirality of the central FS based on
the spin-resolved ARPES results is shown on the right.
(C) Momentum distribution curve of the spin averaged
spectrum at EB= −30 meV (shown in (B) by white line),
together with the Lorentzian peaks of the fit. (D) Mea-
sured spin polarization curves (symbols) for the detector
y′and z′components together with the fitted lines using
the two-step fitting routine [26]. (E) Spin-resolved spec-
tra for the sample y component based on the fitted spin
polarization curves shown in (D). Up (down) triangles
represent a spin direction along the +(-)ˆ y direction. (F)
The in-plane and out-of-plane spin polarization compo-
nents in the sample coordinate frame obtained from the
spin polarization fit. Overall spin-resolved data and the
fact that the surface band that forms the central electron
pocket has ??P? ∝ −ˆ y along the +kxdirection, as in (E),
suggest a left-handed chirality (schematic in (B) and see
text for details).
Spin-texture of topological surface
Fig. S2. (A) Schematic of the bulk BZ of Sb and
its (111) surface BZ. The shaded region denotes the mo-
mentum plane in which the following ARPES spectra
were measured. (B) LEED image of the in situ cleaved
(111) surface exhibiting a hexagonal symmetry. (C) Se-
lect MDCs at EF taken with photon energies from 14 eV
to 26 eV in steps of 2 eV, taken in the TXLU momentum
plane. Peak positions in the MDCs were determined by
fitting to Lorentzians (green curves). (D) Experimental
3D bulk Fermi surface near H (red circles) and 2D surface
Fermi surface near¯Γ (open circles) projected onto the
kx-kz plane, constructed from the peak positions found
in (C). The kz values are determined using calculated
constant hν contours (black curves) (see APPENDIX C

Page 10

10
text).
Fermi surface calculated in (36).
The shaded gray region is the theoretical hole
Fig. S4. (A) Schematic of the surface band structure
of Sb(111) under a time reversal symmetric perturbation
that lifts the bulk conduction (La) band above the Fermi
level (EF). Here the surface bands near¯M are also lifted
completed above EF. (B) Alternatively the surface band
near¯M can remain below EF in which case it must be
doubly spin degenerate at¯M. (C) ARPES intensity plot
of the surface states along the -¯K−¯Γ−¯K direction. The
shaded green regions denote the theoretical projection of
the bulk valence bands, calculated using the full poten-
tial linearized augmented plane wave method using the
local density approximation including the spin-orbit in-
teraction (method described in 40). Along this direction,
it is clear that the outer V-shaped surface band that was
observed along the -¯M−¯Γ−¯M now merges with the bulk
valence band.
Fig.S6. Implications of k-space mirror symmetry
on the surface spin states. (A) 3D bulk Brillouin zone
and the mirror plane in reciprocal space. (B) Schematic
spin polarized surface state band structure for a mirror
Chern number (nM) of +1 and (C) -1. Spin up and down
mean parallel and anti-parallel to ˆ y respectively. The
upper (lower) shaded gray region corresponds to the pro-
jected bulk conduction (valence) band. The hexagons are
schematic spin polarized surface Fermi surfaces for differ-
ent nM, with yellow lines denoting the mirror planes. (D)
Schematic representation of surface state band structure
of insulating Bi1−xSbx and (E) semi metallic Sb both
showing a nM = −1 topology. Yellow circles indicate
where the spin down band (bold) connects the bulk va-
lence and conduction bands.