Ming-Che ChangDept of Physics, NTNU

2011/11/18 @ NTU

Basics of topological insulator

2D TI is also called QSHI

1930 1940 1950 1960 1970 1980 1990 2000 2010

Band insulator(Wilson, Bloch)

Mott insulator

Anderson insulator

Quantum Hall insulator

Topological insulator

A brief history of insulators

Peierlstransition

Hubbard model

Scaling theory of localization

• Gauss-Bonnet theorem for a 2D surface with boundary

( ),2gM MM Mda G ds k π χ

∂∂+ =∫ ∫

2χ = 0χ = 1χ =

• Quantum Hall effect (2D lattice fermion in magnetic field)

21

2

2D B

ZZ

d k Cπ=Ω∫2

1H hC eσ =

• Without B field, Chern number C1= 0

• Bloch states at k, -k are not independent

2D Lattice fermion with time reversal symmetry (TRS)

• C1 of closed surface may depend on caps

• C1 of the EBZ (mod 2) is independent of caps

2 types of insulator, the “0-type”, and the “1-type”

• EBZ is a cylinder, not a closed torus.

∴ No obvious quantization.

Moore and Balents PRB 07

(topological insulator, TI)

BZ

• 2D TI characterized by a Z2 number (Fu and Kane 2006 )

0-type 1-type

How can one get a TI?

: band inversion due to SO coupling

2

( )

1 mod 22 EBZ EBZ

d k dk Aνπ ∂⎡ ⎤= Ω−⎢ ⎥⎣ ⎦∫ ∫

~ Gauss-Bonnet theorem with edge

Topological Goldstone theorem?

Bulk-edge correspondence

Different topological classes

Semiclassical (adiabatic) picture: energy levels must cross (otherwise topology won’t change).

→ gapless states bound to the interface, which are protected by topology.

Eg

helical edge states

Bulk-edge correspondence in TI

TRIM

Dirac point

Fermi level

Bulk states

Edge states

(2-fold degeneracy at TRIM due to Kramer’s degeneracy)

backscattering by non-magnetic impurity forbidden

robust

Topological insulators in real life

• Graphene (Kane and Mele, PRLs 2005)

• HgTe/CdTe QW (Bernevig, Hughes, and Zhang, Science 2006)

• Bi bilayer (Murakami, PRL 2006)

• …

• Bi1-xSbx, α-Sn ... (Fu, Kane, Mele, PRL, PRB 2007)

• Bi2Te3 (0.165 eV), Bi2Se3 (0.3 eV) … (Zhang, Nature Phys 2009)

• The half Heusler compounds (LuPtBi, YPtBi …) (Lin, Nature Material 2010)

• thallium-based III-V-VI2 chalcogenides (TlBiSe2 …) (Lin, PRL 2010)

• GenBi2mTe3m+n family (GeBi2Te4 …)

• …

SO coupling only 10-3 meV

2D

3D

• strong spin-orbit coupling

• band inversion

2D TI

A stack of 2D TI

Helical edge state

z

x

y

3 TI indices

Helical SSfragile

3D TI: 3 weak TI indices:

Fu, Kane, and Mele PRL 07 Moore and Balents PRB 07 Roy, PRB 09

ν0 = z+-z0

(= y+-y0 = x+-x0 )

z0 x0

y0

z+

x+ y+

Eg., (x0, y0, z0 )

1 strong TI index: ν0

difference between two 2D TI indices

A stack of 2D TIScrew dislocation of TI

• not localized by disorder

• half of a regular quantum wire

Ran Y et al, Nature Phys 2009

From Vishwanath’s slides

Weak TI index

S.Y. Xu et al Science 2011

Band inversion, parity change, spin-momentum locking (helical Dirac cone)

Graphene Topological insulator

Even number Odd number (on one side)

located at Fermi energy not located at EF

spin is locked with kSpin is not locked with k

can be opened by substrate cannot be opened

vs.Dirac point:

half integer QHE (if EF is located at DP)

half integer QHE (×4)

k k’

Top

bottom

σH=+1 σH=－1

σH=+1

σH=－1

Electromagnetic response

Axion electrodynamics

JHTI

2

2

12 4axioneL E B E Bh c π

α= ⋅Θ

= ⋅

“magneto-electric” coupling

2

2HeJ Eh

=

“axion” coupling

• Hall current

• Induced magnetization

Effective Lagrangian for EM wave

For systems with time-reversal symmetry, Θ can only be 0 (usual insulator) or π (TI)

Surface state ～2 DEG

E

2

2eM Eh

=

EF

→ half-IQHE

2 2 1note: 2 137

4 eh cc

eα π= = ∼

0EM axionL L L= +2

20 2 2

18

EL Bcπ

⎛ ⎞= −⎜ ⎟

⎝ ⎠

Cr2O3: θ～π/24 (TRS is broken)

First, a heuristic argument:

Maxwell eqs with axion coupling

4

4

0

E

B J Ec c t

B

E Bc t

B

E B

απ

απ

ρ

ππ

π

α

⎛ ⎞∇ ⋅ + =⎜ ⎟⎝ ⎠

∂⎛ ⎞ ⎛ ⎞∇× − = + +⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠∇ ⋅ =

∂∇×

Θ

Θ Θ

= −∂

( )

( )

( )

( ) ( )

2

2 2

4

4

4

4

4 1

E

EB Jc c t

B

J

cJ E Bt

π ρ

π

ραρπ

α απ π

Θ

Θ

Θ

Θ

∇ ⋅ = +

∂∇× = + +

= − ∇ ⋅ Θ

∂= ∇× Θ

∂

+ Θ∂

Effective charge and effective current21ˆ( )

4 2xyc eJ z z E

hα δ σπΘ = − × → =

( )4 zz Bαρ δπΘ =

Θ=π E

Θ=π B

+ + + + ++ + + + +

+ + + + +z

z

Magnetic monopole in TI

A point charge

An image charge and an image monopole

Circulating current

Optical signatures of TI?

• Snell’s law

• Fresnel formulas

• Brewster angle

• Goos-Hänchen effect

• …

axion effect on

Qi, Hughes, and Zhang, Science 2009

Chang and Yang, PRB 2009

D

Longitudinal shift of reflected beam (total reflection)

Static: Dynamic:

(Magnetic overlayer not included)

Dimensional reduction

Topological field theory

• When the flux is changed by Φ0, integer charges are transported from edge to edge → IQHE

Laughlin’s argument (1981):

2D quantum Hall effect 1D charge pump

ψxx

y☉ B

ψx

compactification

2

1

1

4

2

CS

CS

S d xdt A A

S Cj AA

C μντμ ν τ

μ μντν τ

μ

επ

δ εδ π

= ∂

= = ∂

∫

( )

( , ) ( , ), a parameter

polarization1

2

y

x x

A x t x t

P dk a

S dxd A

j P

Pt αβα β

α αβ β

θ

θπ

ε

ε

=

= ∂

= − ∂

∫

∫

∼

Qi, Hughes, and Zhang PRB 2008

21

Berry curvature:

Berry connection:

1 ( )2

ij i j j i

k k

xy

f a a

a i u u

C d k f kπ

= ∂ −∂

= ∂

= ∫

3D topological insulator4D quantum Hall effect

ψ

(x,y,z)

w compactificationψ

Qi, Hughes, and Zhang PRB 2008

42

2

2

2

24

8

S d xdt A A A

Cj A

C

A

μνρστμ ν ρ σ τ

μ μνρστν ρ σ τ

επ

επ

= ∂ ∂

= ∂ ∂

∫

nonlinear response.

32

2

18

14

S d xdt A A

j A

αβγδα β γ δ

α αβγδβ γ δ

επ

επ

= ∂ ∂

= ∂ Θ∂

Θ∫

31= tr ,8 3

ijki jk i j kBZ

id k a f a a aεπ

⎛ ⎞⎡ ⎤Θ +⎜ ⎟⎣ ⎦⎝ ⎠∫( )42 2

1 tr32

[ , ]

ijklij kl

ij i j j i i j

mnk m k n

C d k f f

f a a i a a

a i u u

επ

=

= ∂ −∂ −

= ∂

∫

1

2

3

4

5

QHE

Charge pump

Without TRS With TRS

4D QHE

TI

QSHE

Re-enactment from Qi’s slide

Chernelevator

NTU Phys bldg

1. Semiclassical approach (Xiao Di et al, PRL 2009)

• Z. Wang et al, New J. Phys. 2010

Alternative derivations of Θ

2. A. Essin et al, PRB 2010

3. A. Malashevich et al, New J. Phys. 2010

i

j

PB∂∂

j

i

ME

∂

∂

Explicit proof of Θ=π for strong TI:

2

2

jiij ij ij

j i

MPB E

eh

θ

θ

α α α δ

απ

∂∂= = = +

∂ ∂

Θ=

Physics related to the Z2 invariant

Topological insulator

• generalization of Gauss-Bonnett theorem

4D QHE

Spin pump

Helical surface state

Magneto-electric response

Majorana fermionin TI-SC interface

Time reversal inv, spin-orbit coupling, band inversion

Interplay with SC, magnetism

QC

Quantum SHE

graphene

…

Thank you!