Topological Insulators

and the Quantum Spin Hall Effect

I. Introduction- Topological band theory and topological insulators

II. Two Dimensions : Quantum Spin Hall Effect- Time reversal symmetry and edge states

- Experiment: Transport in HgTe quantum Wells

III. Three Dimensions : Topological Insulators

- Topological Insulator Surface States

- Experiment: Photoemission on BixSb1-x, Bi2Se3, Bi2Te3

IV. Superconducting proximity effect, Majorana fermions- Majorana fermions

- A route to topological quantum computing?

Thanks to Gene Mele, Liang Fu, Jeffrey Teo

Charles L. Kane, University of Pennsylvania

The Insulating State

The Integer Quantum Hall State

IQHE with zero net magnetic field

Graphene with a periodic magnetic field B(r) (Haldane PRL 1988)Band structure

B(r) = 0

Zero gap,

Dirac point

B(r) � 0

Energy gap

σxy = e2/h

Eg

k

Eg ~ 1 eV

e.g. Silicon

g cE ω= �E

2D Cyclotron Motion,

Landau Levels

σxy = e2/h

+ − + − + − +

+ − + − +

+ − + − + − +

Topological Band Theory

21( ) ( )

2 BZn d u u

iπ= ⋅ ∇ × ∇� k k

k k k

The distinction between a conventional insulator and the quantum Hall state

is a topological property of the manifold of occupied states

2| ( ) : ( ) Brillouin zone Hilbert spacek T�

�

Insulator : n = 0

IQHE state : σxy = n e2/h

The TKNN invariant can only change

when the energy gap goes to zero

Classified by the TKNN (or Chern) topological invariant (Thouless et al, 1984)

QHE staten=1

Vacuumn=0

Edge States at a domain wall Gapless Chiral Fermions

E

kKK’

EgHaldane Model

Topological Insulator : A New B=0 Phase 2D Time reversal invariant band structures have a Z2 topological invariant, ν ν ν ν = 0,1

ν=0 : Conventional Insulator ν=1 : Topological Insulator

Kramers degenerate at

time reversal invariant momenta

k* = −−−−k* + G

E

k*=0 k*=π/a

E

k*=0 k*=π/a

Edge States

1. Sz conserved : independent spin Chern integers : (due to time reversal) n n↑ ↓

= −

ν is a property of bulk bandstructure. Easiest to compute if there is extra symmetry:

, mod 2nν

↑ ↓=

2. Inversion (P) Symmetry : determined by Parity of occupied 2D Bloch states at Γ1,2,3,4

( ) ( ) ( )

( ) 1

n i n i n i

n i

P ψ ξ ψ

ξ

Γ = Γ Γ

Γ = ±

Γ4 Γ3

Γ1 Γ2

4

2

1

( 1) ( )n i

i n

υ ξ=

− = Γ∏∏Bulk BrillouinZone

Quantum spin Hall Effect :J� J�

E

Two dimensions : Quantum Spin Hall Insulator

Theory: Bernevig, Hughes and Zhang, Science ’06

Experiement: Konig et al. Science ‘07

HgTe

HgxCd1-xTe

HgxCd1-xTed

d < 6.3 nm

Normal band orderd > 6.3 nm:

Inverted band order

Conventional Insulator QSH Insulator

E

k

E

II. HgCdTe quantum wells

I. Graphene

• Intrinsic spin orbit interaction

� small (~10mK-1K) band gap

• Sz conserved : “| Haldane model |2”

• Edge states : G = 2 e2/h

Kane, Mele PRL ‘05

0 π/a 2π/a

�

�

�

� �

�

Eg

Γ6 ~ s

Γ8 ~ p Γ6 ~ s

Γ8 ~ pNormal

Inverted

G ~ 2e2/h in QSHI

3D Topological InsulatorsThere are 4 surface Dirac Points due to Kramers degeneracy

Surface Brillouin Zone

2D Dirac Point

E

k=Λa k=Λb

E

k=Λa k=Λb

ν0 = 1 : Strong Topological Insulator

Fermi circle encloses odd number of Dirac points

Topological Metal

ν0 = 0 : Weak Topological Insulator

Fermi circle encloses even number of Dirac points

Related to layered 2D QSHI

OR

Λ4

Λ1 Λ2

Λ3

EF

How do the Dirac points connect? Determined

by 4 bulk Z2 topological invariants ν0 ; (ν1ν2ν3)

(ν1ν2ν3) : Miller indices for stacking direction

Determine location of surface Dirac Points

kx

ky

Bi1-xSbx

Theory: Predict Bi1-xSbx is a topological insulator by exploiting

inversion symmetry of pure Bi, Sb (Fu,Kane PRL’07)

Experiment: ARPES (Hsieh et al. Nature ’08)

• Bi1-x Sbx is a Strong Topological

Insulator ν0;(ν1,ν2,ν3) = 1;(111)

• 5 surface state bands cross EF

between Γ and M

ARPES Experiment : Y. Xia et al., Nature Phys. (2009).

Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3

• ν0;(ν1,ν2,ν3) = 1;(000) : Band inversion at Γ

• Energy gap: ∆ ~ .3 eV : A room temperaturetopological insulator

• Simple surface state structure :Similar to graphene, except only a single Dirac point

EF

Unique Properties of Surface States

Spin polarized Fermi circle (points)

• “Half” an ordinary 2DEG (1DEG)• Spin and velocity correlated

- Charge Current ~ Spin Density

- Spin Current ~ Charge Density

1D Edge States of QSHI

• Elastic backscattering forbidden• No 1D Anderson localization• Weak interactions irrelevant

2D Surface States of STI

• π Berry’s phase

• Weak antilocalization• Impossible to localize (Klein paradox)

Exotic States when broken symmetry leads to surface energy gap:

k

E

k

E

EF

EF

Proximity Effects : Open energy gap at surface

†

0( v )H iψ σ µ ψ= − ∇ −

���

† † *ψ ψ ψ ψ↑ ↓ ↑ ↓

∆ + ∆

µDirac Surface States :

Protected by Symmetry

1. Magnetic : (Broken Time Reversal Symmetry)

• Orbital Magnetic field :

• Zeeman magnetic field :

• Half Integer quantized Hall effect :

2. Superconducting : (Broken U(1) Gauge Symmetry)

proximity induced

superconductivity

p p eA→ −†

zMψ σ ψ

21

( )2

xy

en

hσ = +

• S-wave superconductor

• Resembles spinless p+ip superconductor

• Supports Majorana fermion excitations

Fu,Kane PRL 07Qi, Hughes, Zhang PRB (08)

Fu,Kane PRL 08

M. �

T.I.

S.C.

T.I.

Majorana Fermion

• Particle = Antiparticle : γ = γ†

• Real part of Dirac fermion : γ = Ψ+Ψ†; Ψ = γ1+i γ2 “half” an ordinary fermion

• Mod 2 number conservation / Z2 Gauge symmetry : γ � ± γ

Potential Hosts :

Particle Physics :

• Neutrino (maybe) Allows neutrinoless double β-decay

Condensed matter physics : Possible due to pair condensation

• Quasiparticles in fractional Quantum Hall effect at ν=5/2• h/4e vortices in p-wave superconductor Sr2RuO4

• s-wave superconductor / Topological Insulator

Current Status : NOT OBSERVED

† †0Ψ Ψ ≠

Majorana Fermions and

Topological Quantum Computation

• 2 separated Majoranas = 1 fermion : Ψ = γ1+i γ2

2 degenerate states (full or empty)

1 qubit

• 2N separated Majoranas = N qubits

• Quantum information stored non locally

Immune to local sources decoherence

• Adiabatic “braiding” performs unitary operations

Kitaev, 2003

a ab bUψ ψ→

Non-Abelian Statistics

Majorana Bound States on Topological Insulators

SC

h/2e

1. h/2e vortex in 2D superconducting state

2. Superconductor-magnet interface at edge of 2D QSHI

Quasiparticle Bound state at E=0 Majorana Fermion γ0 “Half a State”

TI

0

∆

−∆

E

†

0 0γ γ=

†

Eγ

†

E Eγ γ− =

S.C.M

QSHI

Egap =2|m|

Domain wall bound state γ0

m<0

m>0

| | | |S Mm = ∆ − ∆

1D Majorana Fermions on Topological Insulators

2. S-TI-S Josephson Junction

SC

TI

SC

φ = πφ ≠ π

SCM

1. 1D Chiral Majorana mode at superconductor-magnet interface

TI

kx

E

†

k kγ γ −= : “Half” a 1D chiral Dirac fermion

Fv x

H i γ γ= − ∂�

φ 0

Gapless non-chiral Majorana fermion for phase difference φ = π

( ) cos( / 2)Fv L x L R x R L R

H i iγ γ γ γ φ γ γ= − ∂ − ∂ + ∆�

Manipulation of Majorana FermionsControl phases of S-TI-S Junctions

φ1φ2

0

Majorana

present+

−

Tri-Junction :A storage register for Majoranas

Create

A pair of Majorana bound

states can be created from

the vacuum in a well defined

state |0>.

Braid

A single Majorana can be

moved between junctions.

Allows braiding of multiple

Majoranas

Measure

Fuse a pair of Majoranas.

States |0,1> distinguished by

• presence of quasiparticle.

• supercurrent across line

junction

E

φ−π

00

1E

φ−π

E

φ−π

0

0

0

0

0

0

0

A Z2 Interferometer for Majorana Fermions

A Signature for Neutral Majorana Fermions Probed with Charge Transport

2

hN

eΦ =

e e

e h

γ1

γ2

γ1

γ2 −γ2

N even

N odd

†

1 2

1 2

c i

c i

γ γ

γ γ

= −

= +

• Chiral electrons on magnetic domain wall split

into a pair of chiral Majorana fermions

• “Z2 Aharonov Bohm phase” converts an

electron into a hole

• dID/dVs changes sign when N is odd.

Fu and Kane, PRL ‘09Akhmerov, Nilsson, Beenakker, PRL ‘09

Conclusion

• A new electronic phase of matter has been predicted and observed

- 2D : Quantum spin Hall insulator in HgCdTe QW’s- 3D : Strong topological insulator in Bi1-xSbx , Bi2Se3, Bi2Te3

• Superconductor/Topological Insulator structures host Majorana Fermions

- A Platform for Topological Quantum Computation

• Experimental Challenges

- Charge and Spin transport Measurements on topological insulators- Superconducting structures :

- Create, Detect Majorana bound states- Magnetic structures :

- Create chiral edge states, chiral Majorana edge states- Majorana interferometer

• Theoretical Challenges

- Further manifestations of Majorana fermions and non-Abelian states- Effects of disorder and interactions on surface states