PHYSICAL REVIEW B
109
, 075125 (2024)
Phonon and defect mediated quantum anomalous Hall insulator
to metal transition in magnetically doped topological insulators
Akiyoshi Park
,
1
,
2
Adrian Llanos,
2
Chun-I Lu
,
1
Yinan Chen
,
1
,
2
Sebastien N. Abadi
,
1
Chien-Chang Chen,
1
,
2
Marcus L. Teague,
1
,
2
Lixuan Tai
,
3
Peng Zhang
,
3
Kang L. Wang,
3
and Nai-Chang Yeh
1
,
2
,
*
1
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
3
Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA
(Received 3 October 2023; accepted 19 January 2024; published 12 February 2024)
Quantum anomalous Hall (QAH) state in six quintuple layer Cr
0
.
1
(Bi
0
.
2
Sb
0
.
8
)
1
.
9
Te
3
thin films were studied
through scanning tunneling spectroscopy (STS) and electrical transport measurements. While the surface state is
gapless above the Curie temperature (
T
C
≈
30 K), scanning tunneling spectroscopy (STS) of the sample reveals
a topologically nontrivial gap with an average value of
≈
13
.
5 meV at 4.2 K below the ferromagnetic transition.
Nonetheless, areal STS scans of the magnetic topological insulator exhibit energy modulations on the order of
several meV’s in the surface bands, which result in the valence band maximum in some regions becoming higher
than the energy of the conduction band minimum of some other regions that are spatially separated by no more
than 3 nm. First-principles calculations demonstrate that the origin of the observed inhomogeneous energy band
alignment is an outcome of many-body interactions, namely electron-defect interactions and electron-phonon
interactions. Defects play the role of locally modifying the energy landscape of surface bands while electron-
phonon interactions renormalize the surface bands such that the surface gap becomes reduced by more than
1 meV as temperature is raised from 0 to 4.2 K. These many-body interactions at a finite temperature result in
substantial increase of electron tunneling across the spatially separated conduction band pockets even for finite
temperatures well below
T
C
, thus driving the magnetic topological insulator out of its QAH insulating phase into
a metallic phase at a relatively low temperature.
DOI:
10.1103/PhysRevB.109.075125
I. INTRODUCTION
Experimental observations of the quantum anomalous Hall
effect (QAHE) among Cr-doped (Bi
,
Sb)
2
Te
3
, a system of
magnetic topological insulators (MTIs), have been bound to
below sub-Kelvin temperatures [
1
–
3
]. Such constraint of the
QAHE to low temperatures is unexpected given that these
MTIs exhibit long-range ferromagnetic ordering at much
higher temperatures [
4
]. Specifically, transport measurements
of Cr-doped (Bi
,
Sb)
2
Te
3
have shown bulk Curie temperatures
(
T
C
) ranging from 20–30 K, whereas the appearance of QAHE
has been limited to temperatures below 0.1 K [
4
].
It has been proclaimed that the bottleneck in the tempera-
ture range of the QAHE is the lack of spatially homogeneous
ferromagnetic ordering such that the region with gapped
topological surface states is beneath the percolation thresh-
old at higher temperatures [
5
–
8
]. Nonetheless, significantly
improved magnetic homogeneity using either magnetic mod-
ulation doping in heterostructures [
9
] or codoping Cr with
Vatoms[
10
] could only raise the threshold temperature for
QAHE to approximately 2 K.
Fundamentally for QAHE to ensue, the effective Zeeman
field emanating from ferromagnetic ordering closes the topo-
logically trivial hybridization gap through band inversion and
opens up a nontrivial magnetic gap, thus creating a Chern
*
Corresponding author: ncyeh@caltech.edu
insulator phase characterized by the topological invariant,
C
=±
1, referred to as the Chern number. Although some
experiments have reported the presence of the magnetic gap
opening in the surface states of MTIs [
7
,
11
–
14
], there are
contradictory results that indicate otherwise [
15
,
16
]. There are
also reports of opening of a topological nontrivial gap being
masked by midgapped impurity states [
17
,
18
] or bulk states
[
19
]. In addition, it has been argued that the Dirac point in
MTIs is degenerate with the bulk valence band, thereby com-
promising the conductivity [
20
,
21
]. Hence, the issue regarding
whether magnetic doped (Bi
,
Sb)
2
Te
3
are true topological in-
sulators has not been settled.
In particular, the lack of general agreement among reports
concerning the electronic structure of these materials is at-
tributed to inadequate distinctions between MTI samples in
the bulk and the two-dimensional (2D) limit. The two are
fundamentally different, given that the QAHE has only been
observed in epitaxial films [
6
,
21
,
22
]. The absence of quan-
tized transport in bulk single crystal is attributed to higher
defect concentration such that the chemical potential falls
outside the surface gap [
7
,
23
], and a larger structural inversion
asymmetry that appears to be an obstacle for a topological
phase transition to the Chern insulator state [
24
]. The lack
of a consensus thus calls for a more thorough investigation
to map out the energies of the surface and bulk electronic
states above and below the topological phase transition tem-
perature in 2D MTI-epitaxial films, which duly exhibit a QAH
phase.
2469-9950/2024/109(7)/075125(14)
075125-1
©2024 American Physical Society
AKIYOSHI PARK
et al.
PHYSICAL REVIEW B
109
, 075125 (2024)
-4
-2
0
2
4
-15
-10
-5
0
5
10
15
R
xy
(k
)
30 K
22 K
20 K
18 K
10 K
2 K
0
10
20
30
40
50
60
70
80
90
100
T
(K)
0
2
4
6
8
10
R
xy
(k
)
0 Oe
750 Oe
4000 Oe
(a)
(b)
(c)
(d)
T
C
-4
-2
0
2
4
H
(kOe)
0
5
10
15
20
R
xx
(
h/e
2
)
-1.5
-1
-0.5
0
0.5
1
1.5
R
xy
(
h/e
2
)
H
(kOe)
H || c
T
= 100 mK
0123
T
(K)
0
0.5
1
1.5
2
σ
(
e /h
2
)
σ
xx
σ
xy
10 kOe
H || c
FIG. 1. Temperature and magnetic field dependence of longitudi-
nal and Hall resistance in Cr
0
.
1
(Bi
0
.
2
Sb
0
.
8
)
1
.
9
Te
3
(a) The temperature
dependence of
R
xy
measured at various magnetic fields. A rapid
increase in zero-field
R
xy
appears around
T
=
30 K, suggesting the
onset of ferromagnetism. (b) Hysteric behavior in
R
xy
dependence
of magnetic field (
H
) at different temperatures between 30 K and
2K.(c)
σ
xy
and
σ
xx
upon field cooling with
H
=
10 kOe exhibiting
QAHE at
T
=
100 mK. (d) Magnetic field sweep of
R
xy
and
R
xx
at
T
=
100 mK.
II. EXPERIMENTAL AND COMPUTATIONAL METHODS
A. Sample growth
Cr
y
(Bi
1
−
x
Sb
x
)
2
−
y
Te
3
thin films with an average thickness
of six quintuple layers (QLs) were grown by molecular beam
epitaxy (MBE) on (111) GaAs substrates in an ultrahigh
vacuum Perkin-Elmer instrument. The details of the method
have been outlined in Ref. [
21
]. The thickness of the sample
is determined through the oscillation of reflection high-
energy electron diffraction (RHEED) patterns over the growth
process.
Although there are terracelike structures and thickness
variation over different regions of on the thin film [Fig.
2(a)
],
the average thickness is determined to be six QLs, which is
sufficiently thick to suppress the hybridization gap between
the bottom and top surface states to
≈
5 meV, but still thin
enough to reduce the effect of structural inversion asymmetry
that prevents band inversion to a topological nontrivial phase
[
25
,
26
].
B. Scanning tunneling microscopy/spectroscopy
Scanning tunneling microscopy/spectroscopy (STM/STS)
measurements were performed in a homemade microscope
connected to a RHK R9 controller. To achieve liquid helium
temperatures, small traces of ultrahigh purity He (99
.
999%)
exchange gas were added to the vacuum chamber. Tunneling
0
1
2
3
4
5
6
7
8
(a)
5 nm
50 nm
z
(nm)
(b)
FIG. 2. STM Topographic images at
T
=
4
.
2 K. (a) Images of
terracelike structures were observed over a larger area of (300 nm
×
300 nm) due to the layered nature of Cr
0
.
1
(Bi
0
.
2
Sb
0
.
8
)
1
.
9
Te
3
, obtained
with scanning parameters of
I
t
=
100 pA and
V
b
=
1V.(b)Asmaller
area of (11 nm
×
11 nm), where individual atoms were resolved with
scanning parameter of
I
t
=
1nAand
V
b
=
100 mV. The atomic scale
image is processed through Fourier filtering to improve the signal-to-
noise ratio of the image.
spectroscopy was measured through an ac lock-in amplifier
with a bias modulation of 10 mV amplitude. Mechanically
cleaved Pt-Ir alloy wires were used as the STM tips.
C. Transport measurements
Thin film samples for transport measurements were first
patterned into a Hall bar geometry. Temperature and mag-
netic field dependence of longitudinal and Hall resistance was
measured by using a Physical Property Measurement System
(PPMS) by Quantum Design. Three SRS SR830 Lock-in Am-
plifiers were employed: two were used to monitor the Hall
voltage and the longitudinal voltage, and one was used to
supply a current by applying an ac voltage of
V
rms
=
10 mV
and frequency between
f
=
5–15 Hz across a 1 M
reference
resistor.
D. First-principles calculations
Density functional theory (DFT) calculations were per-
formed by
QUANTUM ESPRESSO
(QE) using a fully relativistic
pseudopotential and setting up noncollinear calculations to
incorporate the effects of spin-orbit coupling [
27
,
28
]. For
calculating the electronic structure of the parent compound
Sb
2
Te
3
, initially a (4
×
4
×
1)
k
-point mesh was con-
structed to obtain self-consistent electron densities through
self-consistent field (SCF) calculations followed by non-self-
consistent field (NCF) calculations to obtain eigenvalues of a
finer (40
×
40
×
1)
k
-point mesh.
Furthermore, to implement and investigate various per-
turbative disorder to topological insulators such as defects,
magnetic dopants, and electron-phonon coupling, which break
translational lattice symmetry, a change from reciprocal-space
to real-space basis is required. This was performed through
using Bloch states obtained from QE to construct a real-space
lattice Hamiltonian by computing maximally localized Wan-
nier functions (MLWFs) via the
WANNIER
90 package [
29
]. For
each atom in the unit cell, initial projections were represented
as
p
x
,
p
y
, and
p
z
orbitals with spin up and down, given that the
p
orbitals of (Bi,Sb) and Te are the most relevant bands lying
075125-2
PHONON AND DEFECT MEDIATED QUANTUM ANOMALOUS ...
PHYSICAL REVIEW B
109
, 075125 (2024)
close to the Fermi level (
E
F
)[
30
]. Hence, a three-QL unit
cell consisting of 30 atoms would yield a (90
×
90)-matrix
Hamiltonian, and a (180
×
180)-matrix Hamiltonian for a six-
QL unit cell.
The electronic structures of the doped system were calcu-
lated through construction of supercells with various doping
configurations. The supercell band structures were determined
through SCF and NCF calculations using QE followed by
Wannerization, as with the case for the unit cell calculations.
Since the Brillouin zone (BZ) of the supercell is smaller
than the BZ of the unit cell, an extra step of band unfolding
was necessitated, which was done through the
WANNIERTOOLS
postprocessing software [
31
].
To realize the effects of finite temperatures, electron-
phonon coupling calculations were performed through a
density functional perturbation theory (DFPT) approach. Dy-
namical matrices were computed using the phonon module
in QE for a (3
×
3
×
1)
q
-point mesh. The electron-phonon
coupling parameters were then calculated up to the lin-
ear response regime through the electron-phonon Wannier
(EPW) code using the dynamical matrices computed from
QE [
32
].
III. RESULTS
A. Sample characterization
Six-QL Cr
y
(Bi
1
−
x
Sb
x
)
2
−
y
Te
3
thin films investigated in this
work consist of a Cr-doping level of
y
=
0
.
10, which is the
optimal doping for ferromagnetic order [
21
], and a Sb-doping
level of
x
=
0
.
80 to tune the Fermi energy (
E
F
) right at the
Dirac point (
E
D
)[
1
,
33
]. The presence of bulk ferromagnetic
order due to Cr doping is manifested by the emergence of
a finite Hall resistance (
R
xy
) across the sample upon zero-
field cooling below
T
C
at approximately 30 K [Fig.
1(a)
],
indicative of time-reversal symmetry breaking. The bulk fer-
romagnetism below
T
C
is further verified by the opening of
a hysteresis loop in the field-dependent Hall resistance data
[Fig.
1(b)
], which reveals a finite coercive field (
H
C
) that
decreases with increasing temperature and diminishes to zero
at approximately 30 K. At an even lower temperature of
T
=
100 mK,
σ
xy
reaches a quantized value of
e
2
/
h
, while the lon-
gitudinal resistivity (
σ
xx
) is suppressed to zero [Figs.
1(c)
and
1(d)
], verifying that the sample is indeed a QAH insulator.
Large area STM topographic images indicate that the
sample surface is atomically flat, exhibiting islands with
terracelike features [Fig.
2(a)
], whereas atomically resolved
images in smaller areas reveal a triangular lattice pattern,
representative of the crystalline structure of (Bi
,
Sb)
2
Te
3
in
the
ab
plane. Noticeably, the unevenness in the topography
over a nm scale represented by the darker color patches in
Fig.
2(b)
may be attributed to collective contributions from
Bi/Sb alloying [
34
], Cr dopants (with an atomic radius much
smaller than those of Bi and Sb atoms), or dilute crystalline
defects, including interstitial and vacancies.
To understand the change in the properties of the samples
across the ferromagnetic transition, where time-reversal sym-
metry is broken, scanning tunneling spectroscopy (STS) was
performed both above
T
C
at liquid nitrogen temperature (77 K)
and below
T
C
at liquid helium temperature (4.2 K).
FIG. 3. Scanning tunneling spectra taken with different tunneling
resistance at
T
=
77 K. (a) Schematic representation of tip-sample
distance changing as a consequence of changing the tunneling re-
sistance from 100 M
to 1 G
. The higher the tunneling current,
the closer the tip-sample distance in which the point spectra were
measured. (b)–(e) Tunneling resistance-dependent point spectra were
obtained with a fixed bias of
V
b
=
100 mV and variable tunneling
currents of
I
t
=
300 pA,
I
t
=
600 pA,
I
t
=
800 pA,
I
t
=
1000 pA.
The horizontal dotted lines are the baselines, which indicate the
minima of each spectrum. The insets exhibit the low-energy region
of the spectra with the corresponding Dirac point.
B. STM investigation above the Curie temperature
Information associated with the local density of states
(LDOS) is acquired by conducting differential conductance
(
dI
/
dV
) spectroscopy, in which the height of the STM tip
relative to the sample surface is kept constant while measuring
the tunneling current (
I
t
) as a function of the bias voltage
(
V
b
) between the sample and the tip with a lock-in ampli-
fier. The spatial separation between the tip and the sample is
determined by the tunneling resistance (
R
G
=
V
b
/
I
t
), where
a higher
R
G
corresponds to a larger tip-sample separation
[Fig.
3(a)
]. Figures
3(b)
–
3(e)
exhibit the tunneling spectra
obtained from an atomically flat surface area at 77 K with
varying
R
G
. With
R
G
=
333 M
(
I
t
=
300 pA,
V
b
=
100
mV), a
V
-shaped energy dependence was evident from the
dI
/
dV
spectrum, which was representative of the Dirac sur-
face states with the minimum
dI
/
dV
appearing at the
E
D
[Fig.
3(b)
], and the finite LDOS at and around
E
D
indicates a
075125-3
AKIYOSHI PARK
et al.
PHYSICAL REVIEW B
109
, 075125 (2024)
FIG. 4. Calculated electronic structure of (Bi
,
Sb)
2
Te
3
from
DFT. (a) The band structure and (b) the DOS of Sb
2
Te
3
with a bulk
configuration and a six QL slab configuration. (c) Crystalline
ab
plane of a 2
×
2
×
1 supercell for the parent compound, Sb
2
Te
3
,and
the doped compound, (Bi
0
.
1
Sb
0
.
9
)
2
Te
3
. The dashed lines represent the
supercell and the solid lines represent the unit cell. (d) The unfolded
band structure of both Bi-doped and parent bulk supercell structure.
virtually gapless surface state as expected above
T
C
. Remark-
ably,
E
D
nearly coincides with the Fermi level (corresponding
to
E
=
0 in the tunneling spectrum) [
33
],whichisinstark
contrast to most single-crystal samples with
E
D
more than
100 meV above
E
F
[
7
,
23
,
35
].
As
R
G
is reduced such that the STM tip is brought closer
to the sample surface, the
dI
/
dV
curve loses its
V
-shaped
spectrum and exhibits kinklike features [Figs.
3(c)
–
3(e)
].
Particularly, at low energies near
E
F
, an energy region of
vanishing LDOS is recognizable. This energy region of van-
ishing LDOS corresponds to the bulk band gap (
b
)[
30
],
as suggested by band structure obtained by first-principles
calculations (Fig.
4
). When reducing the tunneling resistance,
the STM tip is sufficiently close to the sample surface, such
that electron tunneling is no longer restricted exclusively be-
tween the tip and the surface states, but also between the
tip and the bulk states that are only a few atomic layers
below the surface. Tuning
R
G
, accordingly, allows selectively
probing either (both) the surface or (and) bulk states in the
Cr
0
.
1
(Bi
0
.
2
Sb
0
.
8
)
1
.
9
Te
3
thin films of our investigation [
36
].
Notably, the Dirac point is located within
b
[Fig.
3(d)
],
and the value of
b
quantified through STS is on the same
energy scale as that calculated from first principles for the
parent compound, Sb
2
Te
3
as well as that of the doped
(Bi
0
.
1
Sb
0
.
9
)
2
Te
3
compound [Fig.
4(d)
]. Comparing the bulk
band structure of Sb
2
Te
3
and (Bi
0
.
1
Sb
0
.
9
)
2
Te
3
, we note that
the
E
F
value of the Bi-doped bulk band structure is 70 meV
02040
-20
0
20
40
E
D
(meV)
0
0.01
0.02
0.03
0.04
0.05
0.06
fraction
10 nm
E
D
(meV)
(a)
(b)
050100
10 nm
(c)
(d)
(meV)
mean: 9.45 meV
0
50
100
150
0
0.01
0.02
0.03
0.04
fraction
mean: 62.7 meV
b
(meV)
b
FIG. 5. Spatial distribution of
E
D
(
r
)and
b
(
r
)at77K.(a)The
spatial map of
E
D
(
r
) measured at 77 K, (b) with the corresponding
histogram. (c) The spatial map of
b
(
r
) measured at 77 K, (d) with
the corresponding histogram.
higher than the
E
F
of the bulk band structure of the parent
compound, owing to electron donation from the Bi atoms.
Otherwise, the bulk band structures of the two compounds are
nearly identical, with a consistent
b
.
Moreover, we address the issue of the absence of spectral
evidence for the edge states near the step edges shown in
Fig.
2(a)
. We attribute the lack of spectral characteristics of
the edge state to local wave-function hybridization between
the surface states above and below the step edge, similar to
the finite-size effect encountered in the ultrathin film (
<
six
QL) limit, which hinders the manifestation of spectral charac-
teristics of the edge states at a step edge.
To evaluate the spatial distribution of
E
D
(
r
) and
b
(
r
), the
tunneling resistance was tuned to 167 M
(
I
t
=
600 pA,
V
b
=
100 mV), such that information from both the surface and the
bulk are probed. With this fixed resistance, the
dI
/
dV
spec-
tra were taken over a (30 nm
×
30 nm) area. The resulting
distributions are depicted in Fig.
5
. The average Dirac point
lies at
E
D
(
r
)
=
(9
.
45
±
7
.
52) meV, where the uncertainty
here and henceforth represents the first standard deviation of
the spatial variation. In this nm-scale scanned area,
E
D
(
r
) has
a narrow distribution, tightly bound to around
E
F
[Figs.
5(a)
and
5(b)
], further signifying how well the Bi/Sb doping was
tuned in our samples to its ideal energy location to optimize
the QAHE. In addition, the magnitude of the mean bulk
band gap was at
b
(
r
)
=
(62
.
7
±
16
.
5) meV [Figs.
5(c)
and
5(d)
], with the mean energies of the bulk valence band max-
ima and the bulk conduction band minima being
E
b
,
V
(
r
)
=
075125-4