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Topological Insulators and Quantum Anomalous Hall Effect
Tsinghua University
Stockholm, June 21, 2019
Qi-Kun Xue
• Introduction
• MBE-STM-ARPES of topological insulators
• Realization of Quantum Anomalous Hall Effect
• Summary
OUTLINE
Linear dependence
R B
I
Hall Effect: 1879
(non-magnetic materials)
Magnetic property
-1.0 -0.5 0.0 0.5 1.0-10
-5
0
5
10
Ryx (
)
0H (T)
Anomalous Hall Effect: 1881
(magnetic materials)
Edwin H. Hall
Hall Effect and Anomalous Hall Effect
R. Karplus, J. M. Luttinger, Phys. Rev. 95, 1154 (1954)
J. Smit, Physica 24, 39 (1958)
L. Berger, Phys. Rev. B2, 4559 (1970)
Spin-orbit coupling:intrinsic
Skew scattering:extrinsic
Side jump: extrinsic
Anomalous Hall Effect: Mechanism
Nagaosa, Sinova, Onoda, MacDonald, Ong,
Review of Modern Physics 2009
Applications
Hall effect + IC
Integer Quantum Hall Effectmetal
Si
Klaus von Klitzing rxy = h / ie2
rxx = 0
2D electron gas
H
(1980)
Fractional Quantum Hall EffectTsui
Stormer
Laughlin
(1982)
AlGaAsGaAs
2DEG
H
1879
Hall Effect
E. Hall
H
RH
1881
Anomalous HE
H
RH
H
RH h/e2
h/2e2
1980
Integer QHE (Si)
1982
Fractional QHE(GaAs)
K. von Klitzing D. TsuiB. Laughlin H. Stormer
19981985
IQHE FQHE
2010
Half-integerQHE
(graphene)
A. Geim K. Novoselov
From Hall Effect to Quantum Hall Effects (QHE)
1879
Hall Effect
E. Hall
H
RH
1881
Anomalous HE
H
RH
H
RH h/e2
h/2e2
1980
Integer QHE (Si)
1982
Fractional QHE(GaAs)
K. von Klitzing D. TsuiB. Laughlin H. Stormer
19981985
IQHE FQHE
2010
Half-integerQHE
(graphene)
A. Geim K. Novoselov
From Hall Effect to Quantum Hall Effects (QHE)
Quantum Anomalous Hall Effect?
The first Theoretical Proposal for Quantum Hall Effect
without Magnetic Field
Graphene with broken TRS
• Haldane conceived a model that will show QHE in zero magnetic field, it is now called the “Chern insulator”
• It is very abstract and way ahead of its time, but it is highly influential about 20 years later in the field of topological insulators
Topological States of Matter
…
Haldane Kitaev Moore ReadX.-G. Wen
Zoo of quantum-topological phases of matter X. –G. Wen Rev. Mod. Phys. 89, 041004 (2017)
S. –C Zhang
Quantum anomalous Hall effect, Haldane phase, Non-abelion anyons, Topological order, String-net condensation…...
Gauss-Bonnet theorem
K: Gauss curvaturec : Euler number
c
S
KdA2
1
c = 2 c = 0
Topology
Gauss Bonnet
: Berry’s curvature
C : Chern number
CdBZ
k2
1EE
C = 0 C = 1
Topological property of the electronic structure of a 2D insulator
Berry Chern
“TKNN”
T: Thouless
Nobel laureatein Physics 2016
2005: Topological Insulators
Topological Insulators (2005—)
Dark matter on desktopWilczek, Nature 2009
Qi & Zhang, Science 2009
J. Moore, Nature 2010
Hasan & Kane: Rev. Mod. Phys. 2010Qi & Zhang: Rev. Mod. Phys. 2011
Quantum Anomalous Hall Effect
Quantum Spin Hall Effect
Majorana Fermions
Magnetic Monopole and Dyon
TME Effect and Axion
……………...
Ordinary versus Topological Insulators
Valence Band
Conduction Band
Rashba Spin-Orbit Splitting of Surface States
Valence Band
Conduction Band
Ordinary Insulator
Time reversal symmetry, Strong S-O coupling
Spin up
Spin down
Strong spin-orbit coupling
Topological Insulator
“band twisting”
kx ky
E
pcH
mcpcH
2
(m=0)
Conductor Insulator
Classification of Materials (new)
TopologicalInsulator
Insulating (bulk)
conducting (surface)
Spin-Orbital
Coupling
Zhang et al., Nat. Phys. 5, 438 (2009) Xia et al., Nat. Phys. 5, 398 (2009)
Sb2Te3 Bi2Te3
Bi2Se3
Bi2Se3
3D Topological Insulators: Bi2Se3, Bi2Te3, Sb2Te3
Hasan group Shoucheng Zhang group
300 meV
Se
Bi
Chen et al., Science 2009
Bi2Te3
Fisher (Stanford)
Bi2Se3
Cava (Princeton)
Dirac Cone
Xia et al., Nat. Phys. 2009
Zhixun Shen (Stanford) Hasan (Princeton)
Electron Band Structure of 3D TI by ARPES
n-type conductor(Se vacancies)
(Similar to that in ZnO)
Topological Insulator Material
“insulator” by definition: Bulk insulatingSurface metallic (2D)
(real space)
bulk
High quality: low defect/impurity
density
If the bulk is conducting, it is difficult to measure the transport property of its surface with exotic topological property.
Molecular Beam Epitaxy (MBE)(Cho & Arthur, 1970)
Atomic-Level
Scanning Tunneling Microscope (STM)(Binnig & Rohrer, 1981)
Ek : kinetic energy
hu : photon energy
W : work function
Ek = hu – W – E (k//)
E(k//): band dispersion
Angle-Resolved Photoemission Spectroscopy (ARPES)
+
MBE-STM-ARPES
STM
MBE
ARPES
Omicron + VG Scienta
• Introduction
• MBE-STM-ARPES of topological insulators
• Realization of Quantum Anomalous Hall Effect
• Summary
OUTLINE
Establishment of MBE growth conditions
RHEED
Real time RHEED intensity oscillation
TBi >> TSub > TTe/Se
High VI (Te/Se) fluxGrowth rules:
Y. Y. Li et al., Adv. Mater. 2010
(1) Stoichiometric: low impurities(2) Layer-by-layer: flat & single crystalline
Atomically flat Bi2Te3 films by MBE
Y. Y. Li et al., Adv. Mater. (2010)G. Wang et al., Adv. Mater. (2011)X. Chen et al., Adv. Mater. (2011)
16 nm x 16 nm
Te atom
kx ky
EEF
ARPES: Bi2Te3 band structure
Bi2Te3
Si substrate
Experimentally confirmed:
Massless Dirac Cone
Insulating topological insulator
Atomically flat Bi2Se3 films on graphene by MBE
200 nm x 200 nm
-120mV
50 QL
Yi Zhang et al., Nature Physics 6, 584 (2010)
Figure 2
k// (Å-1) k// (Å-1) k// (Å-1)
3 QL 5 QL 6 QLBin
din
g E
nerg
y (
eV
)
1QL 2 QL
k// (Å-1)
EF
k// (Å-1)
Bi2Se3 Band Structure: layer-by-layer
1 QL
Yi Zhang et al., Nature Phys. 6, 584 (2010)
• The thickness and band structure can be controlled with atomic-layer precision by MBE
• Applied to FeSe, MoSe2 and other layered materials
Critical thickness
APRES test for thin Bi2Se3 film grown on graphene SiC surface B
indin
g E
nerg
y (
eV
)
k// (Å-1) k// (Å-1) k// (Å-1) k// (Å-1) k// (Å-1)
EF
7QL 8QL 9QL 10QL 15QL
50QL
Bin
din
g E
nerg
y (
eV
)
EF
k// (Å-1) Position of Dirac PointGap size
Sb2Te3
Y. P. Jiang, PRL 108, 016401 (2012)Y. P. Jiang PRL 108, 066809 (2012)
0.5 m x 0.5 m
STM study of fundamental properties of TIs
01
2 3 4 6587
9
10
11
12
B = 10T
Quantum Interference
Zhang et al., PRL 103, 266803 (2009)
Cheng et al., PRL 105, 076801 (2010)
Absence of backscattering
Jiang et al., PRL 108, 016401 (2012)
Jiang et al., PRL 108, 066809 (2012)
Massless Dirac fermion
(Landau Quantization)
Chang et al., PRL 115, 066809 (2015)
Song et al., PRL 114, 176602 (2015)
With MBE-STM, we are able to prepare high quality
epitaxial thin films and demonstrate their exotic
electronic structure…
New Effect or Law!
• Introduction
• MBE-STM-ARPES of topological insulators
• Realization of Quantum Anomalous Hall Effect
• Summary
OUTLINE
QAHE in magnetic topological insulator
Chaoxing Liu et al. proposed that a 2D topological insulator with ferromagnetic order,
but this compound cannot be made ferromagnetic
• TI could remain ferromagnetic when it is insulating (van Vleck mechanism)
• The Bi2Se3 family topological insulator was proposed to be perfect candidate
Science (2010)
Term: QAHE
2D TI: helical edge states
QAHE in 2D magnetic TIs
QAHE: chiral edge state
Requirements for QAHE: 2D Ferromagnetic Topological Insulator
• It must be magnetic, so there is anomalous Hall effect at B = 0
• It must be topological, so there are spontaneous edge states
• It must be insulating, so there is only edge state transport
The QAHE puts stringent requirements for materials:
• Most ferromagnetic materials are metallic
• Magnetic order is difficult to realize in 2D
• Magnetism and topology may be against each other
QAHE in 2D magnetic TIs
RH = h/e2 = 25812.807449 Ω
2011.05
2012.12
2012.01
2012.10
year
• Sharpen your tools
• Work hard
Quantum Anomalous Hall Effect in Cr0.15(Bi0.1Sb0.9)1.85Te3
-55 V 220 V0 V
30 mK
experiment
20 samples at T = 1.5 K 6 samples at T = 90 mK (zero-field r = 0.87 to 0.98 h/e2) 2 samples at T = 30 mK (full quantization at h/e2)
C. Z. Chang et al., Science 340, 167 (2013)
-1.5 V
1. At different gate voltage, nearly no
change in the shape and coercivity.
(van Vleck mechanism)
2. ryx is nearly independent of H.
(perfect ferromagnetic ordering and
charge neutrality)
3. At -1.5V, ryx = h/e2
H
h/e2
theory
0
Y. Tokura (Tokyo/RIKEN)
K. L. Wang (UCLA)
J. Moodera (MIT)
D. Gordhaber-Gondon (Stanford)
QAHE by other groups
N. P. Ong (Princeton)
N. Sarmath (Penn State)
D. Thouless F. Haldane J. Kosterlitz
Topological Insulators (2005—present)
Dark matter on desktopWilczek, Nature 2009
Qi & Zhang, Science 2009
J. Moore, Nature 2010
Reviews: Qi & Zhang: Phys. Today 2009
Hasan & Kane: Rev. Mod. Phys. 2010
Qi & Zhang: Rev. Mod. Phys. 2011
(Dirac/Weyl semimetals)
Quantum Anomalous Hall Effect
Quantum Spin Hall Effect
Majorana Fermions
Magnetic Monopole and Dyon
TME Effect and Axion
……………...
R. Karplus, J. M. Luttinger, Phys. Rev. 95, 1154 (1954)
J. Smit, Physica 24, 39 (1958)
L. Berger, Phys. Rev. B2, 4559 (1970)
Spin-orbit coupling:intrinsic
Skew scattering:extrinsic
Side jump: extrinsic
Anomalous Hall Effect: Mechanism
Nagaosa, Sinova, Onoda, MacDonald, Ong,
Review of Modern Physics 2009
Material Driven Discoveries
1879
Hall Effect
H
RH
1881
Anomalous HE
H
RH
H
RH h/e2
h/2e2
1980
Integer QHE
1982
FractionalQHE
2016
Topological Phase Transitions Topological Phases of Matter
2005
Half-integerQHE
Si GaAs Graphene
Quantum Anomalous Hall Effect
TI
2013Next?
• QAHE at higher temperatures
• Other novel topological states of matter
New progresses in QAHE
QAHAxion
insulator
MTITI
MTI
MTI
MTINI
C=2 QAH QSH
C=N QAHtune
thickness
Magnetic Weyl semimetal
MTINI…
“Penta-layer” Cr-doped (Bi,Sb)2Te3
Perfect quantization at 0.5 K and zero field
Tokura Group: APL 107, 182401 (2015)
MIT/PSU/Stanford
V-doped Sb2Te3
Quantized Anomalous Hall Effect in V-Sb2Te3
(~4%)
Moodera Group (MIT)25 mK
QAHE at higher T
Cr+V co-doped (BiSb)2Te3
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.0
-0.5
0.0
0.5
1.0
r (
h/e
2)
μ0H (T)
ρyx
ρxx
300 mK
Vg = V
CNP
-150 -100 -50 0 50 100 150
0.0
0.2
0.4
0.6
0.8
1.0
r (
h/e
2)
Vg-V
CNP (V)
ρxx
ρyx
300 mK
0 T
Perfect quantization at T = 300 mK
Y. B. Ou et al., APL Materials 4, 086101 (2016)
300 mK
300 mK0 T
MnBi2Te4: 3D TI by MBE
5SL MnBi2Te4:
• Chern No. =1
• Gap: ~52meV
QAHE
Y. Gong et al., Chin. Phys. Lett. 36, 076801 (2019) (June 2, 2019)
QAHE in single crystal flakes of MnBi2Te4
7 SL
Xianhui Chen (USTC) and Yuanbo Zhang (Fudan): arXiv: 1904.11468Yayu Wang (Tsinghua): arXiv: 1905.00715
Requires a strong magnetic field
“Spin valve” based on QAH edge states
MTI (Cr/V: 0.16/0.84—larger coercivity)
MTI (Cr/V: 0.4/0.6—smaller coercivity)
TI (non-magnetic)
-1.0
-0.5
0.0
0.5
1.0
T = 50 mK
Vg = 110 V
ry
x (
h/e
2)
P1
P2
P'
1
-1.0
-0.5
0.0
0.5
1.0
P'
1
T = 50 mK
Vg = 110 V
x
y (
e2/h
)
P1
P2
-1 0 1
0
10
20
T = 50 mK
Vg = 110 V
rx
x (
h/e
2)
0H (T)
-1 0 1
0.0
0.2
0.4
0.6
T = 50 mK
Vg = 110 V
x
x (
e2/h
)
0H (T)
5 QL
5 QL 3 QL
• When the magnetization directions
of the top and bottom layers are
parallel, QAH (ρxx=0, ρxy= h/e2).
• Its longitudinal resistance becomes
very large (ρxx> 20 h/e2) when anti-
parallel.
Spin valve
Sample
Structure
Synthetic Quantum Spin Hall Effect
-1 0 1
-0.5
0.0
0.5
R14,35
R14,26
Ryx (
h/e
2)
H (T)
0.5
1.0
1.5
R1
4,1
4 (
h/e
2)
0.0
0.5
R14,23
R14,65
Rxx (
h/e
2)
-1 0 1
0.0
0.5R
14,5
4 (
h/e
2)
H (T)
1
2 3 4
56
C=2QAHQSH
• When two QAH sub-systems have the same
magnetization direction (strong field), the system
become a QAH insulator with Chern number 2.
• In the case of opposite magnetization, it becomes a
QSH system.
Summary
• QAHE is well-established quantum state of matter, independently realized by many groups.
• QAHE forms a platform for other exotic states of matter.
Hall Effect Anomalous Hall Effect
1879 1881
IQHE 1980
FQHE 1982QAHE 2013
with Magnetic Field w/o Magnetic Field
science is art
Acknowledgements
Group Members: Ke He, Xucun Ma, Xi Chen, Lili Wang, S. H. Ji (Tsinghua/IOP)
Jinfeng Jia (Shanghai Jiao-Tong Univ.)
Transport: Yayu Wang (Tsinghua), Li Lv, Y. Q. Li (IOP)
Theory: Shoucheng Zhang (Stanford)
Bangfen Zhu, Wenhui Duan (Tsinghua), Zhong Fang, Xi Dai (IOP), X. L. Qi (Stanford), C. X. Liu (Penn State), Shengbai Zhang (RPI), X. C. Xie (PKU), S. Q. Shen (Hong Kong), Feng Liu (Utah)
$$$: NSF, MOST, MOE of China
Thank you very much!