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Magnetism-induced massive Dirac spectra and topological defects in the surface state of Cr-
doped Bi
2
Se
3
-bilayer topological insulators
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2015 New J. Phys. 17 113042
(http://iopscience.iop.org/1367-2630/17/11/113042)
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New J. Phys.
17
(
2015
)
113042
doi:10.1088
/
1367-2630
/
17
/
11
/
113042
PAPER
Magnetism-induced massive Dirac spectra and topological defects
in the surface state of Cr-doped Bi
2
Se
3
-bilayer topological insulators
C-CChen
1
,
2
,MLTeague
1
,
2
,LHe
3
,XKou
3
,MLang
3
,WFan
1
,NWoodward
1
,K-LWang
3
andN-CYeh
1
,
2
,
4
1
Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
3
Department of Electrical Engineering, University of California, Los Angeles, CA 90095, USA
4
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA
E-mail:
ncyeh@caltech.edu
Keywords:
topological insulators, surface state, Dirac fermions, scanning tunneling spectroscopy, proximity effect, ferromagnetism,
impurity resonances
Supplementary material for this article is available
online
Abstract
Proximity-induced magnetic effects on the surface Dirac spectra of topological insulators are
investigated by scanning tunneling spectroscop
ic studies of bilayer structures consisting of
undoped Bi
2
Se
3
thin
fi
lms on top of Cr-doped Bi
2
Se
3
layers. For thickness of the top Bi
2
Se
3
layer
equal to or smaller than 3 quintuple layers, a spatially inhomogeneous surface spectral gap
Δ
opens
up below a characteristic temperature
T
,
c
2D
which is much higher than the bulk Curie temperature
T
c
3D
determined from the anomalous Hall resistance. The mean value and spatial homogeneity of
the gap
Δ
generally increase with increasing
c
-axis magnetic
fi
eld
(
H
)
and increasing Cr doping
level
(
x
)
, suggesting that the physical origin of this surface gap is associated with proximity-induced
c
-axis ferromagnetism. On the other hand, the temperature
(
T
)
dependence of
Δ
is non-
monotonic, showing initial increase below
T
,
c
2D
which is followed by a
‘
dip
’
and then rises again,
reaching maximum at
T
=
T
.
c
3D
These phenomena may be attributed to proximity magnetism
induced by two types of contributions with different temperature dependences: a three-
dimensional contribution from the bulk magnetism that dominates at low
T
,andatwo-
dimensional contribution associated with the RKKY interactions mediated by surface Dirac
fermions, which dominates at
T
c
3D
=
T
<
T
.
c
2D
In addition to the observed proximity magnet-
ism, spatially localized sharp resonant spectra are found along the boundaries of gapped and gapless
regions. These spectral resonances are long-lived at
H
=
0, with their occurrences being most
prominent near
T
c
2D
and becoming suppressed under strong
c
-axis magnetic
fi
elds. We attribute
these phenomena to magnetic impurity-induced to
pological defects in the spin texture of surface
Dirac fermions, with the magnetic impurities being isolated Cr impurities distributed near the
interface of the bilayer system. The long-term stability of these topologically protected two-level
states may
fi
nd potential applications to quantum information technology.
1. Introduction
The research of topological matter is an exciting frontier where the classi
fi
cation of quantum states of matter
beyond the principle of symmetry breaking has stimulated many conceptual advances and experimental
discoveries
[
1
–
3
]
. Among various topological 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. 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 Dirac cones of
TIs by introducing superconductivity
[
9
–
11
]
or magnetism
[
8
,
9
,
12
,
13
]
via either doping or proximity effects
OPEN ACCESS
RECEIVED
19 June 2015
REVISED
1 October 2015
ACCEPTED FOR PUBLICATION
19 October 2015
PUBLISHED
17 November 2015
Content from this work
may be used under the
terms of the
Creative
Commons Attribution 3.0
licence
.
Any further distribution of
this work must maintain
attribution to the
author
(
s
)
and the title of
the work, journal citation
and DOI.
© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
can provide feasible means to realize the elusive Majorana modes
[
9
,
10
,
14
]
and topological magnetoelectric
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-range ferromagnetism has a net magnetization
M
perpendicular to the surface of the
3D-TI, and that the in-plane component of the Hamiltonian
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
HHJMvkkJM
1
2
1
2
,1
zFxyyxz
total
0
eff
eff
()
()
sss s
=+ º - +
where
σ
x,y,z
are the Pauli matrices,
k
x,y
refer to the in-plane momentum of the Dirac fermions,
J
eff
denotes the
ferromagnetic coupling constant in the surface state, and
v
F
is the Fermi velocity. The energy dispersion relation
E
k
for the magnetism-induced massive Dirac fermions can be obtained by diagonalizing
H
total
in equation
(
1
)
,
which yields
EvkJM
2,
2
k
F
2
ex
2
() (
)
()
= +
where
kkk
.
xy
2
22
º+
Hence, an energy gap
Δ
=
(
J
eff
M
)
opens up at the Dirac point for a
fi
nite
c
-axis
magnetization
|
M
|
>
0 according to equation
(
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
–
18
]
and the con
fi
rmation of quantized anomalous Hall effect
[
19
,
20
]
in magnetic 3D-TIs
(
Bi
1
−
x
Cr
x
)
2
Te
3
. However, the microscopic mechanism that mediates long-range ferromagnetism in the surface state
has not been fully established
[
18
,
21
–
26
]
. A number of puzzling phenomena, such as a surface gap opening at a
temperature
T
c
2D
much higher than the onset temperature of bulk magnetization
T
c
3D
[
15
–
17
,
22
–
25
]
, the
absence of gap formation by direct surface magnetic doping
[
27
]
, and the lack of direct scanning tunneling
spectroscopic
(
STS
)
evidence for either magnetism-induced surface gaps
[
30
]
or spectroscopic magnetic
impurity resonances
[
21
]
, all suggest 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 molecular 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
fi
xed 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 suf
fi
ciently small
d
1
values, which prevented possible complications due
to Cr-doping induced changes in the electronic bandstructures of Bi
2
Se
3
, and thus enabled direct observation of
the effect of global time-reversal symmetry breaking on the surface state of TIs. Finally, we discuss the
implications of our
fi
ndings on applications of magnetically doped TIs to spintronics and quantum information
technology.
2. Methods
The samples investigated in this work consisted of MBE-grown bilayer structures as schematically shown in
fi
gure
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
[
18
,
28
,
29
]
. Bulk
electrical transport measurements on these samples revealed the appearance of anomalous Hall resistance at
T
TT
,where
30K
c
3D
c
3D
~
and 20 K for
x
=
10% and 5%, respectively, as exempli
fi
ed in
fi
gure
1
(
b
)
.
Additionally, surface magneto-optic Kerr effect measurements have been performed on similar bilayer samples,
showing
fi
nite Kerr angles
θ
K
at
T
<
T
,
c
3D
with temperature and magnetic
fi
eld dependence similar to those of
the anomalous Hall resistance
[
18
]
, as exempli
fi
ed in supplementary
fi
gures 1
(
a
)
–
(
c
)
.
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 423 K
∼
473 K
for 90 min 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
fi
lled with Ar gas and loaded into an Ar-
fi
lled
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 available to provide magnetic
fi
elds up to
∼
7 Tesla.
2
New J. Phys.
17
(
2015
)
113042
C-C Chen
et al
3. Results and analysis
In this work both topographic and spectroscopic studies 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
[
28
,
29
]
. For an averaged, nominal top layer thickness
d
1
-QL, the local thickness of
the top layer could vary up to 1-QL. Atomically resolved topographic images exhibited triangular lattice patterns
that were always consistent with that of pure Bi
2
Se
3
, as exempli
fi
ed in
fi
gure
2
(
a
)
and
fi
gure
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 structure
(
fi
gure
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 exempli
fi
ed in
fi
gure
2
(
f
)
for a two-dimensional projection of the two Bi-layers within one-QL yields
a FT pattern
(
fi
gure
2
(
e
))
similar to that in
fi
gure
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
(
fi
gure
2
(
h
))
is also feasible
within experimental uncertainties of the superlattice constant and its angle relative to the Bi lattice
(
fi
gure
2
(
g
))
.
In contrast, the FT in
fi
gure
2
(
d
)
for the 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
fi
gure
2
(
d
)
, suggesting random Cr substitutions of Bi for
a smaller Cr concentration, which is consistent with the randomly distributed Cr clusters found with STM
studies directly on 2% Cr-doped Bi
2
Se
3
[
30
]
.
3.2. Zero-
fi
eld spectroscopic studies
For the zero-
fi
eld studies, tunneling conductance
(
d
I
/
d
V
)
versus 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 distinctly 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
T
<
T
,
c
2D
and the temperature evolution for all samples are exempli
fi
ed in
fi
gures
3
(
a
)
–
(
d
)
. Third, the surface gap
Δ
(
r
,
T
)
, obtained by the spectral analysis illustrated in
fi
gure
3
(
e
)
,
appeared to be spatially inhomogeneous where
r
denotes the two-dimensional spatial coordinate.
For a given
r
,
Δ
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
T
<
T
c
3D
=
T
,
c
2D
as exempli
fi
ed by the
T
evolution
of the gap maps and the corresponding gap histograms in the left and middle panels of
fi
gures
4
(
a
)
–
(
c
)
for the
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
normalized Hall resistance
(
R
xy
)
of the Bi
2
Se
3
bilayer samples at
H
=
0, with each set of data normalized to its maximum Hall
resistance value
(
R
xy
-Max
)
. The
R
xy
-Max
values are 1.35
Ω
, 1.26
Ω
and 0.32
Ω
for the
(
3
+
6
)
-10%,
(
1
+
6
)
-10% and
(
1
+
6
)
-5%
samples, respectively. The
T
c
3D
values are estimated from the temperatures associated with the 1% values of
R
xy
-Max
, which yield
∼
30 K,
∼
30 K and
∼
20 K for the
(
3
+
6
)
-10%,
(
1
+
6
)
-10% and
(
1
+
6
)
-5% samples, respectively.
3
New J. Phys.
17
(
2015
)
113042
C-C Chen
et al
(
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
̄
D
(
T
)
. Here the mean gap value
̄
D
at a given
T
was
determined from the peak value of Gaussian
fi
tting to the gap histogram, and the errors were determined from
the one sigma linewidth of the Gaussian
fi
tting. Based on the data for
T
,
̄
()
D
the onset
T
for the surface gap
opening was found to be
T
240 10 K
c
2D
()
=
for
x
=
10% and
T
210
10 K
c
2D
()
=
for
x
=
5%,
signi
fi
cantly higher than the bulk Curie temperatures
T
30
20 K
c
3D
=~
obtained from the onset temperature
of the anomalous Hall resistance. This
fi
nding of
T
c
2D
?
T
c
3D
is in fact consistent with previous reports on
other families of 3D-TIs
[
15
–
17
]
. As we shall elaborate further in Discussion, the non-monotonic temperature
dependence of the surface gap and the relation of
T
c
2D
?
T
c
3D
may be the result of multiple magnetic
interaction components in the bilayer samples, with each component having a different temperature
dependence and interaction range.
The inhomogeneous gap distribution may be attributed to multiple reasons. First, the Cr-substitution of Bi
may not be uniform due to the size mismatch, which could induce lattice strain and inhomogeneous
ferromagnetism. Second, the magnetic moments of Cr ions may not be well aligned along the sample
c
-axis
without an external magnetic
fi
eld. Given that only
c
-axis magnetization component can induce a surface gap in
3D-TIs according to equations
(
1
)
and
(
2
)
, varying spin alignments in different magnetic domains would result
in varying surface gaps
[
31
]
. Third, the sample surface exhibited terrace structures with
∼
1-QL thickness
variations
[
28
]
, which could give rise to varying proximity-induced gaps from the RKKY interaction mediated by
surface Dirac fermions
[
18
,
22
–
24
]
.
To better understand how the aforementioned components contribute to the gap inhomogeneity, we
compare the surface topography of the bilayer samples with their corresponding gap maps. As exempli
fi
ed in
supplementary
fi
gures 2
(
a
)
–
(
b
)
and discussed further in the supplementary information, we
fi
nd that the
correlation between a typical surface topography map and the corresponding gap map over the same area of a
bilayer sample is
fi
nite although relatively weak. The weak correlation is primarily due to small height variations
over the typical areas of our STM studies. Thus, the primary cause of spatially inhomogeneous gaps in the bilayer
systems may be attributed to inhomogeneous Cr distributions and the misalignment of Cr magnetic moments.
Experimentally, the latter effect may be minimized by applying a strong external magnetic
fi
eld along the
c
-axis,
which is the subject of our investigation in the next subsection.
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 hexagonal reciprocal lattice structure and a secondary superlattice of a much weaker intensity, probably coming
from the underlying Cr-doped Bi
2
Se
3
layer. Here
‘
a
’
in the reciprocal space scale
(
2
π
/
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
)
, showing 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.
4
New J. Phys.
17
(
2015
)
113042
C-C Chen
et al
3.3. Finite-
fi
eld spectroscopic studies
We further investigated the effect of increasing
c
-axis magnetic
fi
eld
(
H
)
on the gap distribution over the same
area of each sample at a constant
T
. As exempli
fi
ed in
fi
gure
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
̄
D
derived from the histogram also increased slightly with increasing
H
. This
fi
nding suggests that the
observed surface gap is consistent with
c
-axis ferromagnetism induced by Cr-doping and proximity effect, and
the characteristic
fi
eld for saturating the surface state ferromagnetism appears to be larger than that for
saturating the bulk ferromagnetism
[
18
]
, probably due to the helical spin textures of the former. The small but
fi
nite residual gap inhomogeneity in high
fi
elds may be primarily attributed to spatially inhomogeneous Cr-
distributions.
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
T
<
T
,
c
2D
spatially localized and intense conductance peaks were occasionally
observed along the borders of gapless and gapped regions, as exempli
fi
ed in
fi
gures
6
(
a
)
–
(
d
)
for a
(
1
+
6
)
-5%
sample and in
fi
gure
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
(
fi
gure
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
[
25
]
. Further, the numbers of both single- and double-peak impurity resonances at
H
=
0 were
found to increase rapidly near
T
c
2D
(
fi
gure
6
(
f
))
. In contrast, all resonances disappeared under a large
c
-axis
magnetic
fi
eld at low
T
when gapless regions diminished.
We attribute the sharp impurity resonances to isolated Cr-impurities that were far away from neighboring
Cr ions and probably had partially diffused from the
d
2
-layer into the interfacial region, because they were only
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
fi
rst taking the spatially resolved tunneling spectra over a
fi
xed area
of the sample, plotting the histogram of the gap values to determine the mean gap
̄
D
over this area, and then averaging those spectra
with gap values within one sigma of
.
̄
D
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,
fi
nite gaps were consistently found to develop at low temperatures for samples with
d
1
=
1 and 3 as exempli
fi
ed in
(
a
)
–
(
c
)
, whereas all spectra were gapless for samples with
d
1
=
5 and 7, as exempli
fi
ed in
(
d
)
.
(
e
)
Schematic illustration showing how the gap is estimated from a realistic normalized tunneling conductance spectrum: de
fi
ning the
conductance of the in
fl
ection point in the tunneling spectrum as
h
and the corresponding energy difference between the spectral
inception points as
Δ
h
, we identify
Δ
2
h
for the tunneling conductance at 2
h
, and then extrapolate an effective gap
Δ
at zero
conductance from the formula
Δ
=
2
Δ
h
−
Δ
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
[
30
]
.
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found in zero-
fi
eld along the borders between gapless and gapped regions, and then disappeared under a large
c
-
axis magnetic
fi
eld when gapless regions diminished and long-range ferromagnetism was established. The
strongly non-monotonic temperature dependence of the number of impurity resonances at
H
=
0
(
see
fi
gure
6
(
f
))
may be understood as the result of weakening surface ferromagnetism near
T
,
c
2D
so that more Cr-
impurities became decoupled and acted like isolated impurities. The strong spatial localization and long lifetime
Figure 4.
Temperature evolution and spatial distribution of the surface gap at
H
=
0:
(
a
)
gap maps and the corresponding histograms
of a
(
1
+
6
)
-5% sample taken at
T
=
80 K
(
left panels
)
and
T
=
164 K
(
middle panels
)
, and the
T-
dependence of the mean gap
̄
D
(
second right panel
)
.
(
b
)
Gap map and the corresponding histograms of a
(
1
+
6
)
-10% sample taken at
T
=
79 K
(
left panels
)
and
T
=
153 K
(
middle panels
)
, and the
T-
dependence of the mean gap
̄
D
(
third right panel
)
.
(
c
)
Gap maps and the corresponding
histograms of a
(
3
+
6
)
-10% sample taken at
T
=
20 K
(
left panels
)
and
T
=
79 K
(
middle panels
)
, and the
T-
dependence of the
mean gap
̄
D
(
fourth right panel
)
. The
fi
rst right panel is a schematic illustration of the spatially varying electronic structure
experienced by the tunneling current, showing the dominance of the surface state gap
Δ
over the bulk gap
Δ
B
in determining the
measured spectral gap in STS.
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of these magnetic impurity resonances at
H
=
0 may be attributed to topological protection of the surface state
in 3D-TIs when the Dirac energy
E
D
is relatively close to
E
F
, similar to the case for non-magnetic
impurities
[
25
,
32
]
.
We further remark that our
fi
nding of atomically sharp impurity resonances due to isolated Cr ions differs
from direct STM studies of bare 2% Cr-doped Bi
2
Se
3
[
30
]
: in the latter case, direct atomic imaging revealed that
Cr substituted Bi in clusters so that the resulting spectroscopy did not behave like isolated magnetic impurities
embedded in the surface state of pure Bi
2
Se
3
. In contrast, our identi
fi
cation of isolated Cr-impurities in the
bilayer systems was solely based on spatially resolved
spectroscopic
evidences without directly imaging the Cr
impurities, because Cr ions could not diffuse all the way to the top surface and were still buried by the undoped
Bi
2
Se
3
layer. Therefore, STM
imaging
of all bilayer samples always revealed the atomic structure of pure Bi
2
Se
3
,
Figure 5.
Evolution of the surface gap distribution at
T
=
18 K with applied
c
-axis magnetic
fi
eld:
(
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
(
47
×
47
)
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
(
24
×
24
)
nm
2
area.
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New J. Phys.
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et al
as exempli
fi
ed in
fi
gures
2
(
a
)
and
(
b
)
. Additionally, we have found no evidences for the formation of Cr clusters
at the interface of our bilayer samples: The STEM studies
[
18
]
of typical bilayer samples revealed atomically
sharp interfaces without any cluster-induced variations in the c-axis lattice spacing; our bilayer samples also
exhibited well de
fi
ned 3D ferromagnetism below an appreciable
T
c
3D
(
20
∼
30 K
)
, which was consistent with
more dispersed and uniform distributions of Cr in order to achieve long-range ferromagnetic order. The latter
situation was in stark contrast to the absence of long-range ferromagnetism down to 1.5 K in the 2% Cr-doped
Bi
2
Se
3
sample that exhibited randomly distributed surface Cr-clusters
[
30
]
. In fact, a recent report based on
direct STM studies on a similar TI system
(
Bi
0.1
Sb
0.9
)
1.92
Cr
0.08
Te
3
(
with
T
18 K
c
3D
~
and 4% Cr-doping
)
Figure 6.
Spectral characteristics of isolated magnetic impurities:
(
a
)
gap map of a
(
1
+
6
)
-5% bilayer sample taken at 80 K and
H
=
0
over an area of
(
13
×
13
)
nm
2
, showing both spatially inhomogeneous gaps and some gapless regions
(
dark 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
(
d
I
/
d
V
)
versus
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 impurity 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
T
c
2D
for both the single- and double-peak resonances.
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revealed completely dispersed distributions of Cr and inhomogeneous gaps without evidences for cluster
formation
[
33
]
. These
fi
ndings are consistent with what we have inferred from our experimental studies of the
(
Bi
1
−
x
Cr
x
)
2
Se
3
/
Bi
2
Se
3
bilayer systems.
4. Discussion
In this session we address issues associated with the physical mechanism for proximity magnetism, the physical
origin and the magnitude of the surface energy gap, the feasibility of high surface
T
c
2D
values relative to the bulk
T
,
c
3D
and the topological spin textures in the presence of isolated magnetic impurities.
4.1. Proximity magnetism in the bilayer system
Our experimental results have demonstrated that the dependence of the spectral characteristics on
T
,
H
,
x
and
d
1
for the bilayer samples are all consistent with the scenario that a surface-state gap was induced by the proximity
effect of predominantly
c
-axis ferromagnetism in the Cr-doped bottom layer. The appearance
(
absence
)
of
gapped tunneling spectra below
T
c
2D
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 a critical thickness
d
c
∼
4-QL.
In our investigation of the Cr-doped bilayer system, it is important to realize that the undoped Bi
2
Se
3
layer
should
not
be considered as an isolated ultrathin
fi
lm on a
‘
substrate
’
of Cr-doped Bi
2
Se
3
. This realization is
because of the continuous and seamless growth of undoped Bi
2
Se
3
layer on top of the isostructure, Cr-doped
Bi
2
Se
3
layer
[
18
]
, leading to an effective total thickness of
(
d
1
+
d
2
)
-QL. In contrast, for ultra-thin TI
fi
lms on
dissimilar substrates, an energy gap can open up due to coupling between the top and bottom topological surface
states, and a Rashba-like coupling with further energy splitting in momentum can also be expected due to the
asymmetric chemical potentials between the surface of the thin
fi
lm and its interface with a dissimilar substrate
[
34
]
. This Rashba-like coupling has indeed been con
fi
rmed by ARPES and STS studies for 2-QL to 5-QL Bi
2
Se
3
on various dissimilar substrates
[
35
,
36
]
. In particular, the energy gap induced by the Rashba-like coupling for
ultra-thin TI
fi
lms on dissimilar substrates does
not
exhibit any discernible temperature or magnetic
fi
eld
dependence, which is in sharp contrast to the behavior of the proximity-magnetism induced surface gaps
described in this work.
Having established proximity magnetism as the physical origin of the surface gap in the bilayer system, we
consider next the primary components that contribute to the surface ferromagnetism in the undoped Bi
2
Se
3
layer of
d
1
3-QL. A natural component is the dipole
fi
elds associated with the magnetic moments of Cr ions.
Additionally, RKKY-like magnetic interactions mediated by the bulk carriers and the surface Dirac fermions are
also feasible contributions to the surface ferromagnetism. Therefore, we may express the effective ferromagnetic
coupling constant
J
eff
in terms of the sum of these three components,
J
eff
=
(
J
dipole
+
J
bulk
)
+
J
Dirac
, where
(
J
dipole
+
J
bulk
)
≡
J
3D
may be considered as a 3D component and
J
Dirac
≡
J
2D
a 2D component. We expect
J
3D
to
vary slowly with
d
1
because it primarily represents the bulk magnetic properties of the Cr-doped Bi
2
Se
3
.In
contrast,
J
2D
∝
exp
(
−
2
d
1
/
d
c
)
is strongly dependent on
d
1
because it involves the wavefunction overlaps
between the surface Dirac fermions of the undoped Bi
2
Se
3
layer and those of the
d
-electrons in Cr-doped Bi
2
Se
3
layer
[
18
,
22
–
24
]
.
Concerning the temperature dependence of
J
eff
, we expect
J
3D
to dominate at low
T
because of the strong
con
fi
nement of Dirac fermions to the surface layer and therefore negligible RKKY interaction between surface
Dirac fermions and bulk
d
-electrons. On the other hand,
J
3D
is sensitive to 3D long-range order of magnetic
moments and so should diminish signi
fi
cantly above
T
.
c
3D
In contrast,
fi
nite temperature can enhance
wavefunction overlaps between the surface Dirac fermions and the interfacial
d
-electrons. Therefore, we expect
J
2D
to dominate at elevated temperatures, and have further elaborated our rationale for this temperature
dependence in supplementary information. Moreover, the long Fermi wavelength of surface Dirac fermions
could result in much enhanced RKKY interaction in the bilayer system so that
J
2D
>
J
3D
[
18
,
22
–
24
]
, which may
account for the empirical observation of
T
c
2D
?
T
c
3D
.
4.2. The magnitude and temperature dependence of the surface gap
Based on the scenario described above and noting that the proximity-induced surface gap is given by
Δ
=
(
J
eff
M
)
where
M
is the surface magnetization that increases monotonically below
T
,
c
2D
we
fi
nd that with
decreasing
T
, the surface gap
Δ
(
T
<
T
c
2D
)
can indeed exhibit a non-monotonic
T
-dependence that is
consistent with our experimental
fi
ndings depicted in
fi
gure
4
. As qualitatively exempli
fi
ed in supplementary
fi
gure 3,
Δ
(
T
)
fi
rst increases with decreasing
T
for
T
<
T
,
c
2D
and then exhibits a
‘
dip
’
at
T
x
where
T
c
2D
>
T
x
>
T
,
c
3D
and
T
x
represents a dimensional crossover temperature below which
J
2D
becomes negligible.
While we do not have suf
fi
cient information to model
Δ
(
T
)
quantitatively, the qualitative agreement of
9
New J. Phys.
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C-C Chen
et al
supplementary
fi
gure 3 with experimental results in
fi
gure
4
suggests that our proposed scenario for proximity
magnetism in the bilayer system may provide useful guide for future investigation.
Next, we comment on the magnitude of the proximity-induced surface gaps in Bi
2
Se
3
, which were found to
be comparable or even larger than the bulk gap
(
∼
0.3 eV
)
of Bi
2
Se
3
. While one may question whether it is
feasible for a magnetism-induced surface gap to exceed its bulk gap of a TI, we point out that the bulk gap
∼
0.3 eV of Bi
2
Se
3
is an indirect gap; the direct gap at the
Γ
-point where the Dirac cone resides is in fact
>
0.5 eV
based on bandstructure calculations
[
37
]
. In this context, the largest mean gap values
(
0.4 eV
∼
0.5 eV
)
found in
our work are well within the range of the bulk gap. We further note that the gap values determined from STS
studies would be dominated by the local gap of the surface layer, whereas the gap value determined from ARPES
would be sensitive to the smallest gap over an extended surface area. Therefore, the surface gap determined from
STS tends to be larger than that determined from ARPES for an inhomogeneous sample. Indeed, ARPES studies
on Bi
1.8
Cr
0.2
Se
3
have revealed a surface gap of
∼
0.2 eV
[
18
]
, which is slightly smaller than but still in reasonable
agreement with our
fi
nding of
Δ
=
(
0.25
±
0.09
)
eV for a
(
1
+
6
)
-10% sample at
H
=
0 and
T
=
79 K, as
shown in
fi
gure
4
(
b
)
.
4.3. The feasibility of a much higher surface
T
c
2D
than the bulk
T
c
3D
Concerning the physical feasibility of a relatively high
T
,
c
2D
we note that the energy splitting of the double-peak
spectrum associated with isolated magnetic impurities in the surface state is comparable to the effective
ferromagnetic coupling constant
J
eff
[
21
]
. Noting that the mean-
fi
eld Curie temperature may be estimated from
J
eff
via the relation
k
B
T
c
2D
∼
S
(
S
+
1
)(
J
eff
/
3
)
[
38
]
and that
J
eff
∼
0.1 eV as determined from the measured
energy splitting of the double resonances, we obtain
T
250 K
c
2D
~
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
T
240 K
c
2D
~
for
x
=
10% and
T
210 K
c
2D
~
for
x
=
5% for the observed surface
ferromagnetism, and is much higher than the bulk
T
25 K.
c
3D
~
The disparity of
T
c
2D
and
T
c
3D
may be
attributed to the different microscopic mechanisms for mediating ferromagnetism via the surface-state Dirac
fermions versus via the bulk-state carriers, particularly given the diverging Fermi wavelength of the Dirac
fermions when the Fermi energy approaches the Dirac point
[
18
,
22
–
24
]
. While quantitative details of the
microscopic mechanism responsible for
T
c
2D
?
T
c
3D
require further investigation, the relatively high
T
c
2D
values are promising for realistic spintronic applications, particularly if our
fi
ndings may be generalized to more
homogeneously Cr-doped 3D-TIs such as
(
Bi
1
−
x
Cr
x
)
2
Te
3
[
18
]
.
4.4. Topological spin textures
The appearance of both single- and double-peak magnetic impurity resonances is suggestive of two types of
topological defects associated with an isolated magnetic impurity: The single-peak resonance may be attributed
to a helical Dirac spin texture coupling to a magnetic moment pointing along the
c
-axis, whereas the double-
peak resonance is associated with two opposite chiral spin textures of Dirac fermions coupling to an in-plane
magnetic moment
[
21
]
, as illustrated in
fi
gure
6
(
e
)
. Our assignment of the single-peak resonances in this work to
the helical Dirac spin texture coupled with isolated
c
-axis magnetic moments can be justi
fi
ed by their strong and
non-monotonic temperature dependence near
T
c
2D
(
fi
gure
6
(
f
))
, and also by their suppression under large
c
-axis
magnetic
fi
elds. In contrast, our previous studies of non-magnetic impurity resonances in pure Bi
2
Se
3
MBE-
grown thin
fi
lms
[
32
]
did not
fi
nd any dependence of their occurrences on either the temperature or the applied
magnetic
fi
eld, implying that the physical origin of the single-peak resonances in pure Bi
2
Se
3
is fundamentally
different from that of the single-peak resonances observed in the bilayer systems.
Our
fi
nding of occurrences of impurity resonances only along the borders of gapped and gapless regions is
consistent with suppressed long-range magnetic order along the boundaries between the
c
-axis oriented
(
gapped
)
and in-plane oriented
(
gapless
)
magnetic domains
[
31
]
. Thus, proximity-induced magnetization in the
top Bi
2
Se
3
layer is also much suppressed for regions above the boundaries of
c
-axis and in-plane magnetic
domains
[
31
]
, and Cr-ions located in these regions were more likely to decouple from long-range magnetization
and became effectively isolated magnetic impurities, particularly with the assistance of thermal
fl
uctuations at
suf
fi
ciently high temperatures
(
fi
gure
6
(
f
))
.
Finally, we note that the spin textures associated with an in-plane magnetic impurity may be considered as a
stable
‘
topological bit
’
with two levels associated with the two opposite spin chirality. In principle, for a
completely isolated topological bit, the two-level states may be tuned by a local
c
-axis magnetic
fi
eld. Further, for
E
F
→
E
D
these topological bits are long-lived as the result of topological protection. On the other hand, tuning
E
F
away from
E
D
will result in increasing coupling among spatially separated topological bits
[
21
]
, which may be
viewed as inducing an effective entanglement of wavefunctions among these topological bits. The feasibility of
tuning the two levels within a topological bit by an external magnetic
fi
eld and the coupling among spatially
10
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et al