Basics of topological insulator

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insulator. Anderson insulator. Quantum Hall insulator. Topological insulator. A brief history of insulators. Peierls transition. Hubbard model. Scaling theory.
2011/11/18 @ NTU

Basics of topological insulator Ming-Che Chang Dept of Physics, NTNU

A brief history of insulators

Band insulator (Wilson, Bloch) Mott insulator

Anderson insulator

Peierls transition

1930

1940

1950

Hubbard model

1960

1970

Quantum Hall insulator

Topological insulator

Scaling theory of localization

1980

1990

2000

2010

2D TI is also called QSHI

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



M

da G + ∫ ds k g = 2π χ ( M , ∂M )

χ =2

∂M

χ =0

χ =1

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



2 D BZ

d 2 k Ω Z = 2π C1

e2 σ H = C1 h

2D Lattice fermion with time reversal symmetry (TRS) BZ • Without B field, Chern number C1= 0 • Bloch states at k, -k are not independent

• EBZ is a cylinder, not a closed torus. ∴ No obvious quantization.

Moore and Balents PRB 07

• C1 of closed surface may depend on caps • C1 of the EBZ (mod 2) is independent of caps (topological insulator, TI) 2 types of insulator, the “0-type”, and the “1-type”

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

1 ⎡ 2 ⎤ mod 2 ν= d k dk A Ω − ∫ ∫ ⎥⎦ ∂ ( EBZ ) 2π ⎢⎣ EBZ ~ Gauss-Bonnet theorem with edge How can one get a TI? : band inversion due to SO coupling 0-type

1-type

Bulk-edge correspondence Different topological classes

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

Eg

→ gapless states bound to the interface, which are protected by topology. Topological Goldstone theorem?

Bulk-edge correspondence in TI Bulk states Dirac point

Edge states Fermi level

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

helical edge states robust backscattering by non-magnetic impurity forbidden

Topological insulators in real life

SO coupling only 10-3 meV

• Graphene (Kane and Mele, PRLs 2005) 2D

• 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)

3D

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

• GenBi2mTe3m+n family (GeBi2Te4 …) •… • strong spin-orbit coupling • band inversion

A stack of 2D TI

z

fragile SS Helical

2D TI Helical edge state

y

x

3 TI indices

3D TI: 3 weak TI indices:

Eg., (x0, y0, z0 )

z+ z0

x0

x+

y0

y+

1 strong TI index: ν0 ν0 = z+-z0 (= y+-y0 = x+-x0 ) difference between two 2D TI indices

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

Weak TI index Screw dislocation of TI

A stack of 2D TI

• not localized by disorder • half of a regular quantum wire Ran Y et al, Nature Phys 2009

From Vishwanath’s slides

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

S.Y. Xu et al Science 2011

Dirac point: Graphene

vs.

Topological insulator

Even number

Odd number (on one side)

located at Fermi energy

not located at EF

half integer QHE (×4)

half integer QHE (if EF is located at DP)

Spin is not locked with k

spin is locked with k

can be opened by substrate

cannot be opened

Top

k

σH=+1

k’

σH=-1

σH=+1

bottom

σH=-1

Electromagnetic response Axion electrodynamics

First, a heuristic argument:

Effective Lagrangian for EM wave

EF

LEM = L0 + Laxion → half-IQHE

⎞ 1 ⎛ E2 L0 = 2 ⎜ 2 − B 2 ⎟ 8π ⎝ c ⎠

E

Laxion TI

JH

Θ e2 1 = E⋅B =α E⋅B 2h c 4π 2 4π e 2 e 2 1 note: α = = ∼ c 2h c 137

Surface state ~2 DEG • Hall current

“axion” coupling

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

e2 JH = E 2h

• Induced magnetization

M=

2

e E 2h

“magneto-electric” coupling

Cr2O3: θ~π/24 (TRS is broken)

Maxwell eqs with axion coupling Θ ⎞ ⎛ ∇ ⋅ ⎜ E + α B ⎟ = 4πρ π ⎠ ⎝ Θ ⎞ 4π Θ ⎞ ∂ ⎛ ⎛ ∇×⎜ B −α E ⎟ = + B J+ E α ⎜ π ⎠ c c∂t ⎝ π ⎟⎠ ⎝ ∇⋅B = 0 ∇× E = −

∂ B c∂t

ρΘ =

α δ ( z ) Bz 4π

+ + + + + + + + + + + + + + + B

Θ=π

z

Effective charge and effective current cα 1 e2 JΘ = − δ ( z ) zˆ × E → σ xy = 4π 2 h

∇ ⋅ E = 4π ( ρ + ρΘ )

ρΘ = − ∇× B =

α ∇ ⋅ ( ΘB ) 4π 2 4π 1 ∂E c

( J + J ) + c ∂t Θ

α ∂ cα ΘB J Θ = 2 ∇ × ΘE + 2 4π 4π ∂t

( )

( )

Θ=π

E

z

Static:

Dynamic:

Magnetic monopole in TI

Optical signatures of TI? axion effect on

A point charge • Snell’s law • Fresnel formulas Circulating current

• Brewster angle • Goos-Hänchen effect •…

An image charge and an image monopole

Qi, Hughes, and Zhang, Science 2009

D

Longitudinal shift of reflected beam (total reflection) Chang and Yang, PRB 2009

(Magnetic overlayer not included)

Dimensional reduction Topological field theory

2D quantum Hall effect

1D charge pump

Laughlin’s argument (1981):

y

☉B x x

compactification ψ

Berry • When the curvature: flux is changed by Φ0, = ∂ i a j − ∂are integerfijcharges j ai transported from edge edge → IQHE Berryto connection: ak = i u ∂ k u C1 =

1 d 2 k f xy (k ) ∫ 2π

C1 2 μντ d xdt ε Aμ ∂ν Aτ ∫ 4π δS C j μ = CS = 1 ε μντ ∂ν Aτ δ Aμ 2π

SCS =

ψ x

Ay ( x, t ) ∼ θ ( x, t ), a parameter polarization P (θ ) =

1 dk x ax 2π ∫

S = ∫ dxdt Pε αβ ∂α Aβ jα = −ε αβ ∂ β P

Qi, Hughes, and Zhang PRB 2008

4D quantum Hall effect

3D topological insulator compactification

w

ψ

ψ (x,y,z)

C2 d 4 xdtε μνρστ Aμ ∂ν Aρ ∂σ Aτ 2 ∫ 24π C j μ = 22 ε μνρστ ∂ν Aρ ∂σ Aτ 8π

S=

S=

(

)



jα =

nonlinear response.

1 C2 = d 4 kε ijkl tr fij f kl 2 ∫ 32π fij = ∂ i a j − ∂ j ai − i[ai , a j ]

1

Θ=

2

3 αβγδ Θ ε ∂α Aβ ∂ γ Aδ d xdt ∫

1 4π

2

ε αβγδ ∂ β Θ∂ γ Aδ

1 i ⎛ ⎞ 3 ijk ⎡ ⎤ + ε d k tr a f a a , a i j k ⎦⎟ ⎜ i jk 8π ∫BZ 3 ⎣ ⎝ ⎠

akmn = i um ∂ k un Qi, Hughes, and Zhang PRB 2008

Without TRS

NTU Phys bldg

With TRS

5 4D QHE

4

TI

3

QHE

Chern elevator

QSHE

2 Charge pump

1 Re-enactment from Qi’s slide

Alternative derivations of Θ 1. Semiclassical approach (Xiao Di et al, PRL 2009) 2. 3.

∂Pi ∂B j ∂M j ∂Ei

A. Essin et al, PRB 2010 A. Malashevich et al, New J. Phys. 2010

∂Pi ∂M j = = α ij = α ij + αθ δ ij ∂B j ∂Ei Θ e2 αθ = 2π h Explicit proof of Θ=π for strong TI: • Z. Wang et al, New J. Phys. 2010

Physics related to the Z2 invariant

Topological insulator graphene Quantum SHE Spin pump

• generalization of Gauss-Bonnett theorem

4D QHE

Helical surface state Magneto-electric response Interplay with SC, magnetism Majorana fermion in TI-SC interface



Time reversal inv, spin-orbit coupling, band inversion

QC

Thank you!