of 6
Observation of Fermi-energy dependen
t unitary impurity resonances in
a strong topological insulator Bi
2
Se
3
with scanning tunneling
spectroscopy
M. L. Teague
1
, H. Chu
1
, F.-X. Xiu
2,3
, L. He
2
, K.-L. Wang
2
, N.-C. Yeh
1,4*
1
Department of Physics,California Institute of Technology, Pasadena,CA 91125, USA
2
Department of Electrical Engineering, University of California, Los Angeles, CA 90095, US
3
Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA
4
Kavli Nanoscience Institute, California Institute of Technology, Pasadena,CA 91125, USA
Received January 15 2012
Abstract
Scanning tunneling spectroscopic studies of Bi
2
Se
3
epitaxial films on Si (111) substrates
reveal highly locali
zed unitary impurity
resonances associated with
non-magnetic quantum impurities. The strength of
the resonances depends on
the energy difference bet
ween
the Fermi level (
E
F
) and the Dirac point (
E
D
) and diverges as
E
F
approaches
E
D
. The Dirac-cone surface state of the host recovers within
׽
2Å spatial distance from impurities, suggesting robust topological protection of the surface state of topological insulators a
gainst high-
density impurities that preserve time reversal symmetry.
PACS:
73.20.-r, 68.37.Ef, 72.10.Fk
Keywords:
topological insulators; Dirac fermions; impurity resonances; scanning tunneling spectroscopy
*
Corresponding author. E-mail address: ncyeh@caltech.edu
An exciting development in modern condensed matter
physics is the beautiful manifestation of topological field
theories in strongly correlated electronic systems, where
topological field theories [1] are shown to provide a
classification of order due to macroscopic entanglement that is
independent of symmetry breaking [2]. The fractional
quantum Hall (FQH) state is the first known example of such a
quantum state that exhibits no spontaneous broken symmetry
and has properties depending only on its topology rather than
geometry [2]. Recently, a new class of time-reversal symmetry
protected topological states known as the quantum spin Hall
(QSH) states or the topological insulators (TI) has emerged
and stimulated intense research activities [3,4].
One of the novel properties associated with the TI is the
presence of a Dirac spectrum of chiral low-energy excitations,
which is a salient feature of the Dirac materials that exploits
the mapping of electronic band structures and an embedded
spin or pseudo-spin degree of freedom onto the relativistic
Dirac equation [3–9]. These materials, including graphene [9]
and the surface state (SS) of three dimensional (3D) strong
topological insulators (STI) [3–8], have emerged as a new
paradigm in condensed matter for investigating the topological
phases of massless and massive Dirac fermions. In the case of
3D-STI, an odd number of massless Dirac cones in their SS is
ensured by the Z
2
topological invariant of the fully gapped
bulk [3–8]. Backscattering of Dirac fermions is suppressed
due to topological protection that preserves the Dirac
dispersion relation for any time
-reversal invariant perturbation
[3,4]. Thus, 3D-STI are promising materials for applications in
areas of spintronics [3,4,10] and topological quantum
computation [3,4,11] if their SS exhibit sufficient stability to
impurities [12,13].
While direct backscattering is prohibited in both the SS of
3D-STI and in graphene, sharp resonances are not excluded
because Dirac fermions with a finite parallel momentum may
be confined by potential barriers [9]. In fact, theoretical
calculations for Dirac fermions in the presence of non-
interacting impurities have predicted the occurrence of strong
impurity resonances [12, 13]. Nonetheless, no direct empirical
observation of strong resonances has been demonstrated to
date despite numerous reports of spectral evidences for
quasiparticle interferences associated with impurity or step-
edge induced scattering [14–16]. In this letter we report direct
scanning tunneling spectroscopic (STS) observation of
impurity resonances in a 3D-STI system, Bi
2
Se
3
. We find that
the strength of non-magnetic impurity resonances appears
strongly dependent on the energy difference between the
Fermi level (
E
F
) and the Dirac point (
E
D
) and diverges as
E
F
E
D
. The impurity resonances occur near
E
D
and are
localized within a small region of a radius
r
0
׽
2Å so that the
SS spectra of the host remain undisturbed even for high-
density unitary impurities. These findings suggest that the SS
of a 3D-STI is topologically well protected against impurities
that preserve time reversal symmetry. Moreover, the absence
of strong impurity resonances in other spectroscopic studies
[14–17] may be attributed to the large energy difference
between
E
F
and
E
D
that led to substantial screening of the
impurity states.
2
The samples investigated in this work are epitaxial Bi
2
Se
3
films grown on Si(111) by molecular beam epitaxy (MBE).
Details of the growth process have been described elsewhere
[18]. Transmission electron microscopy (TEM) on these films
exhibited perfect triangular lattice structures, and ARPES
(angle resolved photoemission spectroscopy) studies revealed
a single Dirac cone [18]. Figure 1(a) shows an atomic force
microscope (AFM) image of an as-grown Bi
2
Se
3
epitaxial film
with an average thickness of 44 quintuple layers (QLs). The
film surface consists of large triangle-shaped flat terraces,
reflecting the hexagonal crystalline structure inside the (0001)
plane. The height of each terrace is
׽
0.95 nm, corresponding
to a single QL thickness. The typical lateral dimension of the
top layer ranges from 150 to 350 nm, and the width of each
subsequent terrace is 70
׽
90 nm.
Fig. 1. Characteristics of MBE-grown Bi
2
Se
3
epitaxial films on
Si(111):
(a)
AFM image of a sample with a mean thickness of 44
Fig. 1. Characteristics of MBE-grown Bi
2
Se
3
epitaxial films on
Si(111):
(a)
AFM image of a sample with a mean thickness of 44
QLs, showing triangle-shaped flat terraces.
(b)
Main panel:
Comparison of the typical tunneling conductance spectra of two
Bi
2
Se
3
films of 60-QL and 7-QL thicknesses. The Dirac point
E
D
shifts away from the Fermi level
E
F
= 0 with decreas
ing thicknesses.
Inset: A representative tunneling spectrum for a 3-QL sample,
showing opening of an energy gap around
E
F
.
(c)
Histogram of
E
D
in
the 60-QL sample.
(d)
Histogram of
E
D
in the 7-QL sample
comparable results.
After MBE-growth, samples were transferred to the
cryogenic probe of a homemade scanning tunneling
microscope (STM). The sealed STM assembly was evacuated
and cooled to either 6 K or 77 K in ultra-high vacuum. Both
spatially resolved topography and normalized tunneling
conductance (
dI
/
dV
)/(
I
/
V
) vs. energy (
E
=
eV
) spectroscopy
were acquired pixel-by-pixel simultaneously, with tunneling
currents perpendicular to the sample surface, and the typical
junction resistance was
׽
1 G
. Detailed survey of the
surface topography and tunneling conductance spectra was
carried out over typically (8×8) nm
2
areas, and each area was
subdivided into (128×128) pixels.
Generally the normalized tunneling conductance spectra
in our STS studies were found consistent throughout a flat
area. Representative point spectra for the 60-QL and 7-QL
samples are given in the main panel of Fig. 1(b), and the
ranges of the Dirac energy for all areas investigated are
E
D
=
(
73±38) meV and
E
D
= (
100±25) meV, respectively, as
summarized by the histograms of the Dirac energies shown in
Figs. 1(c) and 1(d). In contrast, a typical spectrum for the 3-
QL sample (inset of Fig. 1(b)) reveals apparent opening of an
energy gap (
׽
0.4 eV) around
E
F
= 0.
Fig. 2. (Color online)
(a)
Atomically resolved (
dI
/
dV
) map of a 60-
QL sample at
E
=
89 meV.
(b)
Atomically resolved (
dI
/
dV
) map of
a 7-QL sample at
E
=
130 meV.
(c)
Fourier transformation (FT) of
the (
dI
/
dV
) map for
E
=
30 meV over the same area shown in (a) for
the 60-QL sample, showing a circular
diffraction ring consistent with
the SS dispersion relation for
E
v
<
E
<
E
c
where
E
v
and
E
c
refer to the
top of the bulk valence band and the bottom of the bulk conduction
band, respectively, and (
E
c
E
v
)
׽
300 meV, and the reciprocal space
units are given in the convention of (2
) over lattice constants.
(d)
FT
conductance map for
E
=
50 meV over the same area shown in (b)
for the 7-QL sample, showing a circular diffraction ring consistent
with the SS dispersion relation for
E
v
<
E
<
E
c
.
(e)
Schematic
illustration of the energy dispersion relations associated with the bulk
and surface states of the 3D-STI Bi
2
Se
3
, showing an apparently
circular Fermi surface for
E
v
<
E
<
E
c
.
(f)
FT conductance map for
E
=
300 meV over the same area shown in (a) for the 60-QL sample,
showing clear first-order and weak second-order Bragg diffraction
spots for
E
<
E
v
.
(g)
FT conductance map for
E
=
300 meV over the
same area shown in (b) for the 7-QL sample, showing first-order
Bragg diffraction spots for
E
<
E
v
.
(h)
Schematic energy dispersion
relations associated with the bulk and surface states of the 3D-STI
Bi
2
Se
3
, showing dominating bulk contributions for
E
<
E
v
.
Despite the relatively consistent tunneling spectra for
most areas in view, we note the presence of a few atomic
impurities, as manifested by the localized high conductance
spots in Fig. 2(a) for the 60-QL sample and in Fig. 2(b) for the
3
7-QL sample. On the other hand, unlike other 3D-STI (e.g.,
Bi
2
Te
3
and Bi
1
x
Sb
x
) with more complicated Fermi surfaces
that lead to SS deviating from a perfect Dirac cone as well as
impurity-induced quasiparticle interferences (QPI) for
sufficiently high-energy quasiparticles [14–16], the SS of
Bi
2
Se
3
does not exhibit discernible QPI because the perfect
single Dirac cone prevents backscattering. Hence, the Fourier
transformation (FT) of the
conductance maps primarily
exhibited round spots in the center of the FT conductance
maps and very faint Bragg diffraction spots at low energies
within the surface state, as show
n in Figs. 2(c)-(e). Eventually
significant Bragg diffraction peaks appear in the reciprocal
space for energies merged into the bulk state, as manifested in
Figs. 2(f)-(h) for the FT conductance maps taken at
E
=
300
meV on the 60-QL and 7-QL samples, respectively. In
contrast, QPI with wave-vectors smaller than the reciprocal
lattice constants were found in samples thinner than 6 QLs
[21] due to modified SS as the result of wave-function
overlapping and Rashba-type spin-orbit splitting between the
top and bottom surfaces of the thin film [19,20]. Given that the
energy dispersion relation for the 3-QL sample deviates from
that of a Dirac cone, in the following we focus our studies of
the impurity resonances only on the 60-QL and 7-QL samples.
To investigate the spectral evolution associated with the
presence of these quantum impurities, we show in Figs. 3(a)-
(f) different line-cuts across a (5×8) nm
2
constant-bias
conductance map and the corresponding spectra for the 60-QL
sample. For a line-cut along an area without impurities as
exemplified in Fig. 3(a), the tunneling spectra are generally
consistent everywhere, showing a Dirac energy
E
D
= (
35±10)
meV slightly below the Fermi level. On the other hand, the
tunneling spectra directly above quantum impurities reveal
strong resonant conductance peaks at
E
׽
E
D
. Moreover, these
resonant peaks are spatially confined to a region of
׽
2Å in
radius, as shown by the spectra along various line-cuts in Figs.
3(b)-(d) and further elaborated in Fig. 5(a). These spectral
characteristics clearly reveal that the SS of the host recovers
rapidly from impurities.
Interestingly, for quantum impurities separated by only one
lattice constant, the spectral characteristics for the inter-
impurity region exhibit strong interferences for energies deep
into the bulk valence band while the SS spectra have restored
to that of the host, as exemplified in Figs. 3(e)-(f). These
findings therefore imply strong topological protection of the
SS against impurities even in the limit of significant effects on
the bulk state. For comparison, while similar quantum
impurities are observed in the 7-QL sample, the intensity of
the impurity resonances is much reduced, as illustrated in Figs.
4(a) and 4(b). As discussed later, this weakened impurity
resonance may be attributed to the larger energy separation
(
E
F
E
D
)
E
(see Fig. 1(b) and Fig. 4(c)).
To better quantify the spatial confinement and energy
dependence of the impurity resonance, we illustrate in Fig.
5(a) the spatial dependence (r) of the tunneling conductance
near one of the isolated impurities in the 60-QL sample. For
E
׽
E
D
, we find strong resonance in the tunneling conductance
over a very narrow spatial range
r
׽
±2Å, as illustrated by the
solid curve in Fig. 5(a). On the other hand, for
E
<
E
D
but still
within the SS, the spectral resonance diminishes rapidly, as
shown by the curve of black symbols in Fig. 5(a). Similarly,
no impurity resonance is visible for energies deep into the
bulk valence band, as exemplified by the (
dI
/
dV
)-
vs.
-
r
curve
taken at
E
=
275 meV (red symbols) in Fig. 5(a). In the case
of two closely located impurities, we find that the impurity
resonances at
E
׽
E
D
remains strongly localized spatially
(solid blue curve in Fig. 5(b)). Moreover, the SS spectrum of
the small intermediate region between two impurities appears
to fully recover to that of the host (black symbols in Fig. 5(b)),
whereas the bulk spectrum (
E
=
400 meV) for the same
intermediate region exhibits strong interferences, as
exemplified by the red symbols in Fig. 5(b) and also in Fig.
3(e). The rapid recovery of the SS spectrum from impurities
may be understood as the result of topological protection of
the SS in Bi
2
Se
3
, even in the limit of high-density impurities.
Fig. 3. (Color online) Spectral evolution along various line-cuts of a
(5×8) nm
2
area of a 60-QL sample at
T
= 77 K, where the white
dotted line in each upper panel represents a line-cut across an
atomically resolved constant-energy conductance (
dI
/
dV
) map, and
the corresponding (
dI
/
dV
)-
vs
.-
E
spectra along the line-cut are given in
the lower panel:
(a)
Across an impurity-free region;
(b)
Across a
single impurity;
(c)
Across two impurities;
(d)
Across an isolated
impurity;
(e)
Between two closely spaced impurities along the
horizontal direction;
(f)
Between two closely spaced impurities along
the vertical direction.
Similarly, for the 7-QL sample
with a larger value of |
E
|,
the impurity resonance at
E
׽
E
D
for either an isolated
impurity or two closely spaced impurities is also highly
localized, as exemplified in Figs. 5(c)-(d). Moreover, the SS
spectrum recovers rapidly and the effect of adjacent impurities
4
on the bulk valence band diminishes significantly (Fig. 5(d))
relative to that of the 60-QL sample (Fig. 5(b)), probably due
to stronger screening associated with a larger |
E
| value in the
7-QL sample.
Fig. 4. (Color online) Spectral evolution along various line-cuts of a
(5.1×8.8) nm
2
area of a 7-QL sample at
T
= 77 K:
(a)
Atomically
resolved constant-bias conductance map for
E
= 5 meV;
(b)
(
dI
/
dV
)-
vs.
-
E
spectra along the slanted dotted line in (a) that cuts across two
point impurities;
(c)
(
dI
/
dV
)-
vs.
-
E
spectra along an impurity-free
region represented by the vertical dashed line in (a).
Fig. 5. (Color online) Spatial distribution and energy dependence of
the impurity resonances for 60-QL and 7-QL samples:
(a)
(
dI
/
dV
)
vs.
spatial distance (
r
) spectrum of a 60-QL sample from the center of an
isolated impurity for
E
׽
E
D
(blue solid curve), <
E
D
(within the SS,
black solid circles) and
ا
E
D
(in the bulk valence band, red open
diamonds).
(b)
(
dI
/
dV
)-
vs.
-
r
spectrum of a 60-QL sample from the
center of two adjacent impurities for
E
׽
E
D
(blue solid curve), <
E
D
(black solid circles) and
ا
E
D
(red open diamonds).
(c)
(
dI
/
dV
)-
vs.
-
r
spectrum of a 7-QL sample from the center of an isolated impurity
for
E
׽
E
D
(blue solid curve), <
E
D
(black solid circles) and
ا
E
D
(red open diamonds). All spectra reveal slight conductance
modulations associated with the underlying atomic lattice structure.
(d)
(
dI
/
dV
)-
vs.
-
r
spectrum of the 7-QL sample from the center of two
adjacent impurities for
E
׽
E
D
(blue solid curve), <
E
D
(black solid
circles) and
E
ا
E
D
(red open diamonds).
To understand the quantitative dependence of impurity
resonances on
E
, we follow the Keldysh Green function
formalism detailed in Ref. [13] for tunneling conductance
above a non-magnetic impurity in graphene, which may be
applied to the SS tunneling conductance
g
imp
in 3D-STI by
reducing the conductance contributions from two sublattices in
graphene to one Dirac cone in Bi
2
Se
3
. Specifically, the
Hamiltonian for the low energy Dirac quasiparticles of
topological insulators may be modeled by considering the
following contributions [13]:
(1)
where
H
TI
= (
σ
p
) is the Dirac Hamiltonian for the SS of a
3D-STI (with
σ
and
p
denoting the spin and momentum
operators, respectively),
H
imp
is the impurity Hamiltonian,
H
tip
is the Hamiltonian for the STM tip, and the Hamiltonians
H
TI
imp
,
H
tip
TI
and
H
tip
imp
describe hopping between TI and
the impurity electrons, between TI and the STM tip electrons,
and the STM tip electrons and the impurity, respectively [13].
Given the Hamiltonian
H
in Eq. (1), the time (
t
) dependent
tunneling current
I
(
t
) may be expressed by the formula:
(2)
where
N
tip
denotes the number operator of the tip electrons.
Assuming non-interacting Dirac fermions and non-interacting
impurities, and taking a cutoff energy
beyond which the
bulk states dominate, the
g
imp
vs.
ω
(
E
/
) spectrum may be
derived from Eq. (1) and Eq. (2) and by using the Keldysh
Green functions [13]. Thus, at
T
= 0 the tunneling conductance
g
imp
(
ω
) above a nonmagnetic impurity in the single Dirac cone
system Bi
2
Si
3
becomes [13]:
(3)
where
ρ
tip
is the density of states of the STM tip, and
imp
(
E
) is
the self-energy of impurity. The quantities
B
(
E
),
q
(
E
) and
χ
(
E
)
in Eq. (3) are related to th
e unperturbed retarded Green
function of Dirac fermions via the following relations
[13]:
(4)
where
σ
denotes the spin index [13]. In Eqs. (3) and (4) the
parameters
U
0
,
V
0
and
W
0
correspond to the interaction
energies between the STM tip and the host TI, between the
impurity and the TI, and between the STM tip and the
impurity, respectively [13]. Introducing the dimensionless
parameters
δ
ω
+ (
F
D
),
F
(
E
F
/
),
D
(
E
D
/
),
u
0
(
U
0
/
),
v
0
(
V
0
/
),
w
0
(
W
0
/
), and
ω
imp
(
imp
E
D
)/
,
TI
imp
tip
TI-imp
tip-TI
tip-imp
,
   
HH H H H H H

tip
tip
,,
t
e dN
dt
ie
N



IH
imp
,
d
g
dV
I



 

22
2
tip
2
imp
12Re
2
,
Im
1
BE
qE
qE
E
e
h
E
E



  



0,
R
G


00
0
0
12
,
qWUVIE VIE







imp
imp
imp
Re
Im
,
EE
E
E

  

00
2
,
BE UV I E



0,
1
Tr Re
,
,
R
IE
E
k
k
G



0,
2
Tr Im
,
,
R
IE
E
k
k
G
5
and imposing the condition |
δ
| < 1 so that the energy range of
impurity resonance spectra is
restricted to that of the SS, we
simplify Eqs. (3) and (4)
into the following expressions:
(5)
where
q
and
χ
are given by
.
(6)
In the limit of |
δ
|
0 and for unitary impurities where
ω
imp
0,
g
imp
diverges with an asymptotic form [|
δ
|(ln|
δ
|)
2
]
1
.
Using Eqs. (5) and (6), we illustrate the impurity resonant
spectra for (
E
/
) = 0, 0.1 and 0.3 in Fig. 6(a), where we have
taken
ω
imp
= 0 and
T
= 0. For comparison, we illustrate in Fig.
6(b) two empirical impurity resonant spectra taken from the
60-QL and 7-QL samples together with their respective
theoretical simulations in Fig. 6(c), where thermal smearing at
T = 77 K has been included in the theoretical curves. We find
that the theoretical peak positions are consistent with
ω
imp
0
(with
ω
imp
= 0 for the 60-QL sample and
ω
imp
= 0.002 for the
7-QL sample) so that the impurity resonant energy
imp
for
both samples nearly coincides with the corresponding Dirac
energy
E
D
. This finding suggests that the impurity resonances
for both samples are in the unitary limit [12], where the
impurity strength
U
imp
for
imp
E
D
diverges via the relation
(
imp
E
D
)
׽
5 sgn(
U
imp
)/(|
U
imp
| ln|
U
imp
|). We further note that
empirically the impurity resonant peak positions for both
samples also nearly coincide with their respective Dirac
energies obtained from regions without impurities.
While qualitative and semi-quantitative understanding can
be achieved with the analysis outlined above, we find that the
linewidths of the experimental data are generally broader than
those of the theoretical curves, and the ratio of the relative
peak heights also differs between theory and experiments.
These quantitative discrepancies suggest that the simple non-
interacting Dirac fermion model may not completely account
for our experimental findings.
More specifically, the aforementioned theoretical analysis
of our experimental spectra has the following physical
implications. First, the strong dependence of impurity
resonances on (
E
/
) is a direct consequence of the linear
dispersion relation of the surface Dirac fermions, which gives
rise to an approximate logarithmic divergence in the limit of
E
F
E
D
[12,13]. In contrast, for samples with large |
E
| due
to excess doping, the spectral weight of impurity resonances
may become too small to resolve directly with STS studies
[17]. Second, the occurrence of strong resonance peaks at
E
D
implies that these non-magnetic impurities are in the unitary
limit [12]. Finally, in the
E
F
E
D
limit the broader linewidth
and higher intensity of the experimental resonance peak than
theoretical predictions [12,13] may imply the necessity to
consider interacting Dirac fermions when the fermion density
of states approaches zero.
In summary, we have demonstrated scanning tunneling
spectroscopic evidence of impurity resonances in the surface
state of a strong topological insulator, Bi
2
Se
3
. The impurities
are in the unitary limit and the spectral resonances are
localized spatially (within a radius
׽
2Å). The spectral weight
of impurity resonances diverges as the Fermi energy
approaches the Dirac point, and the rapid recovery of the
surface state from non-magnetic impurities suggests robust
topological protection against perturbations that preserve time-
reversal symmetry.
Fig. 6. Dependence of non-magnetic impurity resonances on
E
in a Dirac material:
(a)
Simulated STS on top of a non-
magnetic impurity for (
E
/
) = 0, 0.1, 0.3 and
T
= 0. The
parameters used for calculations are similar to those in Ref.
[13]:
imp
= 0,
u
0
= 0.00025,
v
0
= 0.05 and
w
0
= 0.0005.
(b)
Comparison of representative empirical impurity resonance
spectra of the 60-QL and 7-QL samples.
(c)
Theoretical curves
generated by using Eq. (3) and the parameters
imp
= 0
(
0.002) and (
E
/
) = 0.01 (0.03) for the 60-QL (7-QL)
samples. We have taken
T
= 77 K and
= 3.0 eV for creating
the spectra in (c), and have used the same values of
u
0
,
v
0
and
w
0
as those in (a). We further note that the absolute values of
the tunneling conductance are shown in arbitrary units so that
only the relative values of the tunneling conductance under
varying conditions are physically significant.
Acknowledgements
This work at Caltech was jointly supported by the Center
on Functional Engineered Nano Architectonics (FENA), the
Institute for Quantum Information and Matter, an NSF Physics
Frontiers Center with support of the Gordon and Betty Moore
Foundation, and the Kavli Foundation through the Kavli
Nanoscience Institute (KNI) at Caltech. The work at UCLA


2
2
0
imp
2
12
,
1
qq
gu








00
0
2
0
4
2 ln
ln 1
4
,
qwu
v
v


 


imp
2
2
0
2ln
ln 1
4
v







6
was supported by FENA. We thank K. Sengupta and A. V.
Balatsky for valuable discussion.
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