Experimental Demonstration of Topological Surface States Protected by Time-Reversal Symmetry

Abstract

We report direct imaging of standing waves of the nontrivial surface states
of topological insulator Bi$_2$Te$_3$ by using a low temperature scanning
tunneling microscope. The interference fringes are caused by the scattering of
the topological states off Ag impurities and step edges on the
Bi$_2$Te$_3$(111) surface. By studying the voltage-dependent standing wave
patterns, we determine the energy dispersion $E(k)$, which confirms the Dirac
cone structure of the topological states. We further show that, very different
from the conventional surface states, the backscattering of the topological
states by nonmagnetic impurities is completely suppressed. The absence of
backscattering is a spectacular manifestation of the time-reversal symmetry,
which offers a direct proof of the topological nature of the surface states.

Full text

arXiv:0908.4136v3 [cond-mat.mes-hall] 10 Oct 2009
Experimental demonstration of the topological surface states protected by the
time-reversal symmetry
Tong Zhang,1, 2 Peng Cheng,1Xi Chen,1, ∗Jin-Feng Jia,1Xucun Ma,2Ke He,2Lili
Wang,2Haijun Zhang,2Xi Dai,2Zhong Fang,2Xincheng Xie,2and Qi-Kun Xue1, 2, †
1Department of Physics, Tsinghua University, Beijing 100084, China
2Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
(Dated: October 10, 2009)
We report direct imaging of standing waves of the nontrivial surface states of topological insula-
tor Bi2Te3by using a low temperature scanning tunneling microscope. The interference fringes are
caused by the scattering of the topological states off Ag impurities and step edges on the Bi2Te3(111)
surface. By studying the voltage-dependent standing wave patterns, we determine the energy dis-
persion E(k), which confirms the Dirac cone structure of the topological states. We further show
that, very different from the conventional surface states, the backscattering of the topological states
by nonmagnetic impurities is completely suppressed. The absence of backscattering is a spectacular
manifestation of the time-reversal symmetry, which offers a direct proof of the topological nature of
the surface states.
PACS numbers: 68.37.Ef, 73.20.-r, 72.10.Fk, 72.25.Dc
The strong spin-orbital coupling in a certain class of
materials gives rise to the novel topological insulators in
two [1, 2] and three dimensions [3, 4, 5, 6, 7] in the
absence of an external magnetic field. The topological
states on the surfaces of three dimensional (3D) mate-
rials have been studied recently in Bi1−xSbx[6, 8, 9],
Bi2Te3and Bi2Se3[7, 10, 11, 12, 13], which possess
insulating gaps in the bulk and gapless states on sur-
faces. The surface states of a 3D topological insulator
comprise of an odd number of massless Dirac cones and
the crossing of two dispersion branches with opposite
spins is fully protected by the time-reversal-symmetry
at the Dirac points. Such spin-helical states are ex-
pected to bring forward exotic physics, such as magnetic
monopole [14] and Majorana fermions [15, 16]. To date,
the experimental study of topological insulators is pre-
dominantly limited to the determination of their band
structure by angle-resolve photoemission spectroscopy
(ARPES) [8, 9, 10, 11, 12, 13]. Distinct quantum phe-
nomena associated with the nontrivial topological elec-
tronic states still remain unexplored. Particularly, there
is no direct experimental evidence for the time reversal
symmetry that protects the topological property. Here,
using the low temperature scanning tunneling microscopy
(STM) and spectroscopy (STS), we report the direct ob-
servation of quantum interference caused by scattering of
the 2D topologically nontrivial surface states off impuri-
ties and surface steps. Our work strongly supports the
surface nature of the topological states, and provides a
way to study the spinor wave function of the topological
state. More significantly, we find that the backscattering
of topological states by a nonmagnetic impurity is forbid-
den. This result directly demonstrates that the surface
states are indeed quantum-mechanically protected by the
time reversal symmetry.
The interference patterns in STM experiments [17, 18,
19, 20] result from the 2D surface states perturbed by
surface defects. A surface state is uniquely characterized
by a 2D Bloch wave vector ~
kwithin the surface Brillouin
zone (SBZ). During elastic scattering, a defect scatters
the incident wave with a wave vector ~
kiinto ~
kf=~
ki+~q,
with ~
kiand ~
kfbeing on the same constant-energy con-
tour (CEC). The quantum interference between the ini-
tial and final states results in a standing wave pattern
whose spatial period is given by 2π/q . When the STM
images of a standing wave are Fourier transformed [21],
the scattering wave vector ~q (~~q is the momentum trans-
fer) becomes directly visible in the reciprocal space. In
contrast, for bulk states, there will be continuous ranges
of wave vectors on the projected SBZ for a given energy.
Usually, no distinct interference fringe can be produced
by bulk states and visualized by STM. In this sense, the
standing wave is surface-states-sensitive and particularly
suitable for studying topological insulators.
Our experiments were conducted in an ultra-high vac-
uum low temperature (down to 0.4 K) STM system
equipped with molecular beam epitaxy (MBE) for film
growth (Unisoku). The stoichiometric Bi2Te3film, a ro-
bust topological insulator, was prepared on single crystal
substrate Si(111) by MBE. Details of sample prepara-
tion are described elsewhere [22]. Shown in Fig. 1(a)
is a typical STM image of the Bi2Te3film with a thick-
ness of ∼100 nm. The atomically flat morphology of
the film is clearly observed. The three steps seen in
Fig. 1(a) all have the height (0.94 nm) of a quintuple
layer. The steps are preferentially oriented along the
three close-packing ([100], [110] and [010]) directions.
The image with atomic resolution [Fig. 1(b)] exhibits
the two-dimensional hexagonal lattice structure of the
Te-terminated (111) surface of Bi2Te3. Our STM obser-
vation further reveals a small density of clovershaped de-
fects on the surface (see supporting material [23]). Simi-

2
FIG. 1: (a) The STM topograph of the Bi2Te3(111) film. The
imaged area, 250 nm by 250 nm, was scanned at a sample
bias of 3 V and tunneling current of 50 pA. (b) The atomic-
resolution image. Tellurium atom (pink colored) spacing is
about 4.3 ˚
A. The image was scanned at a sample bias of -
40 mV and tunneling current of 0.1 nA. (c) dI/dV spectrum
taken on bare Bi2Te3(111) surface. The spectrum was mea-
sured with setpoint V=0.3 V, I=0.1 nA. The arrows indicate
the bottom of conduction band and the top of valence band,
respectively. (d) Calculated band structure of Bi2Te3(111)
along high-symmetry directions of SBZ (see the insert). The
red regions indicate bulk energy bands and the purple regions
indicate bulk energy gaps. The surface states are red lines
around the ¯
Γ point.
lar to Bi2Se3[13, 24, 25], these structures can be assigned
to the substitutional Bi defects at the Te sites by examin-
ing their registration with respect to the 1 ×1-Te lattice
in the topmost layer.
The surface states of Bi2Te3were investigated by STS
and the first-principles calculations [7]. The STS de-
tects the differential tunneling conductance dI/dV [Fig.
1(c)], which gives a measure of the local density of states
(LDOS) of electrons at energy eV. The two shoulders in-
dicated by arrows in Fig. 1(c) are the bottom of the
bulk conduction band and the top of the bulk valence
band, respectively. The Fermi level (zero bias) is within
the energy gap, indicating that the film is an intrin-
sic bulk insulator [22]. The differential conductance in
the bulk insulating gap linearly depends on the bias and
is attributed to the gapless surface states. These fea-
tures in STS are in good agreement with those obtained
by the first-principles calculations (see supporting mate-
rial [23]). According to the calculations [Fig. 1(d)], the
topological states of Bi2Te3form a single Dirac cone at
the center (¯
Γ point) of the SBZ [10, 11], giving rise to
a vanishing DOS in the vicinity of k= 0. However, the
surface states around ¯
Γ point overlap in energy with the
bulk valence band. For this reason, the Dirac point is
invisible in STS.
On the aforementioned surface, we deposited a small
amount (0.01 ML) of Ag atoms, which form trimmers on
the surface, as shown in Fig. 2(a) and more clearly in
supporting material [23]. The atomically resolved STM
image (see supporting material [23]) reveals that the Ag
atom in a trimmer adsorbs on the top site of a surface
Te atom [26]. This situation is schematically shown in
Fig. 2(b). In addition, the Fermi level shifts upwards
in energy by 20∼30 meV after Ag deposition, suggest-
ing electron transfer from the Ag atoms to the substrate
(see supporting material [23]). The dI/dV mapping was
then carried out in a region containing Ag trimmers. At
each data point, the feedback was turned off and the bias
modulation was turned on to record dI/dV . This proce-
dure resulted in a series of spatial mapping of LDOS at
various bias voltages.
Figures 2(c) to 2(g) summarize the dI/dV maps for
bias voltages ranging from 50 mV to 400 mV from the
area shown in Fig. 2(a). The first striking aspect of these
images is the existence of standing wave [17, 18, 19] in
the vicinity of the Ag trimmers. The spatial modulation
of LDOS by an Ag trimmer forms a hexagonal pattern,
whose edges are perpendicular to the ¯
Γ−¯
M directions in
SBZ. This situation is more clearly resolved at large bias
voltages [Figs. 2(f) and 2(g)]. As expected, the interfer-
ence pattern is anisotropic as a result of the hexagram
CEC [10]. The spatial period of the standing wave scales
inversely with the bias voltage and is determined by the
momentum transfer during scattering at a given energy.
Below 50 meV, the fringes become obscured. It results
from a combination of two effects: (i) The wavelength in-
creases rapidly as the bias voltage approaches the Dirac
point, where k= 0. (ii) At low energy, more topological
surface states with different wavelengths are involved in
the formation of standing wave as indicated by the first-
principles calculations [Fig. 1(d)]. The superposition of
waves with various wavelengths smears out the interfer-
ence fringes. With increasing bias, especially after the
surface states in the ¯
Γ−¯
M direction merge into the bulk
conduction band at ∼0.2 eV above the Dirac point ac-
cording to calculation [Fig. 1(d)], the contribution of
states in the ¯
Γ−¯
M direction vanishes and the states in
the vicinity of ¯
Γ−¯
K direction gradually gain more weight,
leading to more distinct interference patterns. After the
surface states in the ¯
Γ−¯
K direction merge into the bulk
conduction band at ∼0.6 eV above the Dirac point [Fig.
1(d)], the standing waves fade out again.
To quantify the standing waves and obtain the scat-
tering wave vectors, we performed Fourier transforma-
tion of the dI/dV maps into the ~q-space [Figs. 2(h) to
2(l)]. One important feature in the power spectra can be
immediately discerned by comparing the six-fold sym-
metric pattern in the ~q-space with SBZ (the red hexagon
in Fig. 2(h)): the regions with high intensity are al-
ways oriented toward the ¯
Γ−¯
M directions, while the

3
FIG. 2: Standing waves induced by Ag trimmers on Bi2Te3(111) surface. (a) STM image (28 nm by 28 nm) of a region with
four Ag trimmers adsorbed on Bi2Te3(111) surface. (b) The adsorption geometry of Ag trimmer. (c) to (g) The dI /dV maps
of the same area as (a) at various sample bias voltages. Imaging conditions: I=0.1 nA. Each map has 128 by 128 pixels and
took two hours to complete. The interference fringes are evident in the images. The green and red regions indicate modulations
with high intensity. (h) to (k) The FFT power spectra of the dI /dV maps in (c) to (g). The SBZ in (h) is superimposed on
the power spectra to indicate the directions in ~q-space. The resolution of FFT, which is 2π/28 nm−1, is determined by the size
of the STM image.
intensity in the ¯
Γ−¯
K directions vanishes (see support-
ing material [23]). Such phenomena can be understood
by exploring possible scattering processes on the CEC
in the reciprocal space [Fig. 3(a)]. Generally, the ~
ki
and ~
kfpairs with high joint DOS should dominate the
quantum interference. For energies at which the inter-
ference fringes are prominent, the regions on CEC with
high DOS are primarily centered around the ¯
Γ−¯
K di-
rections [10]. Therefore, three scattering wave vectors,
labeled ~q1,~q2and ~q3, might be expected to appear in the
power spectra. Among them, however, only ~q2is along
the ¯
Γ−¯
M directions and can generate the observed stand-
ing waves. Both ~q1and ~q3are invisible in the power spec-
tra. There is a simple argument to account for the disap-
pearance of ~q1: the time-reversal invariance. The topo-
logical states |~
k↑i and | − ~
k↓i are related by the time-
reversal transformation: |−~
k↓i =T |~
k↑i, where Tis the
time-reversal operator. It is straightforward to show that
h−~
k↓ |U|~
k↑i =−h~
k↑ |U| − ~
k↓i∗=−h−~
k↓ |U|~
k↑i = 0
for fermions, where Uis a time-reversal invariant oper-
ator and represents the impurity potential of the non-
magnetic Ag impurity. Therefore, the backscattering
between ~
kand −~
kis quantum-mechanically prohibited.
Most of the observed features in the interference pat-
tern, including the extinction of wave vector ~q3, have
been recently well explained by a full theoretical treat-
ment [27] based on the T-matrix approach for multiband
systems [28]. In addition to the existence of standing
waves, the absence of backscattering represents the sec-
ond and most striking aspect of our experiment, which
makes the topological standing waves more extraordinary
as compared to the conventional surface states on metal
samples [17, 18, 19, 20].
We can obtain the dispersion of the massless Dirac
FIG. 3: (a) The scattering geometry. The CEC is in the
shape of a hexagram. The dominant scattering wave vectors
connect two points in ¯
Γ−¯
K directions on CEC. ~
ki(pink arrow)
and ~
kf(red arrows) denote the wave vectors of incident and
scattered states. ~q1,~q2and ~q3(blue arrows) are three possible
scattering wave vectors. (b) Energy dispersion as a function
of kin the ¯
Γ−¯
K direction. The data (black squares) are
derived from FFT in Fig. 2. The red line shows a linear fit to
the data with vF= 4.8×105m/s. The error bars represent
the resolution of FFT (see the caption of Fig. 2).
fermions in the ¯
Γ−¯
K direction using the interference
patterns and their Fourier transforms. For ~q2, the scat-
tering geometry determines q2=√3k[see Fig. 3(a)],
where kis the wave vector in the ¯
Γ−¯
K direction at a
given energy. The resulting kvalues vary linearly with
energy [Fig. 3(b)]. The slope of the linear fitting provides
a measurement of the Dirac fermion velocity vF, which
is 4.8×105m/s. In addition, the energy of the Dirac
point is estimated to be 0.25 eV by the intercept of the
dispersion with the energy axis. These observations are
in agreement with the results from the first-principles cal-
culation and the ARPES data [7, 22]. More importantly,
the unoccupied states, which are inaccessible to ARPES,

4
can be probed by the standing waves with STM.
Interference fringes are also found at the step edges on
the surface [29] [Figs. 4(a) to 4(h)]. Similar to the case
of Ag trimmers, the standing waves produced by steps
are predominantly propagating along the ¯
Γ−¯
M direc-
tion. The fringes are clearly visible even at the negative
bias voltages probably owing to the stronger scattering
potential compared to that of the Ag trimmers. The dis-
persion curve deduced from these patterns again shows a
linear relation between the scattering wave vector and the
energy [Fig. 4(i)]. Using the slope of the linear fitting to-
gether with the same scattering geometry as that for the
Ag trimmer, the Fermi velocity is found to be 4.8×105
m/s, the same as that obtained from the standing waves
caused by the Ag impurities.
FIG. 4: Standing waves on the upper terrace by a step edge
((a) to (h)). All the images are dI/dV maps at various bias
voltages of an area of 35 nm by 35 nm. Imaging conditions:
I=0.1 nA. (i) Energy dispersion deduced from the standing
waves at the step edge. The dispersion is a function of the
scattering wave vector q. The inserted STM image shows the
step that produces the standing waves in (a) to (h).
The existence of standing wave strongly supports the
surface nature of topological states. An important issue
that immediately arises is whether the topological states
respond differently to the magnetic and the nonmagnetic
impurities. Theoretically, it was pointed out [5, 30, 31]
that a time-reversal breaking perturbation, such as mag-
netic impurities, can induce scattering between the states
|~
k↑i and −|~
k↓i and open up a local energy gap at the
Dirac point. It remains an open question to observe the
distinct signature of time-reversal breaking in topological
insulators.
We thank S.-C. Zhang, X.-L. Qi, Y. Ran and S.-Q.
Shen for valuable discussions. The work is supported
by NSFC and the National Basic Research Program of
China. The STM topographic images were processed us-
ing WSxM (www.nanotec.es).
Note added. At the completion of this manuscript for
submission, we became aware of related work by P. Rou-
san et al. [32]. The authors reported STM study of scat-
tering from disorder in BiSb alloy.
∗Electronic address: xc@mail.tsinghua.edu.cn
†Electronic address: qkxue@mail.tsinghua.edu.cn
[1] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science
314, 1757 (2006).
[2] M. K¨onig et al., Science 318, 766 (2007).
[3] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,
106803 (2007).
[4] J. E. Moore and L. Balents, Phys. Rev. B 75, 121306
(2007).
[5] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B
78, 195424 (2008).
[6] J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78,
045426 (2008).
[7] H. J. Zhang et al., Nature Phys. 5, 438 (2009).
[8] D. Hsieh et al., Nature 452, 970 (2008).
[9] D. Hsieh et al., Science 323, 919 (2009).
[10] Y. L. Chen et al., Science 325, 178 (2009).
[11] D. Hsieh et al., Nature 460, 1101 (2009).
[12] Y. Xia et al., Nature Phys. 5, 398 (2009).
[13] Y. S. Hor et al., Phys. Rev. B 79, 195208 (2009).
[14] X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science 323,
1184 (2009).
[15] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407
(2008).
[16] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys.
Rev. Lett. 102, 187001 (2009).
[17] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature
363, 524 (1993).
[18] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Science
262, 218 (1993).
[19] Y. Hasegawa and Ph. Avouris, Phys. Rev. Lett. 71, 1071
(1993).
[20] G. A. Fiete and E. J. Heller, Rev. Mod. Phys. 75, 933
(2003).
[21] J. E. Hoffman et al., Science 297, 1148 (2002).
[22] Y. Y. Li et al. (unpublished).
[23] See EPAPS for additional materials about the experi-
mental results.
[24] S. Urazhdin et al., Phys. Rev. B 66, 161306 (2002).
[25] S. Urazhdin et al., Phys. Rev. B 69, 085313 (2004).
[26] The other candidate model of the Ag trimmer is that the
Ag atom substitutes a topmost layer Te atom. In this
case, the formation of Ag trimmers is kinetically more
difficult. Although the exact structure of Ag trimmers
does not affect the main conclusion here, it remains an
interesting subject for further study, for example, by first-
principles calculation.
[27] W.-C. Lee, C. J. Wu, D. P. Arovas, and S.-C. Zhang,
arXiv:0910.1668 (2009).
[28] W.-C. Lee and C. J. Wu, arXiv:0906.1973 (2009).
[29] Z. Alpichshev et al., arXiv:0908.0371 (2009).

5
[30] Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang,
Phys. Rev. Lett. 102, 156603 (2009).
[31] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Nature Phys.
4, 273 (2008).
[32] P. Rousan et al., Nature 460, 1106 (2009).