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=
(
)
8π
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!
