of 12
Magnetism-induced massive Dirac spec
tra and topological defects in the
surface state of Cr-doped Bi
2
Se
3
-bilayer topological insulators
C.-C. Chen
1,2
, M. L. Teague
1,2
,
L. He
3
, X. Kou
3
, M. Lang
3
, W. Fan
1
, N. Woodward
1
, K.-L. Wang
3
and N.-C. Yeh
1,2,4*
1
Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Institute of Quantum Matter and Information, Ca
lifornia Institute of Technology, Pasadena, CA
91125, USA
3
Department of Electrical Engineering, Univers
ity of California, Los Angeles, CA 90095, USA
4
Kavli Nanoscience Institute, California Ins
titute of Technology, Pasadena, CA 91125, USA
*
E-mail: ncyeh@caltech.edu
Abstract.
Proximity-induced magnetic effects on the surface Dirac spectra of topological insulators
are investigated by scanning tunneling spectroscopi
c (STS) studies of bilayer structures consisting of
undoped Bi
2
Se
3
thin films on top of Cr-doped Bi
2
Se
3
layers. For the thickness of the top Bi
2
Se
3
layer
equal to or smaller than 3 quintuple layers (QL)
, a spatially inhomogeneous surface spectral gap
opens up below a characteristic temperature
C
2D
T
. The mean value and spatia
l homogeneity of the gap

increase with decreasing temperature (
T
), increasing
c
-axis magnetic field (
H
) and increasing Cr
doping level (
x
), suggesting that the physical origin of this surface gap is associated with proximity-
induced
c
-axis ferromagnetism. Additionally, spatially
localized resonant spectra are found near
isolated Cr impurities along the boundaries of gapp
ed and gapless regions. Th
ese spectral resonances
are long-lived at
H
= 0, with their occurrences being most prominent near
C
2D
T
and becoming
suppressed under strong
c
-axis magnetic fields. We attribute these phenomena to magnetic impurity-
induced topological defects in the spin texture of
surface Dirac fermions, and discuss the feasibility of
applying such “topological bits” to quantum information technology.
1. Introduction
The research of topological matter is an exciting frontier where the classification of quantum states of
matter beyond the principle of symmetry breaking has stimulated many conceptual advances and
experimental discoveries [1-3]. Among various topo
logical matter, topological insulators (TIs) [4-8]
are bulk insulators in two or three dimensions with strong spin-orbit coupling and gapless surface
states protected by the time-reversal invariance (TRI
). The gapless surface state of TIs consists of an
odd number of Dirac cones where the energy-momentum dispersion relation is linear, similar to the
massless Dirac fermions in graphene except for the
odd number of Dirac cones and an additional spin-
momentum locking in the former. Gapping the Dira
c cones of TIs by introd
ucing superconductivity
[9-11] or magnetism [8, 9, 12, 13] via either doping
or proximity effects can pr
ovide feasible means to
realize the elusive Majorana modes [9, 10, 14] and topological magnetoelectric (TME) effect [12, 13]
in condensed matter systems.
The underlying physics for a magnetism-induced surface gap in three-dimensional (3D) TIs is
based on the assumption that the long-r
ange ferromagnetism has a net magnetization
M
perpendicular
to the surface of the 3D-TI, and that th
e in-plane component of the Hamiltonian
H
0
for the surface
Dirac fermions remains intact after the introduction of c-axis magnetization. Hence, the total
Hamiltonian
H
total
for the Dirac fermions becomes

total
0
ex
ex
11
22
z
Fxy yx
z
JM
v k
k
JM


 
HH
,
(1)
where
x,y,z
are the Pauli matrices,
k
x,y
refer to the in-plane momentum of the Dirac fermions,
J
ex
denotes the ferromagnetic exchange coupling constant in the surface state, and
v
F
is the Fermi
velocity. The energy dispersion relation
k
E
for the magnetism-induced massive Dirac fermions can be
obtained by diagonalizing
H
total
in Eq. (1), which yields


22
ex
2
F
k
EvkJM

,
(2)
where
222
x
y
kkk
. Hence, an energy gap
= (
J
ex
M
) opens up at the Dirac point for a finite
c
-axis
magnetization |
M
|
>
0 according to Eq. (2).
Experimental evidences for the occurrence of
long-range ferromagnetism in the surface state
of 3D-TIs have been manifested by the observation of surface gap opening in angle resolved
photoemission spectroscopy (ARPES) [15-17] and th
e confirmation of quantized anomalous Hall
effect (QAHE) [18,19] in magnetic 3D-TIs (Bi
1
x
Cr
x
)
2
Te
3
. However, the microscopic mechanism [20,
21] that mediates long-range ferromagnetism in the surface state remains unclear. A number of
puzzling phenomena, such as a surface gap opening at a temperature
C
2D
T
much higher than the onset
of bulk magnetization
C
3D
T
[15-17,21], the absence of gap formation
by direct surface magnetic doping
[22], and the lack of direct STS evidence for eith
er magnetism-induced surface gaps or spectroscopic
magnetic impurity resonances [20], all sugge
st that further investigation is needed.
We report in this work direct evidence for magnetism-induced surface-state energy gaps and
magnetic impurity resonances in 3D-TIs by STS studies of bilayer structures of Bi
2
Se
3
and Cr-doped
Bi
2
Se
3
. The bilayer samples were grown by molecula
r beam epitaxy (MBE) on InP (111) single
crystalline substrates,
with an updoped Bi
2
Se
3
layer of varying thicknesses,
d
1
quintuple layers (QLs),
on top of a Cr-doped Bi
2
Se
3
layer of a fixed thickness
d
2
= 6-QL. These bilayer structures ensured that
magnetism may be observed in the undoped Bi
2
Se
3
through the proximity effect for sufficiently small
d
1
values, which prevented possible complications du
e to Cr-doping induced changes in the electronic
bandstructures of Bi
2
Se
3
. Finally, we discuss the implications
of our findings on applications of
magnetically doped topological insulators to spintronics and quantum information technology.
2. Methods
The samples investigated in this work consisted
of MBE-grown bilayer stru
ctures as schematically
shown in figure 1(a), where
d
1
= 1, 3, 5, 7-QL for the 10% Cr-doping level, and
d
1
= 1-QL for 5% Cr-
doping. Hereafter we use the nomenclature (
d
1
+
d
2
)
-x
% to denote our samples. Details of the growth
process, structural characterizations and ARPES studies of these bilayer samples have been reported
elsewhere [23-25]. Bulk electrical transport measurements on these samples revealed the appearance
of anomalous Hall resistance at
C
3D
25
TT
K, as exemplified in figure 1(b).
For the STM studies, each bilayer sample was capped with ~
1 nm Se inside the MBE growth
chamber for passivation immediately after the bilayer growth. The sample was subsequently
transferred from the growth chamber via a vacuum suitcase to another vacuum chamber, where the
sample was annealed at 150 to 200
o
C for 90 minutes under vacuum (
<
10
5
Torr) to remove the Se
capping layer. The exposed bilayer sample was cooled to 300 K in vacuum, and then the sample-
containing chamber was filled with Ar gas and load
ed into an Ar-filled glove box, where the sample
was removed from the chamber and transferred to
the STM probe placed in the same glove box. The
STM probe was sealed, transferred to its cryostat, and then evacuated down to ~
10
10
Torr at liquid
helium temperatures. The variable temperature range achievable for our homemade STM system was
from 300 K to ~
10 K, and a superconducting magnet was availa
ble to provide magnetic fields up to ~
7 Tesla.
Figure 1.
(a)
Schematics of the side view of a Bi
2
Se
3
bilayer sample, showing an undoped
Bi
2
Se
3
layer of a thickness
d
1
-QL on top of a Cr-doped Bi
2
Se
3
layer of a thickness 6-QL
grown on InP (111). A gold contact was placed on top of
d
1
. (b) Temperature dependent Hall
resistance measurements on the Bi
2
Se
3
bilayer samples at
H
= 0, showing the onset of
anomalous Hall effect below
C
3D
25
T
K.
3. Results and analysis
In this work both topographic and spectroscopic st
udies were made on all samples as a function of
T
(from 300 K to 15 K) and
H
(from 0 to 3.5 Tesla) using the variable temperature STM.
3.1. Structural characteristics from surface topographic studies
The surface topography on large scales revealed
pyramid-like terraces with steps corresponding to
single atomic layers, as described previously [23]. For an averaged, nominal top layer thickness
d
1
-
QL, the local thickness of the top layer could va
ry up to 1-QL. Atomically resolved topographic
images exhibited triangular lattice patterns th
at were consistent with that of pure Bi
2
Se
3
, as
exemplified in figure 2(a) and figure 2(b) for (1+6
)-10% and (5+6)-10% samples, respectively. On the
other hand, the Fourier transformation (FT) of the
surface topography appeared to be dependent on
d
1
.
We found that FT of the (1+6)-10% topography
showed an expected hexagonal Bragg diffraction
pattern for Bi
2
Se
3
plus an additional, faint superlattice stru
cture (figure 2(c)), which may be attributed
to the underlying Cr-doped Bi
2
Se
3
. For instance, a periodic substitution of Cr for Bi as exemplified in
figure 2(f) for a two-dimensional projection of the two Bi-layers within one-QL yields a FT pattern
(figure 2(e)) similar to that in figure 2(c). This
superlattice structure corresponds to a local Cr
concentration of 1/12. Another similar structure with a local Cr concentration of 1/8 (figure 2(h)) is
also feasible within experimental
uncertainties of the superlattice constant and its angle relative to the
Bi lattice (figure 2(g)). In contra
st, the FT in figure 2(d) for th
e surface topography of a (5+6)-10%
sample only revealed the hexagonal diffraction pattern of pure Bi
2
Se
3
due to the relatively thick
d
1
layer. Interestingly, we note that the FT topography of the (1+6)-5% samples also agreed with figure
2(d), suggesting random Cr substitutions of Bi for a sm
aller Cr concentration, which is consistent with
the randomly distributed Cr clusters found with STM studies directly on 2% Cr-doped Bi
2
Se
3
[26].
3.2. Zero-field spectroscopic studies
For the zero-field studies, tunneling conductance (
dI
/
dV
)
vs
. biased voltage (
V
=
E
/
e
) measurements
were carried out on each sample over multiple areas
, followed by detailed analysis of the spatially
resolved spectral characteristics. While apparent
spatial variations were found in all samples,
systematic investigations led to several general fi
ndings. First, all samples revealed gapless Dirac
tunneling spectra at 300 K. Second, with decreasing
T
there were two distinc
tly different types of
spectral characteristics: For samples with nominal
d
1
= 5 and 7, the tunneling spectra remained
gapless for all
T
, except for occasional areas where the actual
d
1
values in the nominal
d
1
= 5 sample
were ~
4. In contrast, for samples with nominal
d
1
= 1 and 3, the majority spectra revealed gapped
features at
C
2D
TT
, and the temperature evolution for all samp
les are exemplified in figure 3 (a)-(d).
Third, the surface gap
(
r
,
T
), obtained by the spectral analysis illustrated in figure 3(e), appeared to
be spatially inhomogeneous where
r
denotes the two-dimensional spatial coordinate.
Figure 2.
Structural characteristics of MBE-grown Bi
2
Se
3
bilayer samples on InP (111): (a)
Surface topography of a (1+6)-10% sample over an area of (6×6) nm
2
, showing a triangular
lattice structure. (b) Surface topography of
a (5+6)-10% sample over an area of (6×6) nm
2
,
showing a triangular lattice. (c) Fourier transformation (FT) of the surface topography in a
(1+6)-10% sample, revealing a dominant he
xagonal reciprocal lattice structure and a
secondary superlattice of a much weaker inte
nsity, probably coming from the underlying Cr-
doped Bi
2
Se
3
layer. Here “a” in the reciprocal space scale (2
/a
3) refers to the in-plane
nearest neighbor distance between Bi (Se) and Bi (Se). (d) FT of the surface topography on a
(5+6)-10% sample, showing a purely hexagonal reciprocal lattice. (e) Simulated FT of the
1/12 Cr-substituted Bi layer illustrated in (f), show
ing a FT similar the data in (c). Here the
blue dots represent Bi atoms and the red dots
represent Cr substitutions. (g) Simulated FT of
the 1/8 Cr-substituted Bi layer illustrated in (h), showing a FT also similar the data in (c)
within experimental errors.
For a given
r
,
mostly increased with decreasing
T
except near
T
x
~
(110±10) K where a
slight dip appeared, and eventually
saturated to a maximum value at
C
2D
TT
, as exemplified by the
T
evolution of the gap maps and the corresponding gap histograms in the left and middle panels of
figure 4 (a)-(c) for the (1+6)-5%, (1+6)-10% and (3+6)-10% samples, and also summarized in the
right panels for the temperature dependence of the corresponding mean gap
(
T
). Here the mean gap
value
at a given
T
was determined from the peak value of Gaussian fitting to the gap histogram,
and the errors were determined from the one sigm
a linewidth of the Gaussian fitting. Based on the
data for
()
T
, the onset
T
for the surface gap opening was found to be
C
2D
T
= (240 ± 10) K for
x
=
10% and
C
2D
T
= (210 ± 10) K for
x
= 5%, significantly higher than the bulk Curie temperature
C
3D
~
T
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
25 K obtained from the onset temperature of the anomalous Hall effect. We note that our finding of
CC
2D
3D
TT
is consistent with previous reports
on other families of 3D-TIs [15-17].
Figure 3.
Temperature evolution of representative
normalized tunneling conductance spectra
of (
d
1
+
d
2
)-
x
% bilayer samples taken at
H
= 0: (a) (1+6)-5%, (b) (1+6)-10%; (c) (3+6)-10%,
and (d) (5+6)-10%. For samples with
d
1
= 1 and 3, each representative spectrum in (a)-(c) at a
given
T
was determined by first taking the spatially resolved tunneling spectra over a fixed
area of the sample, plotting the histogram of
the gap values to determine the mean gap
over
this area, and then averaging those spectr
a with gap values within one sigma of
. To
compensate for the thermal drift of the STM tip
and to ensure that the spectral analysis was
carried out over the same sample area for all different temperatures, we compared the
topographic images taken at all temperatures
to identify the overlapped areas. With this
procedure, finite gaps were consistently foun
d to develop at low temperatures for samples
with
d
1
= 1 and 3 as exemplified in (a)-(c), wherea
s all spectra were gapless for samples with
d
1
= 5 and 7, as exemplified in (d). (e)
Schematic illustration showing how the gap is
estimated from a realistic normalized tunneling conductance spectrum: Defining the
conductance of the inflection point in the tunneling spectrum as
h
and the corresponding
energy difference between the spectral inception points as
h
, we identify
2
h
for the
tunneling conductance at 2
h
, and then extrapolate an effective gap

at zero conductance from
the formula

= 2
h

2
h
. The maximum gap thus obtained is
consistent with the theoretical
values (0.3
~
0.5 eV) from the densities of states of Se
I,II
in Cr-doped Bi
2
Se
3
[26].
The inhomogeneous gap distribution may be the result of multiple reasons. First, the Cr-
substitution of Bi may not be uniform as the result of the size mismatch that could induce significant
lattice strain and inhomogeneous ferromagnetism. Se
cond, the magnetic moments of individual Cr
may not be perfectly aligned along the sample c-axis
in the absence of an external magnetic field.
Given that only
c
-axis magnetization component can induce an
energy gap in the surface state of 3D-
TIs according to Eqs. (1) and (2), varying spin a
lignments in different magnetic domains would result
in varying surface gaps [27]. Third, the sample surfa
ce exhibited terrace-like structures with thickness
variations up to 1-QL [23], which could give rise to different proximity-induced gaps.
(e)
dI/dV
vs.
Energy
Figure 4.
Temperature evolution and spatial distribution of the surface gap at
H
= 0: (a) Gap
maps and the corresponding gap histograms of a (1+6)-5% sample taken at
T
= 80 K (left
panels) and
T
= 164 K (middle panels), and the temperature dependence of the mean gap
(second right panel). (b) Gap map and the corresponding gap histograms of a (1+6)-10%
sample taken at
T
= 79 K (left panels) and
T
= 153 K (middle panels), and the temperature
dependence of the mean gap
(third right panel). (c) Gap maps and the corresponding gap
histograms of a (3+6)-10% sample taken at
T
= 20 K (left panels) and
T
= 79 K (middle
panels), and the temperature
dependence of the mean gap
(fourth right panel). The first
right panel is a schematic illustration of the sp
atially varying electronic structure experienced
by the tunneling current, showing the dominance of the surface state gap
over the bulk gap
B
in determining the measured spectral gap in STS.
3.3. Finite-field spectroscopic studies
We further investigated the effect of increasing
c
-axis magnetic field (
H
) on the gap distribution over
the same area of each sample at a constant
T
. As exemplified in figure 5(a)-(f) for samples of (1+6)-
5% and (3+6)-10% taken at
T
= 18 K and
H
= 0, 1.5 T and 3.5 T, the gap maps became increasingly
homogeneous and the mean gap value
derived from the histogram also increased slightly with
increasing
H
. This finding suggests th
at the observed surface gap is consistent with
c
-axis
ferromagnetism induced by Cr-doping and proximity ef
fect, and the characterist
ic field for saturating
the surface state ferromagnetism appears to be larger
than that for saturating the bulk ferromagnetism
[24], probably due to the helical spin textures of
the former. The small but finite residual gap
inhomogeneity in high fields may be attributed to
spatially inhomogeneous Cr-distributions and the
d
1
variations.
Figure 5.
Evolution of the surface gap distribution at
T
= 18 K with applied c-axis magnetic
field: (a-c) Gap maps (upper panels) and the corresponding gap histograms (lower panels) of
a (1+6)-5% sample taken at
H
= 0, 1.5 T and 3.5 T over the same (20 ×
20) nm
2
area. (d-f)
Gap maps (upper panels) and the corresponding gap histograms (lower panels) of a (3+6)-
10% sample taken at
H
= 0, 1.5 T and 3.5 T over the same (20 ×
20) nm
2
area.
3.4. Minority spectra
While the majority of the tunneling spectra in the (1+6)-5%, (1+6)-10% and (3+6)-10% samples
revealed gapped characteristics for
C
2D
TT
, spatially localized and intense conductance peaks were
occasionally observed along the border
s of gapless and gapped regions, as exemplified in figure 6 (a)-
(d) for a (1+6)-5% sample and in figure 6 (f) for a (1+6)-10% sample. These long-lived minority
spectra either consisted of a single sharp
conductance peak at a small negative energy
E
=
E
near the
Dirac point
E
D
; or comprised of double conductance peaks (figure 6(b)) at
E
=
E
and
E
=
E
+
, where
E
+
is near the Fermi energy
E
F
= 0. These double-peak spectral characteristics were consistent with
theoretical predictions for magnetic
impurity resonances [20]. Further, the numbers of both single-
and double-peak impurity resonances at
H
= 0 were found to increase rapidly near
C
2D
T
(figure 6(f)).
In contrast, all resonances disappeared under a large c-axis magnetic field at low
T
when gapless
regions diminished.
We attribute the sharp impurity resonances to isolated Cr-impurities that diffused from the
d
2
layer into the
d
1
layer because they were only found in zero-field along the borders between gapless
and gapped regions, and then disappeared under a large
c
-axis magnetic field when gapless regions
diminished. The temperature dependence at
H
= 0 as exemplified in figure 6 (f) may be understood as
the result of weakening ferromagnetism near
C
2D
T
, so that more Cr-impurities became decoupled and
acted like isolated impurities. The strong spatial
localization and long lifetime of these magnetic
impurity resonances at
H
= 0 may be attributed to topological pr
otection of the surface state in 3D-TIs
when the Dirac energy
E
D
is relatively close to
E
F
, similar to the case for n
on-magnetic impurities [20,
28]. Our finding of atomically sharp impurity res
onances due to diffused Cr atoms into the undoped
d
1
layer differs from direct ST
S studies of 2% Cr-doped Bi
2
Se
3
[26], where Cr substituted Bi in clusters
and so did not behave like atomically isolated ma
gnetic impurities in the surface state of pure Bi
2
Se
3
.
4. Discussion
The dependence of the spectral characteristics on
T
,
H
,
x
and
d
1
for the bilayer samples are all
consistent with the scenario that the appearance of a surface-state gap was induced by the proximity
effect of predominantly
c
-axis ferromagnetism in the Cr-dop
ed bottom layer. The appearance
(absence) of gapped tunneling spectra below
C
2D
T
for bilayer samples with
d
1
3-QL (
d
1
5-QL)
suggests that the proximity effect due to
c
-axis magnetic correlation is limited to ~
4-QL.
Concerning the physical feasibility of a relatively high
C
2D
T
, we note that the energy splitting
of the double-peak spectrum associated with isol
ated magnetic impurities in the surface state is
comparable to the ferromagnetic exchange coupling
J
ex
[20]. Noting that the mean-field Curie
temperature may be estimated from
J
ex
via the relation
C
2D
ex
13
~
B
J
kT
S S
[29] and that
J
ex
~
0.1 eV, we obtain
C
2D
T
~
250 K by assuming
S
= 1/2. While this rough estimate does not include the
screening effect from Dirac fermions, it is
still comparable to our STS measurements of
C
2D
T
~ 240 K
for
x
= 10% and
C
2D
T
~
210 K for
x
= 5% for the observed surface ferromagnetism, and is much higher
than the bulk
C
3D
T
~
25 K. The disparity of
C
2D
T
and
C
3D
T
may be attributed to the different microscopic
mechanisms for mediating ferromagnetism between th
e surface state of a helical metal and the bulk
state of an insulator. Additionally, the non-monotonic
T
-dependence of the tunneling gap for all
samples with
d
1
= 1 and 3 (figure 3) may be indicative a crossover from 2D to 3D ferromagnetism.
While the microscopic mechanism responsible for
CC
2D
3D
TT
requires further investigation, the
relatively high
C
2D
T
values are promising for realistic spintr
onic applications, particularly if our
findings may be generalized to more homogeneously Cr-doped 3D-TIs such as (Bi
1
x
Cr
x
)
2
Te
3
[24].
Figure 6.
Spectral characteristics of
isolated magnetic impuritie
s: (a) Gap map of a (1+6)-5%
bilayer sample taken at 80 K over an area of (13×13) nm
2
, showing both spatially
inhomogeneous gaps and some gapless regions (dar
k blue). The arrow indicates a site where a
spatially localized double-resonant spectrum in
(b) is observed. (b) Sharp resonances of an
isolated impurity are manifested in the (
dI
/
dV
) vs.
E
and
X
plot, where
X
is a horizontal
linecut across an isolated impurity indicated by
the arrow in (c). The sharp double resonant
peaks appear at
E
=
E
and
E
=
E
+
, where
E
is near
E
D
and
E
+
is near
E
F
. (c) Tunneling
conductance map taken at
E
=
E
over the same area as in (a), showing spatially isolated
conductance peaks in bright sharp spots. (d) Two-dimensional distribution of the tunneling
conductance at bias voltage
V
= (
E

/
e
) over a (4×4) nm
2
area indicated by the dashed box in
(c), showing three sites with intense impuri
ty resonances. (e) Schematics of a topological
defect (red arrows of opposite helicity) due to an isolated magnetic impurity in the surface-
state Dirac spin textures (counterclockwise black arrows). (f)
T
-evolution of the counts of
single- and double-peak and total impurity resonances over a (20×20) nm
2
area of a (1 + 6)-
10% sample, showing a rapid increase in the number of impurity resonances near
C
2D
T
for
both the single- and double-peak resonances.