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Topological Surface States Protected From
Backscattering by Chiral Spin Texture
Pedram Roushan1, Jungpil Seo1, Colin V. Parker1, Y. S. Hor2, D. Hsieh1, Dong
Qian1, Anthony Richardella1, M. Z. Hasan1, R. J. Cava2 & Ali Yazdani1
1Joseph Henry Laboratories & Department of Physics, Princeton University, Princeton,
NJ 08544
2Department of Chemistry, Princeton University, Princeton, NJ 08544.
Topological insulators are a new class of insulators in which a bulk gap for
electronic excitations is generated by strong spin-orbit coupling1-5. These novel
materials are distinguished from ordinary insulators by the presence of gapless
metallic boundary states, akin to the chiral edge modes in quantum Hall systems,
but with unconventional spin textures. Recently, experiments and theoretical
efforts have provided strong evidence for both two- and three-dimensional
topological insulators and their novel edge and surface states in semiconductor
quantum well structures6-8 and several Bi-based compounds9-13. A key
characteristic of these spin-textured boundary states is their insensitivity to spin-
independent scattering, which protects them from backscattering and localization.
These chiral states are potentially useful for spin-based electronics, in which long
spin coherence is critical14, and also for quantum computing applications, where
topological protection can enable fault-tolerant information processing15,16. Here
we use a scanning tunneling microscope (STM) to visualize the gapless surface
states of the three-dimensional topological insulator Bi1-x Sbx and to examine their
scattering behavior from disorder caused by random alloying in this compound.
Combining STM and angle-resolved photoemission spectroscopy, we show that
despite strong atomic scale disorder, backscattering between states of opposite
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momentum and opposite spin is absent. Our observation of spin-selective
scattering demonstrates that the chiral nature of these states protects the spin of
the carriers; they therefore have the potential to be used for coherent spin
transport in spintronic devices.
Angle-resolved photoemission spectroscopy (ARPES) experiments on the (111)
surface of Bi1-x Sbx crystals have been used to identify surface states within the bulk
band gap of these compounds9,10. The shape of the Fermi surface for these states shows
an odd number of band crossings between time-reversal invariant momentum points in
the first Brillion zone (FBZ) at the Fermi energy, which confirms the identification of
Bi1-x Sbx as a strong topological insulator for x > 7%. The odd number of crossings
protects the surface states from being gapped regardless of the position of the chemical
potential or the influence of non-magnetic perturbations4. Furthermore, spin-sensitive
experiments have established that these surface states have a chiral spin structure and an
associated Berry’s phase10, which makes them distinct from ordinary surface states with
strong spin-orbit coupling17. All these characteristics suggest that backscattering, or
scattering between states of equal and opposite momentum, which results in Anderson
localization in typical low-dimensional systems, will not occur for these two-
dimensional carriers. Random alloying in Bi1-x Sbx ,which is not present in other
material families of topological insulators found to date, makes this material system an
ideal candidate to examine the impact of disorder on topological surface states.
However, to date there have been no experiments that have probed whether these chiral
two-dimensional states are indeed protected from spin-independent scattering.
We performed our experiments using a home-built cryogenic STM that operates
at 4 K in ultrahigh vacuum. Single crystal samples of Bi0.92 Sb0.08 were cleaved in situ in
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UHV at room temperature prior to STM experiments at low temperatures. The
topographic images of the sample are dominated by long wavelength (~20Å)
modulations in the local density of states (Fig. 1a). However, atomic corrugation can
also be observed in the topography (inset Fig. 1a). Spectroscopic measurements show a
general suppression of the density of states near the Fermi level, with spectra appearing
for the most part homogenously across the sample surface (Fig. 1b). ARPES
measurements9,10 and recent band structure calculations13 suggest that within ±30meV of
the Fermi level, where there is a bulk gap, tunneling should be dominated by the surface
states. While tunneling spectroscopy measurements do not distinguish between bulk and
surface states, energy-resolved spectroscopic maps shown in Fig.’s 2a, b and c display
modulations that are the result of scattering of the surface electronic states. As expected
for the scattering and interference of surface states, the observed patterns are not
commensurate with the underlying atomic structure. While we do not have direct
information on the identity of the scattering defects, the random distribution of
substituted Sb atoms is a likely candidate.
Energy-resolved Fourier transform scanning tunneling spectroscopy (FT-STS)
can be used to reveal the wavelengths of the modulations in the local density of states
and to obtain detailed information on the nature of scattering processes for the surface
state electrons18. Previous studies on noble metal surface states18,19 and high-TC
superconductors20 have established the link between modulation in the conductance at a
wavevector q and elastic scattering of quasi-particles and their interference between
different momentum states ( k1 and k2, where q=k1+k2) at the same energy. The FT-
STS maps shown as insets in Fig.’s 1a,b and c for Bi0.92 Sb0.08 display rich qausi-particle
interference (QPI) patterns, which have the six-fold rotational symmetry of the
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underlying lattice, and evolve as a function of energy. These patterns display the
allowed wavevectors q and the relative intensities for the various scattering processes
experienced by the surface state electrons.
Within a simple model of QPI, the interference wavevectors connect regions of
high density of states on contours of constant energy (or the Fermi surface at the
chemical potential). Therefore the QPI patterns should correspond to a joint density of
states (JDOS) for the surface state electrons that can be independently determined from
ARPES measurements20-23. Fig. 2d and e show contours of constant energy (CCE) in the
FBZ, as measured with ARPES at two energies on Bi0.92 Sb0.08 crystals (following
procedures described in Ref 9). The CCE consist of an electron pocket centered on the
Γpoint, hole pockets half way to the
M
point, and two electron pockets that occur very
close to the
M
point9,10. From these measurements, we can determine the JDOS as a
function of the momentum difference between initial and final scattering states, q, using
kdqkIkIqJDOS 2
)()()( ∫+= , where I(k) is the ARPES intensity that is proportional
to the surface states’ density of states at a specific two-dimensional momentum k. Fig.
3a and b shows the results of computation of the JDOS from ARPES data for two
different energies. Contrasting these figures to the corresponding QPI data in Fig. 3b
and 3e, we find a significant suppression of the scattering intensity along the directions
equivalent to Γ-
M
in the FBZ. Backscattering between various electron and hole
pockets around the Γpoint should give rise to a continuous range of scattering
wavevectors along the Γ -
M
direction, a behavior not observed in the data (see also
the expanded view of the JDOS and QPI in Fig. 4a). This discrepancy suggests the
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potential importance of the surface states’ spin texture and the possibility that spin rules
are limiting the backscattering for these chiral electronic states.
To include spin effects, we use the results of spin-resolved ARPES studies and
assign a chiral spin texture to the electron and hole pockets as shown in Fig. 2f. ARPES
studies have resolved the spin structure only for the central electron pocket and the hole
pockets near the
Γ
point in the FBZ; however, we assign a similar chiral structure for
states near
M
. This assignment is consistent with the presence of a π Berry’s phase that
distinguishes the spin topology of the Bi0.92 Sb0.08 surfaces states from that of surface
state bands that are simply split by spin-orbit coupling. In the latter case, the spin-
polarized surface bands come in pairs, while for topological surface states there should
be an odd number of spin-polarized states between two time reverse equivalent points in
the band structure. 1,2, 11,13
To understand scattering and interference for these spin-polarized states, we
determine the spin-dependent scattering probability, kdqkIkqTkIqSSP 2
)(),()()(
∫
+= ,
which in similar fashion to the JDOS uses the ARPES-measured density of states, )(kI ,
but also includes a spin-dependent scattering matrix element ),( kqT . This matrix
element describes the scattering probability as a function of momentum transfer and
spins of states that are connected by the scattering process. Shown in Fig. 3c and f are
the calculated )(qSSP from ARPES data at two different energies using a matrix
element of the form
2
)()(),( qkSkSkqT += . This simple form of spin-selective
scattering reduces scattering between states with nonaligned spins and completely
suppresses scattering between states with opposite spin orientations. Comparison of the
SSP patterns to the QPI measurements in Fig. 3 shows that including spin effects leads
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to remarkably good agreement between the scattering wavevectors measured by STM
and those expected from the shape of the surface CCE as measured by ARPES (features
in the FT-STS and SSP at different wavevectors are categorized and given labels in Fig.
3g). A quantitative comparison between the QPI from the STM data and JDOS and SSP
from ARPES data can be made by computing the cross correlation between the various
patterns. Focusing on the high-symmetry direction, which is shown in Fig. 4a, we find
that the QPI (excluding the central q=0 section, which is dominated by the disorder) is
95% correlated with the SSP in the same region. The cross-correlation is found to be
83% between the QPI and JDOS. Therefore, the proposed form of the spin-dependent
scattering matrix element is the critical component for understanding the suppression of
scattering along the high symmetry directions in the data.
The proposed scattering matrix elements and associated spin-scattering
rules are further confirmed by a more comprehensive analysis of the QPI patterns. An
example of such an analysis is shown in Fig. 4, in which we associate features along the
high symmetry direction in the QPI and SSP with specific scattering wavevectors q that
connect various regions of the CCE. The observed wavevectors in QPI and SSP obey
spin rules imposed by , as illustrated schematically in Fig. 4b. We also depict in
Fig. 4b examples of scattering processes that, while allowed by the band structure and
observed in JDOS, violate the spin scattering rules and are not seen in QPI data in Fig.
4a. A comprehensive analysis of all the features in the QPI data (included in the
supplementary section) demonstrates that allowed set of scattering wavevectors 1
q
through 8
q (Fig. 3g) exclude those that connect states with opposite spin. Finally, in
Fig. 5, we show dispersion as a function of energy for some of the wavevectors q in the
),( kqT
),( kqT
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QPI and compare their energy dispersion to that expected from ARPES results in the
SSP. Remarkably, all the features of the complex QPI patterns and their energy
dependence can be understood in detail by the allowed scattering wavevectors based on
the band structure of the topological surface states and the spin scattering rule. This
agreement provides a precise demonstration that scattering of electrons over thousands
of Angstroms, which underlies the QPI maps, strictly obeys the spin scattering rules and
associated suppression of backscattering.
Other surface states with strong spin-orbit interaction may be expected to show
evidence for spin-selective scattering; however, since spin states come in pairs, the QPI
patterns can rarely probe these rules24. In some situations there is evidence of such
rules25,26, but the precision with which scattering of surface states for Bi0.92 Sb0.08 can be
understood using spin-selective scattering is unprecedented. Unusual scattering of chiral
electronic states is also seen in monolayer graphene, where the underlying two-atom
basis leads to a pseudospin index for quasi-particles and results in suppression of
intravalley scattering27,28. The key difference expected for surface states of a topological
insulator is the degree to which they can tolerate disorder. This aspect is clearly
demonstrated here for surface states of Bi0.92 Sb0.08 , where strong alloying causes
scattering for the surface state electrons yet the spin-selection scattering rules are strictly
obeyed over length scales much longer than that set by the atomic scale disorder. Future
experiments with magnetic scattering centers can further probe the spin scattering rules
for topological surface states and may provide the setting for the manipulation of these
spin-polarized states in device applications.
Methods Summary
The Bi0.92Sb0.08 single crystals were grown by melting stoichiometric mixtures of
elemental Bi(99.999%) and Sb(99.999%) in 4mm inner diameter quartz tubes. The
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samples were cooled over a period of two days, from 650 to 260 ˚C, and then annealed
for a week at 260 ˚C. The samples were cleaved in situ in our home-built cryogenic
STM that operates at 4 K in ultra-high-vacuum. The STM topographies were obtained
in constant-current mode, and dI/dV spectroscopy was measured by a standard lock-in
technique with f=757Hz, and an AC modulation of 3mV added to the bias voltage. The
spatial resolution during the dI/dV mapping was about 2Å, which provided the
capability to resolve k-vectors up to twice the first Brillion zone in momentum space.
The spin-resolved ARPES measurements were performed at the SIS beam line at the
Swiss Light Source using the COPHEE spectrometer with a single 40 kV classical Mott
detector and photon energies of 20 and 22 eV. The typical energy resolution was 80mV,
and momentum resolution was 3% of the surface BZ. The JDOS formula provides a
map of all scatterings by calculating the self-convolution of a given CCE. While JDOS
disregards the spin texture of a CCE, the SSP formula considers a weight factor for each
possible scattering proportional to the square of the projection of the initial spin state
onto the final state.
We gratefully acknowledge K. K. Gomes and A.N. Pasupathy for important suggestions on experimental
procedure and initial analysis. This work was supported by grants from ONR, ARO, NSF-DMR, and
NSF-MRSEC programme through Princeton Center for Complex Materials. P.R. acknowledges support
of NSF graduate fellowship.
Author Contributions Y. S. Hor and R. J. Cava carried out the growth of the single crystals and
characterized them. D. Hsieh, D. Qian and M.Z. Hasan performed the ARPES studies of the samples.
STM measurements as well as data analysis was done by Pedram Roushan, Jungpil Seo, Colin V. Parker,
Anthony Richardella and Ali Yazdani.
Correspondence and requests for materials should be addressed to A.Y. (yazdani@princeton.edu).
References
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Figure 1. STM topography, and dI/dV spectroscopy of the Bi0.92Sb0.08 (111)
surface. a, STM topograph (+50meV, 100pA) of the Bi0.92Sb0.08 (111) surface
over an 800 Å by 800 Å area. The inset shows an area of 80 Å by 80 Å that
displays the underlying atomic lattice (+200mV, 15pA). b, Spatial variation of
the differential conductance (dI/dV) measurements across a line of length 250Å.
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A typical differential conductance measurement over larger energy ranges is
shown in the inset.
Figure 2. dI/dV maps, QPI patterns, and ARPES measurements on
Bi0.92Sb0.08 (111) surface. a, b, and c, Spatially resolved conductance maps of
the Bi0.92Sb0.08 (111) surface obtained at -20 mV, 0 mV, and +20 mV (1000 Å x
1300 Å). In the upper right corner of each map the Fourier transform of the
dI/dV maps are presented. The hexagons have the same size as the FBZ. The
Fourier transforms have been symmetrized in consideration of the three-fold
rotation symmetry of the (111) surface. d and e, ARPES intensity map of the
surface state at -20mV and at the Fermi level, respectively. f, The spin textures
from ARPES measurements are shown with arrows, and high symmetry points
are marked (Γand 3
M
).
Figure 3.Construciton of joint density of states (JDOS) and spin scattering
probability (SSP) from ARPES data and their comparison with FT-STS.
a, the JDOS and SSP calculated at EF, from ARPES data presented in Fig. 2e.
b, the FT-STS at EF. c, the SSP calculated at EF. d, the JDOS calculated at
-20mV, from ARPES data presented in Fig. 2d. e, the FT-STS at -20mV. f, the
SSP calculated at -20mV. g, the schematization of the features associated with
scattering wavevectors 1
q to 8
q in the FT-STS data.
Figure 4. Comparison of the various parts of the QPI patterns along the
Γ
-
M
direction at Fermi level. a, close up view of the QPI pattern from JDOS,
FT-STS, and SSP at Fermi level, along the
Γ
-
M
direction. The last row shows
the schematic representation of q2,q4, and q8 , which correspond to scatterings
shown in b. Two high intensity points which are only seen in JDOS are labeled
as A and B. b, the Fermi surface along the
Γ
-
M
direction, with spin orientation
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of the quasiparticles shown with arrows. The horizontal color-coded arrows
show the sources of the scatterings seen in the STM data. Note that all
highlighted spins have the same orientation. The top row depicts the scatterings
which involve opposite spins and are presented in JDOS, but absent in FT-STS,
and SSP.
Figure 5. Dispersion of various peaks from FT-STS and ARPES. a, the
intensity of the FT-STS maps along the
Γ
-
M
direction for various energies. The
two peak positions correspond to q1 and q2, which become larger with
increasing in energy. Each curve is shifted by 0.6 pS for clarity. b, dispersion of
the position of q1, q2, and q3 from ARPES (open symbols) and STM (solid
symbols). The data were obtained from fitting the peak in the intensity of the
QPI patterns measured in STM, and calculated from the ARPES CCE. Each
STM data point is the averaged value of six independent measurements, and
the error bar represents one standard deviation. The systematic error was
negligible.
Methods
Crystal growth The Bi0.92Sb0.08 single crystals were grown by melting stoichiometric
mixtures of elemental Bi(99.999%) and Sb(99.999%) in 4mm inner diameter quartz
tubes from a stoichiometric mixture of high purity elements. The samples were cooled
over a period of two days, from 650 to 260 ˚C, and then annealed for a week at 260 ˚C.
The obtained single crystals were confirmed to be single phase and identified as having
the rhombohedral A7 crystal structure by X-ray power diffraction using a Bruker D8
diffractometer with Cu Kα radiation and a graphite diffracted beam monochromator.
STM measurement We performed our experiments using a home-built cryogenic STM
that operates at 4 K in ultrahigh vacuum. With our STM, we have examined several
crystals of Bi0.92Sb0.08, grown under the same conditions, and we have not noticed any
sample dependence for any of the results we are presenting in this article. The typical
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size of the crystals we used was 1 mm×1 mm×0.3 mm. Samples were cleaved in situ at
room temperature in UHV prior to STM experiments at low temperatures. The weak
bonding between bilayers in this crystal makes the (111) surface the natural cleavage
plane. A mechanically sharpened Pt–Ir alloy wire was used as an STM tip, and the
quality of the tip apex was examined by scanning an atomically clean Ag(111) surface.
The STM topographies were obtained in constant-current mode, and dI/dV
spectroscopy was measured by a standard lock-in technique with f=757Hz, an ac
modulation of 3mV added to the bias voltage, and the feed-back loop disabled during
the measurement. The spatial resolution during the dI/dV mapping was about 2Å, which
provides the capability to resolve k-vectors up to twice the first Brillion zone in
momentum space. The real space resolution also guarantees inclusion of atomic peaks
in FT-STS maps, providing the most accurate calibration. In addition, the deviation of
these atomic peaks from a perfect hexagon can be used as a measure of the thermal
drift, which for the results presented was negligible, allowing us to symmetrize the FT-
STS results without smearing out features or creating artificial ones.
ARPES measurements High resolution ARPES and spin-resolved ARPES have been
measured in different labs. High-resolution ARPES data have been taken at beamlines
12.0.1 and 10.0.1 of the Advanced Light Source at the Lawrence Berkeley National
Laboratory, as well as at the PGM beamline of the Synchrotron Radiation Center in
Wisconsin, with photon energies ranging from 17 eV to 55 eV and energy resolutions
ranging from 9 meV to 40 meV, and momentum resolution better than 1.5% of the
surface Brillouin zone (BZ). Spin-integrated 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 Scienta 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 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 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 pass energy of 3 eV.
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JDOS and SSP calculation Two plausible assumptions were considered for scattering
between quasiparticles, spin independent (JDOS) and spin dependent scattering (SSP).
The JDOS formula provides a map of all scatterings by calculating the self-convolution
of a given CCE, and hence disregards the spin texture of a CCE. The SSP formula
considers a weight factor for each possible scattering, proportional to the square of the
projection of the initial spin state onto the final state. Therefore, vectors connecting
quasiparticles of opposite spins do not contribute to the SSP mapping. The mean
subtracted correlation of the FT-STS to the JDOS and SSP was calculated in a
rectangular region starting 0.08Å-1 away from the
Γ
point and ending at
M
, and of
width 0.22Å-1. The calculation of SSP using various subsets of the pockets of the Fermi
surface enabled us to identify scattering processes which give rise to the various q’s (see
supplementary Fig. 1S). By using a Gaussian fit to the peak of the intensity, we were
able to follow the energy evolution of several q’s in the FT-STS maps, (see fig. 5 of the
main article). A similar fitting procedure was applied to the ARPES intensity maps at
fixed energies.
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Figure 1
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Figure 2
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Figure 3
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Figure 4
20
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Figure 5
Supplementary figure
Figure S1. Decomposition of the various parts of the QPI pattern at the Fermi
level. a.I, the Fermi level QPI measured with STM. The individual features are
highlighted in a.II to a.V. b.I, the SSP calculation of the QPI pattern from ARPES
intensity maps at Fermi level, and its decomposition into various constituent parts is
shown in b.II to b.V. c.I, the schematization of the various features seen in the FT-STS
data. c.II to c.V, various parts of the Fermi contours measured by ARPES, with arrows
showing the sources of the scatterings seen in STM data. Columns II to V have the
following order: in c, we show a specific part of the Fermi surface, and in b the SSP
based of that part is presented. In a, the corresponding parts of the pattern which are
visible in STM data are in color and the rest are shown in gray. In the legend, the
ARPES intensity map and its spin texture at the Fermi level is shown.
