Introduction to topological insulator

47
Ming-Che Chang Dept of Physics, NTNU 7/29/11 @ NTHU Introduction to topological insulator

Transcript of Introduction to topological insulator

Page 1: Introduction to topological insulator

Ming-Che ChangDept of Physics, NTNU

7/29/11 @ NTHU

Introduction to topological insulator

Page 2: Introduction to 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

Page 3: Introduction to topological insulator

A brief review on topology

K≠0

G≠0

K≠0

G=0

extrinsic curvature K vsintrinsic (Gaussian) curvature G

G>0

G=0

G<0

Positive and negative curvature

Page 4: Introduction to topological insulator

4 (1 )M

G gda π= −∫

• Gauss-Bonnet theorem (for a 2-dim closed surface)

0g = 1g = 2g =

• Gauss-Bonnet theorem for a surface with boundary

( ),2gM MM Mda G ds k π χ

∂∂+ =∫ ∫

Marder, Phys Today, Feb 2007

Page 5: Introduction to topological insulator

2D lattice electron in magnetic fieldB

Cyclotron motion

21

12 Z

BZ

C d k Zπ

Ω= ∈∫

• topological number (1st Chern number)

~ Gauss-Bonnet theorem

2D Brillouin zone

=

Berry connection

( )

Berry curvature

( ) ( )

k k k

k

A k i u u

k A k

≡ ∇

Ω = ∇ ×

Cell-periodic Bloch state

Topology in condensed matter

Page 6: Introduction to topological insulator

• Without B field, Chern number C1= 0

• Bloch states at k, -k are not independent

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

Page 7: Introduction to topological insulator

• 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

Page 8: Introduction to topological insulator

Time reversal operator

• Kramer’s degeneracy (for fermions with TRS)

• Bloch statek k

k k

ψ ψε ε

↑ − ↓

↑ − ↓

Θ =

=

2 1 for fermionsΘ = −

If H commutes with Θ, then are degenerate in energy (Kramer pair)

,ψ ψΘ

k

ε “↑”

“↓”

k

ε↑

w/ inv symm w/o inv symm

• Time Reversal Invariant Momenta (TRIM)

Γ00 Γ10

Γ01 Γ11

k

ε “↑”

“↓”

TRIM

Dirac point

Page 9: Introduction to topological insulator

Bulk-edge correspondence

Different topological classes

Semiclassical (adiabatic) picture: energy levels must cross.

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

Eg

Topological Goldstone theorem?

Page 10: Introduction to topological insulator

Example 1: QHE

Gapless excitations at the edge

Robust against disorder (no back-scattering)

skipping orbit (Chiral edge state)

number of edge modes = Hall conductance

(Classical picture) (Semiclassical picture)

LLs

Page 11: Introduction to topological insulator

Topological insulatorUsual insulator

• 0 or even pairs of surface states • odd pairs of surface states

TRIM

Dirac point

Example 2: Topological insulator

backscattering by non-magnetic impurity forbidden

fragile robust

Page 12: Introduction to topological insulator

Robust chiral edge state

Robust helical edge state

Edge state: Quantum Hall vs topological insulator

Real space Energy spectrum (for one edge)

Page 13: Introduction to topological insulator

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

Page 14: Introduction to topological insulator

2D TI A stack of 2D TI

Helical edge state

z

x

y

3 TI indices

Helical SSfragile

Page 15: Introduction to topological insulator

( )1 1 2 2 3 3 / 2M G G Gν ν ν ν= + +2D TIs stacked along

3 weak TI indices: (ν1, ν2, ν3 )

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

Page 16: Introduction to topological insulator

Weak TI indices

Page 17: Introduction to topological insulator

A stack of 2D TIScrew dislocation of TI

1D helical metal exists iff

• not localized by disorder

• half of a regular quantum wire

Ran Y et al, Nature Phys 2009

From Vishwanath’s slides

Page 18: Introduction to topological insulator

Strong TI index

Page 19: Introduction to topological insulator

1 2 3 4( 1) 1ν δ δ δ δ− ≡ = +• Z2 class

1 2 3 4( 1) 1ν δ δ δ δ− ≡ = −

2 filled

( )i n in

δ ξ≡ Γ∏

• Parity eigenvalue ( ) ( ) ( ) ; ( ) 1

n i n i n i

n i

P ψ ξ ψξ

Γ = Γ Γ

Γ = ±

(normal phase)

(TI phase)

If there is inversion symm

then Bloch state at TRIM Γi has a definite parity

2D

If there is NO inversion symm

( ) ( ) ( )mn m nw k u k u k≡ − Θ

Γ00 Γ10

Γ01 Γ11

T-reverse matrix (antisymm)

Fu and Kane PRB 2006, 2007Or, count zeros of the Pfaffian Pf[w] (mod 2)

[ ][ ]

Pf ( )

det ( )i

ii

w

Γ≡

Γ

Page 20: Introduction to topological insulator

Similar analysis applies to 3D

For example,

( )

( )

0

1 2 3

1 2 3

0,1

1,2,3

0; 0,1

1

1i

i

i j i

n n nn

n n nn n

ν

ν

δ δ δ

δ δ δ≠

=

=

= =

− =

− =

( )1 2 3 1 1 2 2 3 312n n n n b n b n bΓ = + +TRIM:

strong

weak

Page 21: Introduction to topological insulator

ARPES of Bi2Se3

H. Zhang et al, Nature Phys 2009 Helical Dirac cone

Surface state in topological insulator

W. Zhang et al, NJP 2010

Reality check:

Σ1

Page 22: Introduction to topological insulator

S.Y. Xu et al Science 2011

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

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Page 24: Introduction to topological insulator

Dirac point, Landau levels

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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

Page 26: Introduction to topological insulator

Cγ π=

Landau levels of the Dirac cone

= 2n FE v eB n⇒

Cyclotron orbits

BE

k

Dirac cone

P. Cheng et al, PRL 2010

LLs

Berry phase

2 122 2

C eBk nππγπ ⎛ ⎞= + −⎜ ⎟

⎝ ⎠

( ) FE k kν=

Linear energy dispersion

Orbital area quantization

STM experiment

Page 27: Introduction to topological insulator

Magnetoelectric response

Axion electrodynamics

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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

To have the 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)

Θ → -Θ for TRS

Page 29: Introduction to topological insulator

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

Page 30: Introduction to topological insulator

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)

Page 31: Introduction to topological insulator

TI thin film, normal incidence

,

,

24

4

4

iij

i j

i j

ij

i j

ntn n

c

n ncr

n nc

σπ

π σ

π σ

±

±

±

±

±

=+ +

− −=

+ +

12 2323 21

12 12 23 2123 21

11

11

t t tr r

r r t r tr r

=−

= +−

Multiple reflections (λ>>d)

(A.Khandker et al, PRB 1985) • Kerr rotation

• Faraday rotation

( )

1 3

121 =

F

n n

θ θ θ

αξ

+ −= −

++

( )assume 0, 1xx xy xyσ ξσ ξσ= = = ±

( )( )

122 2

3 12

2 11

K

nn n

θαξ

ξ α

+=

− + +

4xyc

α π σ=Agree with calculations from axion electrodynamics

1

2

3

xyσ

xyσTI

substrateit t e θ±

± ±=

ir r e θ±± ±=

Page 32: Introduction to topological insulator

Kerr effect and Faraday effect for a TI thin film

“Giant” Kerr effect

• W.K. Tse and A.H. MacDonald, PRL 2010

• J. Maciejko, X.L. Qi, H.D. Drew, and S.C. Zhang, PRL 2010

1 3

2tan F F n nθ θ α α≈ = →

+

123

221

4 1tan4K

nn n

αα

θα

= →− +

(Free-standing film)

xy xyσ σ=

E

0Kθ =

0Fθ =

xy xyσ σ= −

E

Page 33: Introduction to topological insulator

Electric polarization

Thouless charge pumpZ2 spin pump

Page 34: Introduction to topological insulator

31 ( )P d r rV

rρ= ∫

Electric polarization of a periodic solid

ΔP

… …

PUnit cell+ -Choice 1 … …

P

+-Choice 2 … …

Resta, Ferroelectrics 1992

NOT well defined for an infinite solid

Start from j = dP/dt, which is well-defined, then

1 0dPP d P Pd λ λλλ = =Δ = = −∫ λ: atomic displacement

nevertheless

Page 35: Introduction to topological insulator

( )

( , )

( , )

2Berry potential:

under a "large" GT:

with ( ) (- )=2

k k

B

i kk

k

Z

k

qP dk

P P nq

a k

a k u

u u n

u

e

i λ

φ

λ

λ π

φ

λ

λ

π φ π π

=

→ +

=

King-Smith and Vanderbilt, PRB 1993

Polarization and Berry phase (1D example)

λ

0

1

kπ-π

ΔP has no such ambiguity

( )2qP dk a kλ π

Δ = ⋅∫

Page 36: Introduction to topological insulator

Thouless charge pump (Thouless PRB 1983)

Periodic variation: H(k,t+T)=H(k,t)

( )2

( ,

)

qP dk a k nq

k k t

λ πΔ = ⋅ =

=

t

0

T

kπ-π

n is the first Chern number

Quantized pumping at Temp=0, for adiabatic variation.

Trivial

Pumping charges without using a DC bias

Example 1: a sliding potential

t=0

t=T

Page 37: Introduction to topological insulator

Example 2: SSH model with staggered field (Rice and Mele PRL 1983)

( ) ( )† † †1 1

( ) 1 ( ) 1 . .2 2

i ii i i i st i i

i i i

t tH c c c c h t c c h cδ+ += + − + − +∑ ∑ ∑

0 02 2( ( ), ( )) cos , sinst

t tt h t hT Tπ πδ δ⎛ ⎞= ⎜ ⎟

⎝ ⎠

t=0

t=T/4

t=T/2

t=3T/4

t=T

t=5T/4

……

adiabatic variation

After a period of T, one electron is pumped to the right

Page 38: Introduction to topological insulator

Spin pump: Fu-Kane model (PRB 2006)

( ) ( )† † †1 1

( ) 1 ( ) 1 . .2 2

i ii i i i st i i

i i i

zt tH c c c c h t c c h cα α αα α σ β βσδ+ += + − + − +∑ ∑ ∑

Spin-dep staggered field

After t=T/2, one unpaired up (down) spin appears on the R(L) edge. (Effectively one up-spin is pumped to the right.)

t=0

t=T/4

t=T/2

t=3T/4

t=T

• Sz is conserved

• Sz is not conserved

After t=T/2, again unpaired spins (not necessarily up/down) at each end. (some spin is pumped to the right)

Page 39: Introduction to topological insulator

0 0

( ( ), ( ))2 2cos , sin

stt h tt th

T T

δπ πδ⎛ ⎞= ⎜ ⎟

⎝ ⎠

• H is time-reversal invariant at t=0, T/2, T…

• Kramer degeneracy at t=0, T/2, T…

• Different ground states t=0, T.

→ Z2 symmetry

• no crossing, no pumping

3T/2 2T

Fu and Kane, PRB 2006

( ) ( )† † †1 1

( ) 1 ( ) 1 . .2 2

i ii i i i st i i

i i i

zt tH c c c c h t c c h cα α αα α σ β βσδ+ += + − + − +∑ ∑ ∑

Mid-gap edge states

Fu-Kane model

Page 40: Introduction to topological insulator

Dimensional reduction

Topological field theory

Page 41: Introduction to topological insulator

• 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

= ∂ −∂

= ∂

= ∫

Page 42: Introduction to topological insulator

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

=

= ∂ −∂ −

= ∂

Page 43: Introduction to topological insulator

1

2

3

4

5

QHE

Charge pump

Without TRS With TRS

4D QHE

TI

QSHE

Re-enactment from Qi’s slide

Chernelevator

NCTS bldg

Page 44: Introduction to topological insulator

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=

Page 45: Introduction to topological insulator

• Effective Hamiltonian for the SS [Shen]

• Disorder, non-crystalline TI [Shen]

• Electron-electron interaction [Xie]

• Hybrid materials (TI+M, TI+SC) [Fang]

• TI-SC interface, Majorana fermion [Law, Wan]

• Periodic table of TI/SC

• …

Not in this talk,

Page 46: Introduction to topological insulator

Physics related to the Z2 invariant

Topological insulator

• Gauss-Bonnett theorem

• parity eigenvalue

• zero of Pfaffian4D 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

Page 47: Introduction to topological insulator

Thank you!