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!