Ming-Che ChangDept of Physics, NTNU
7/29/11 @ NTHU
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
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
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
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
=
Cω
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
• 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
• 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
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
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?
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
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
Robust chiral edge state
Robust helical edge state
Edge state: Quantum Hall vs topological insulator
Real space Energy spectrum (for one edge)
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
( )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
Weak TI indices
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
Strong TI index
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
wδ
Γ≡
Γ
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
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
S.Y. Xu et al Science 2011
Band inversion, parity change, spin-momentum locking (helical Dirac cone)
Dirac point, Landau levels
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
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
Magnetoelectric 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
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
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)
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±± ±=
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
Electric polarization
Thouless charge pumpZ2 spin pump
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
( )
( , )
( , )
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 = ⋅∫
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
…
…
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
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)
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
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
NCTS 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=
• 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,
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
…
Thank you!
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