Topological Insulators and Quantum

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

Hall Effect: 1879

(non-magnetic materials)

Magnetic property

-1.0 -0.5 0.0 0.5 1.0

-10 -5 0 5 10

R yx ()

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 Effect

metal Si

Klaus von Klitzing r _xy = h / ie ² r _xx = 0

2D electron gas

H

(1980)

Fractional Quantum Hall Effect

Tsui

Stormer

Laughlin

(1982)

AlGaAs

2DEG GaAs

H

1879 Hall Effect

E. Hall

H R_H

1881 Anomalous HE

H

R_H

H R_H h/e²

h/2e²

1980 Integer QHE (Si)

1982

Fractional QHE (GaAs)

K. von Klitzing B. Laughlin H. Stormer D. Tsui

1985 1998

IQHE FQHE

2010 Half-integer

(graphene)QHE

A. Geim K. Novoselov

From Hall Effect to Quantum Hall Effects (QHE)

1879 Hall Effect

E. Hall

H R_H

1881 Anomalous HE

H

R_H

H R_H h/e²

h/2e²

1980 Integer QHE (Si)

1982

Fractional QHE (GaAs)

K. von Klitzing B. Laughlin H. Stormer D. Tsui

1985 1998

IQHE FQHE

2010 Half-integer

(graphene)QHE

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 ^{X.-G. Wen} Kitaev Moore Read

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 curvature

c : Euler number

  ^ c

S

2 KdA 1

c = 2 c = 0

Topology

Gauss Bonnet

: Berry’s curvature C : Chern number

C d

BZ



  ^k

 2

1

E E

C = 0 C = 1

Topological property of the electronic structure of a 2D insulator

Berry Chern

“TKNN”

T: Thouless

Nobel laureate

in Physics 2016

2005: Topological Insulators

Topological Insulators (2005—)

Dark matter on desktop

Wilczek, Nature 2009

Qi & Zhang, Science 2009 J. Moore, Nature 2010

Hasan & Kane: Rev. Mod. Phys. 2010 Qi & 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”

k

k

E

p c

H

mc p

c

H  

 

















(m=0)

Conductor Insulator Classification of Materials (new)

Topological Insulator

Insulating (bulk)

conducting (surface)

Spin-Orbital

Coupling

Zhang et al., Nat. Phys. 5, 438 (2009) Xia et al., Nat. Phys. 5, 398 (2009)

Sb

Te

Bi

Te

Bi

Se

Bi

Se

3D Topological Insulators: Bi ₂ Se ₃ , Bi ₂ Te ₃ , Sb ₂ Te ₃

Hasan group Shoucheng Zhang group

300 meV

Se

Bi

Chen et al., Science 2009

Bi ₂ Te ₃

Fisher (Stanford)

Bi ₂ Se ₃

Cava (Princeton)

Dirac

Xia et al., Nat. Phys. 2009 Cone

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 insulating

Surface 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)

E_k: kinetic energy hu : photon energy W : work function

E_k = 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

T

>> T

> T

High VI (Te/Se) flux Growth rules:

Y. Y. Li et al., Adv. Mater. 2010

(1) Stoichiometric: low impurities

(2) Layer-by-layer: flat & single crystalline

Atomically flat Bi ₂ Te ₃ 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

k _x k _y

E_F

E

ARPES: Bi ₂ Te ₃ band structure

Bi

Te

Si substrate

Experimentally confirmed：

Massless Dirac Cone

Insulating topological insulator

Atomically flat Bi ₂ Se ₃ 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 QL

Binding Energy (eV)

1QL

2 QL

k_// (Å^-1)

E_F

k_// (Å^-1)

Bi ₂ Se ₃ 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, MoSe

and other layered materials

Critical

thickness

APRES test for thin Bi₂Se₃ film grown on graphene SiC surface

Binding Energy (eV)

k_// (Å^-1) k_// (Å^-1) k_//(Å^-1) k_// (Å^-1) k_//(Å^-1)

E_F

7QL 8QL 9QL 10QL 15QL

50QL

Binding Energy (eV)

E_F

k_//(Å^-1)

Position of Dirac Point Gap size

Sb ₂ Te ₃

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

0 1

2 3 4 65 78 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

R _H = h/e ² = 25812.807449 Ω

2011.05

2012.12

2012.01

2012.10

year

• Sharpen your tools

• Work hard

Quantum Anomalous Hall Effect ^{in Cr}

^(Bi

^Sb

⁾

^Te

-55 V 0 V 220 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/e

)

 2 samples at T = 30 mK (full quantization at h/e

)

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. r

is nearly independent of H.

(perfect ferromagnetic ordering and charge neutrality)

3. At -1.5V, r

= h/e

H h/e

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 desktop

Wilczek, 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 R_H

1881 Anomalous HE

H

R_H

H R_H h/e²

h/2e²

1980 Integer QHE

1982 Fractional

QHE

2016 Topological

Phase Transitions Topological

Phases of Matter

2005 Half-integer

QHE

Si ^{GaAs Graphene}

Quantum Anomalous Hall Effect

TI

Next?

• QAHE at higher temperatures

• Other novel topological states of matter

New progresses in QAHE

QAH Axion

insulator

MTITI MTI

MTI MTINI

C=2 QAH QSH

C=N QAH

tune thickness

Magnetic Weyl semimetal

MTINI

…

“Penta-layer” Cr-doped (Bi,Sb) ₂ Te ₃

Perfect quantization at 0.5 K and zero field

Tokura Group: APL 107, 182401 (2015)

MIT/PSU/Stanford

V-doped Sb

Te

Quantized Anomalous Hall Effect in V-Sb ₂ Te ₃

(~4%)

Moodera Group (MIT)

QAHE at higher T

Cr+V co-doped (BiSb)

Te

-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

μ₀H (T)

ρ_yx ρ_xx 300 mK V_g = V_CNP

-150 -100 -50 0 50 100 150

0.0 0.2 0.4 0.6 0.8 1.0

r (h/e2 )

V_g-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 mK 0 T

MnBi ₂ Te ₄ : 3D TI by MBE

5SL MnBi

Te

:

• 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 MnBi ₂ Te ₄

7 SL

Xianhui Chen (USTC) and Yuanbo Zhang (Fudan): arXiv: 1904.11468 Yayu 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 V_g = 110 V

r yx (h/e2 )

P₁

P₂

P^'₁ -1.0

-0.5 0.0 0.5 1.0

P^'₁

T = 50 mK V_g = 110 V

 xy (e2 /h)

P₁

P₂

-1 0 1

0 10 20

T = 50 mK V_g = 110 V

r xx (h/e2 )

₀H (T)

-1 0 1

0.0 0.2 0.4 0.6

T = 50 mK V_g = 110 V

 xx (e2 /h)

₀H (T)

5 QL 5 QL 3 QL

• When the magnetization directions of the top and bottom layers are parallel, QAH (ρ_xx=0, ρ_xy= h/e²).

• Its longitudinal resistance becomes very large (ρ_xx> 20 h/e²) when anti- parallel.

Spin valve Sample

Structure

Synthetic Quantum Spin Hall Effect

-1 0 1

-0.5 0.0 0.5

R_14,35 R_14,26

R yx (h/e2 )

H (T)

0.5 1.0 1.5

R 14,14 (h/e2 )

0.0 0.5

R_14,23 R_14,65

R xx (h/e2 )

-1 0 1

0.0 0.5

R 14,54 (h/e2 )

H (T)

1

2 3 4

6 5

QAHC=2 QSH

• 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 ¹⁹⁸²

QAHE 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!