Changyoung Kim

Dept of Physics,

Yonsei University

ARPES studies on topological insulators by

angle resolved photoemission

이화 CNRS Lecture (Feb 08, 2011)

Collaborators : S. R. Park, Chul Kim, W. S. Jung, Y. Y. Koh, D. J.

Song, S. Kimura, M. Arita, K. Shimada, S. Kimura, J. H. Han, H. J.

Lee, H. J. Choi, K. D. Lee, N. Hur, J. Y. Kim, B. K. Cho, J. H. Kim,

Y. S. Kwon, J. H. Han

OUTLINE

• Introduction to topology

• Topological insulators

• Metallic surface states & chiral spins

• Verification of existence of TIs

• Quasi-particle dynamics

• Chiral orbital angular momentum states

• Summary

Topology

What is ‘topology’?

• Topologically same (same class)

=

• Topologically different (different class)

=

http://en.wikipedia.org/wiki/File:Mug_and_Torus_morph.gifhttp://en.wikipedia.org/wiki/File:Torus.pnghttp://en.wikipedia.org/wiki/File:Torus.png

Topologically different

• Moebius strip • Simple strip

• ‘Topology’ can be used to sort things out.

That is, it is a way of classifying objects, states or materials.

Defining topology

• How do we mathematically define ‘topology’ of the strip?

+

-

+

-

+ + & - - → +1 topology

+ - & - + → -1 topology

3210; Topological

Quantum Numbers

00

10

Topologically trivial

Topologically non-trivial

• The ‘strip’ can be defined in any space, including momentum

space

• In general, topology is defined by topological quantum number

Topology

Topological number (examples)

+

-• Sign change after 1 turn : -1, non-trivial

• No sign change after 1 turn : +1, trivial

• Moebius strip

• Spins

• +1, Integer spin, Bosons

• -1, Half-integer spin, Fermion

Topological number

http://en.wikipedia.org/wiki/File:Moebius_Surface_1_Display_Small.png

Other concepts

Terminology – time reversal symmetry

• Time reversal ( t → -t )

• Time reversal symmetry

E(k,↑) = E(-k, ↓)

, Position of a particle in three-space

, Acceleration of the particle

, Force on the particle

, Energy of spinless particle

, Electric potential (voltage)

, Electric field

, Density of electric charge

, Electric polarization

, The time when an event occurs

, Velocity of a particle

, Linear momentum of a particle

, Angular momentum of a particle

, Electromagnetic vector potential

, Magnetic induction

, Density of electric current

, Magnetization

Even symmetry (no change after reversal) Odd symmetry (- sign after reversal)

Terminology – inversion symmetry

• Inversion ( r → -r )

• Inversion symmetry

inversion

Even symmetry (no change after inversion) Odd symmetry (- sign after inversion)

, Position of a particle in three-space

, Acceleration of the particle

, Force on the particle

, Electric polarization

, Velocity of a particle

, Linear momentum of a particle

Q, Electric charge

, Angular momentum of a particle

, The time when an event occurs

m, Particle mass

E(k,↑) = E(-k, ↑)

Inversion symmetry & crystal structure

• Inversion symmetry • No inversion symmetry

• In a crystal with inversion symmetry

Things happen on the surface

• Surface reconstruction • STM image

Si 7X7 surface

Surface states

• Potential • States

Bulk states

Surface states

http://upload.wikimedia.org/wikipedia/en/d/d3/FigPotential.PNGhttp://upload.wikimedia.org/wikipedia/en/b/b0/Solution1.PNGhttp://en.wikipedia.org/wiki/File:Solution2.PNG

Spin-orbit interaction

Due to orbital motion, this electron feels magnetic field,

which couples to the spin of the electron

Spin-orbit interaction

• Perturbation Hamiltonian

• For the same atomic level, H’ ~ Z4

• In general, SOI increases with atomic number Z

Various Hall effects

Hall effect

• Metallic system, magnetic field

Anomalous Hall effect

• Metallic system with magnetization

Quantum Hall effect

• Landau levels• Hall effect with

coherent orbital motion

Quantum Hall effect

Edge states in QHE

• No current in the bulk (when EF is between Landau levels)

• Current flow along the edge

• No back scattering (dissipationless current)

• Edge states ‘protected’

B

Spin Hall effect

• Metallic system, no magnetic field,

spin accumulation

Spin Hall effect

Energy

x

B = 0

Optical Kerr rotation used

Edge spin current

e

Quantum spin Hall effect

• Insulating system, current only at the edges, B = 0

B = 0

Edge currents

e

Quantum spin Hall effect (observation)

• Quantum spin Hall effect in HgTe quantum wellsPrediction : Bernevig, Hughes and Zhang, Science ’06

Observation : Konig et al., Science ’07 HgTe

Comparison of various Hall effects

B≠0

Charge

B=0

Spin

Hall Quantum Hall

Spin Hall Quantum spin Hall

Conductor

Magnetic field

ElectronHole

Conductor

Magnetic field

ElectronHole

Bulk conductance &

Charge accumulation

at the edges

Edge currents &

No back scattering

Protected edge states

Conductor

Bulk conduction &

Spin accumulation

at the edges

SOI at the edges

ElectronElectron

Charge current

Spin current

Insulator

Edge currents &

Spin polarized

at the edges

SOI at the edges

ElectronElectron

Charge current

Spin current

Bulk insulatingBulk conducting

Topological insulators

- 3D extension of QSHE

Metallic edge states in QSH???

• QHE : Originally metallic states. Bulk becomes insulating

due to coherent orbital motion.

• QSHE : Originally insulator. Edges somehow need to

become metallic → topology comes in here!

metallic=

Typical band

Bonding

(Symmetric or ‘+’)

Inverted band

Bonding (S or ‘+’)

ABonding (AS or ‘-’)

SOI

Band inversion by SOI

• Spin–orbit interaction (SOI) may invert the band structure

Anti-Bonding

(Anti-Symmetric or ‘-’)

Note : Parity can be defined for certain momentum points

Moebius strip in the phase space

• n=0 state (topologically trivial)

• n=1 state (topologically non-trivial)

k

E

G X

k

E

G X

Conventional strip Conventioanl insulator

Moebius strip Different class of insulator

Topologically trivial insulator

Topologically non-trivial insulator

• Spin degeneracy is lifted due to SOI (each band has only one spin)

• Odd number of bands cross EF• Edge states protected against perturbation

Properties of edge states

G

‘Protected’ states

Chiral spin states

Spin-orbit interaction

Due to orbital motion, this electron

feels magnetic field, which couples

to the spin of the electron

+

B

E

• Perturbation Hamiltonian

Symmetry breaking at the surface

Rashba effects

Solid VacuumSurf

ace

Rashba effects

Spin degeneracy lifted

F. Reinert et al. PRB (2001) J. Henk et al. PRB (2003)

Rashba splitting

Au(111) surface

states

Time reversal pair

: TI

: Trivial

Fermi surface

G

Energ

y

momentum

EF

Time reversal

pair

Experimental Verification

of TIs

BE

Kin

etic

En

ergy

Initial State

Momentum

En

erg

y

Ef

h

Final State

“V

ert

ica

l tr

an

sit

ion

”

Momentum Resolved PES – Mapping Bands

• Measure both energy

and momentum of

electron

• Direct Mapping of

“Band” (ARPES)

Graphite Band StructureB

ind

ing

en

erg

y (

eV

)

G(A) M(L) K(H) G(A)

0

1

2

3

4

5

6

7k

y (Å

-1)

0 1 2 3

0

1

2

3

4

G

M KG

ky (Å-1)

Year Author Done

2006 Kane & Mele Proposes concepts of QSHE in 2D and relevant Z2 invariants

2006 Zhang Proposes CdTe-HgTe heterostructure for QSHE

2007 Wurzburg group Finds QSHE in CdTe-HgTe heterostructure

2007 Fu-Kane Proposes 3D version of Z2 invariants and name 3D candidate

materials (Bi1-xSbx)

2007 Behnia Quantum oscillation study of pure Bi

2008 Hasan First ARPES study of Bi1-xSbx

2009 Fang & Zhang Proposes new TBI materials (Bi2Se3, Bi2Te3, Sb2Te3) with one

Dirac surface

2009 Hasan ARPES study of Bi2Se3

2009 Chen ARPES study of Bi2Te3

2009 Yazdani Observes QPI in Bi1-xSbx (x=0.08)

2009 Manoharan QPI of pure Sb

2009 Ando Quantum oscillation of Bi1-xSbx

Chronology of QSHE => TI

Crystal structure of Sb and Bi

First prediction (Bi1-xSbx)

Nature 452, 970 (2008)

Zhang Fang, NatPhys 5, 438 (2009)

Nature Physic 5, 398 (2009)

Bi2-xCaxSe3

Bi2-xSnxTe3 (x=0.0) Bi2-xSnxTe3 (x=0.0067)Bi2-xSnxTe3 (x=0.0027)

Y. L. Chen et al., Science (2009)

Chiral spins

• ARPES

• Reconstruction of spin states

Joint DOS

Suppression of back scattering

• ARPES on Bi0.92Sb0.08

* SSP : Spin-dependent Scattering Probability

ExpJDOS SSP

From ARPES

P. Roushan et al., Nature (2009)

Quasi-particle dynamics

(intrinsic life time)

• Cared only about the topological nature,

that is, the # of Ef crossings.

• We would like to see the quasi-particle

dynamics, i.e., will do line shape analysis

22 ),(Im)),(Re(

),(Im1),(

e

kk

kkA

k

Spectral function

Energy

AR

PE

S inte

nsity

FWHM:

2Im ~ 1/t

-Im

(meV

)

100

Binding Energy (meV)

200 150 100 50 0

50

0

Sm

1.8

5 Ce

0.1

5 Cu

O4

S. R. Park et al., PRL (2009)

G

,

400 300 200 100 0

ARPES for lifetime measurement

Photohole (> 0)

Phonon

(0)

Resulting spectrum

2”

Photohole (< 0)

X

” 0

Phonon contribution to

Photohole at

’ ”

20

'0 2

1"'~"

ddAt T=0,

"))'"(1)("(')'(~"

dFFdF DDDFinite T,

222

2

1)(

2

1~ kT

for accessible energy and T range

”

”~28060

4020

0x1

0-3

302520151050

kT

80

60

40

20

0x10

-3

302520151050

e-e contribution to

Extracting many-body interactions

Hengsberger et al. 83, 592 (1999)

Be(0001)

T. Valla et al. PRL83, 2085 (1999)

D

Width~ 1/t

Dispersion

Mo(110)

D

e-e

e-ph

-0.4

5-0.4

0-0.3

5-0.3

0-0.2

5-0.2

0-0.1

5-0.1

0-0.0

50.0

00.0

5

eV

-0.20 -0.10 0.00 0.10

Inverse angstrong

-0.4

5-0.4

0-0.3

5-0.3

0-0.2

5-0.2

0-0.1

5-0.1

0-0.0

50.0

00.0

5

eV

-0.20 -0.10 0.00 0.10

Inverse angstrong

-0.4

5-0.4

0-0.3

5-0.3

0-0.2

5-0.2

0-0.1

5-0.1

0-0.0

50.0

00.0

5

eV

-0.20 -0.10 0.00 0.10

Inverse angstrong

0.0

0.1

0.2

0.0-0.1 0.1 0.0-0.1 0.0-0.1 0.1

Bin

din

g E

ner

gy

(eV

)

Momentum (Å-1)

12K 90K 210K

0.1

0.3

0.4

ImΣ

(meV

)

Binding Energy(eV)

0

20

40

80

60

0.000.150.25 0.10 0.050.20

12K90K210K

0.080.060.040.02

-0.25

-0.20

-0.15

-0.10

-0.05

0.0012K

Bin

din

g E

ner

gy

(eV

)

Momentum (Å-1)

Dispersion

WidthT-dependence on Bi2Se3

-0.1

5-0.1

0-0.0

50.0

00.0

5

eV

-0.10 0.00 0.10

Inverse angstrong

-0.1

5-0.1

0-0.0

50.0

00.0

5

eV

0.100.00-0.10

Inverse angstrong

-0.1

5-0.1

0-0.0

50.0

00.0

5

eV

-0.10 0.00 0.10

Inverse angstrong

0.0

0.1

0.2

0.30.0-0.1 0.1 0.0-0.1 0.0-0.1 0.1

Momentum (Å-1)

24K 150K 287K

0.1

• Energy and T independent !

24K

-0.02-0.03-0.04

-0.8

-0.6

-0.4

-0.2

0.0

Bin

din

g E

ner

gy (

eV)

Momentum (Å-1)

0.0 .02 .04 .06-.08 -.06 -.04 -.02

Binding Energy(eV)

ImΣ

(meV

)

0

10

20

30

40

50

24K

150K

287K

Dispersion

WidthT-dependence on Bi2Te3

-0.3

0-0.2

5-0.2

0-0.1

5-0.1

0-0.0

50.0

00.0

50.1

0

eV

-0.10 0.00 0.10

Inverse angstrong

5.8

05.7

55.7

05.6

55.6

05.5

55.5

05.4

55.4

0

eV

-0.10 0.00 0.10

Inverse angstrong

5.8

05.7

55.7

05.6

55.6

05.5

55.5

05.4

55.4

0

eV

-0.10 0.00 0.10

Inverse angstrong

0.0

0.1

0.2

0.30.0-0.1 0.1 0.0-0.1 0.1 0.0-0.1 0.1

Momentum (Å-1)

11K 80K 310K ImΣ

(meV

)

0.000

20

40

80

Binding Energy(eV)0.10 0.050.15

60

11k80k310k

-0.15

-0.10

-0.05

0.00 11K

-0.03-0.04-0.05

Bin

din

g E

ner

gy

(eV

)

Momentum (Å-1)-0.02

T-dependence on Sb

Sb data taken with laser ARPES

EDC fit dispersion

EDC fit peak HWHMat differnt K location

Green line:DE~6meV

Green line: DE~10meV

Sb(111)

0.25 0.20 0.15 0.10 0.05

Binding energy (eV)

0.0

60

50

40

30

20

10

0

Im

(meV

)

Bi2Se3(Our first)

Sb (Y. Chen, Z. X. Shen)

7eV P. E.

Sb

Bi2Te3

Laser ARPES data from Sb(111)

Point I:

• Extremely small, if any, e-e interaction in the SS

independent width

• Extremely small, if any, e-ph coupling in the SS

-no kinks & T-independent peak width at =0

• Probably due to the small FS size (little phase space)

Point II:

• e-e interaction exists in the bulk states

• Defects couple the surface and bulk states

• For a clean system, SSs are decoupled from the bulk.

Observations/Summary

Chiral orbital angular

momentum states

Nature Physics 5, 398 (2009)

Bi2-xCaxSe3

Left circularRight circular

kx

ky

kx

ky

Bin

din

g E

ner

gy (

eV)

Bin

din

g E

ner

gy (

eV)

Polarization dependence

0.0

-5.0

5.0

*phase125

kx

ky

kx

ky

Binding Energy (eV)

0.90.60.0 0.3

0.0

-5.0

5.0

An

gle

(m

om

entu

m)

An

gle

(m

om

entu

m)

Polarization dependence

Photoemission : i f

Right CP : mRCP = 1, therefore mf = mi + 1

Left CP : mLCP = – 1, therefore mf = mi – 1

Initial state : composed of p states (l = 1)

Final state ~ eikr

mi

1

0

-1

mf

RHP

LHP

Dichroism results from orbital angular

momentum of the initial state

Dichroism

Dichroism in ARPES

Initial state

Final state

Dipole selection rules

Dim

LCP

Bright

RCP

Dim

Bright

Orbital

Spin

• Due to symmetry, out of surface orbital angular momentum is constant

• In-plane component contributes to the contrast

Chiral orbital angular momentum

At 0degree

< Lx > < Ly > < Lz >

Se1 0.0000 0.0210 0.0355

Bi1 0.0000 0.1479 0.0078

Se2 0.0000 -0.0354 0.0056

Bi1’ 0.0000 0.0906 0.0099

Se1’ 0.0000 -0.0319 0.0183

Calculated orbital angular momentum

GK M

G

KM

Surface

• Strong coupling between spin and local orbital angular

momentum

• Not only spins but also orbital angular momentum has

chiral structure

• Possibility of scattering from orbital angular momentum

scattering

Summary