Topological Insulators and Superconductors

I. Topological Insulators and Band Theory

Unifying theme: bulk – boundary correspondence

- Integer Quantum Hall Effect

- 2D Quantum Spin Hall Insulator

- 3D Topological Insulator

- Topological Superconductivity, Majorana fermions

II. Summary and Outlook

- What we have accomplished

- Challenges for the Future

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

sxy = e2/h

Eg

k

Eg ~ 1 eV

e.g. Silicon

g cE

E2D Cyclotron Motion,

Landau Levels

sxy = e2/h

+ - + - + - +

+ - + - +

+ - + - + - +

Topological Band TheoryThe distinction between a conventional insulator and the quantum Hall state

is a topological property of the band structure

Insulator : n = 0

IQHE state : sxy = n e2/hThe TKNN invariant can only change at a phase

transition where the energy gap goes to zero

( ) :H k2 B lo c h H a m ilto n a n s

B rillo u in zo n e ( = to ru s T ) w ith e n e

rg y g a p

The set of N occupied Bloch wavefunctions defines a U(N)

vector bundle over the Brillouin zone torus.

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

Closely related to theory of electric polarization

1

( )N

ii

u

k

2

21 1

2 2C T

n d d

A k F k

1

( ) ( ) ( )

N

i i

i

i u u

- kA k k k

( ) ( ) k

F k A k

Berry connection

Berry curvature

1st Chern number

kx

ky

C/a

/a

-/a

-/a

Edge States

Gapless states must exist at the interface between different topological phases

IQHE state

n=1

Egap

Domain wall bound state

Vacuum

n=0

Edge states ~ skipping orbits

n=1 n=0

Dirac Equation :

x

y

M<0

M>0

Gapless Chiral Fermions : E = v k

E

Haldane Model

ky

Egap

KK‟

Smooth transition : band inversion

Jackiw, Rebbi (1976)

Su, Schrieffer, Heeger (1980)

Bulk – Boundary Correspondence : Dn = # Chiral Edge Modes

( )x x y y z

H i M xs s s - + +v

| |

( ') '/

0

x

y

M x d x

ik y

e e

- v

Time Reversal Invariant 2 Topological Insulator

n=0 : Conventional Insulator n=1 : Topological Insulator

Kramers degenerate at

time reversal

invariant momenta

k* = -k* + G

k*=0 k*=/a k*=0 k*=/a

Even number of bands

crossing Fermi energy

Odd number of bands

crossing Fermi energy

Understand via Bulk-Boundary correspondence : Edge States for 0<k</a

2 topological invariant (n = 0,1) for 2D T-invariant band structures

Time Reversal Symmetry :

Kramers‟ Theorem :

1

( )H H-

-k k *y

i s

21 - All states doubly degenerate

2D Quantum Spin Hall Insulator

Theory: Bernevig, Hughes and Zhang, Science ‟06

Experiement: Konig et al. Science „07

HgTe

HgxCd1-xTe

HgxCd1-xTed

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 /a 2/a

↑

↓

↑

↑ ↓

↓

Eg

Normal

Inverted

G ~ 2e2/h in QSHI

G6 ~ s

G8 ~ p

k

E

Conventional Insulator

2( ) 1

n a +

d < 6.3 nm : Normal band order

G6 ~ s

G8 ~ p k

E

QSH Insulator

2( ) 1

n a -

d > 6.3 nm : Inverted band order

band

inversion

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

Surface Brillouin Zone

2D Dirac Point

E

k=a k=b

E

k=a k=b

n0 = 1 : Strong Topological Insulator

Fermi circle encloses odd number of Dirac points

Topological Metal :

1/4 graphene

Robust to disorder: impossible to localize

n0 = 0 : Weak Topological Insulator

Fermi circle encloses even number of Dirac points

Related to layered 2D QSHI

OR

4

1 2

3

EF

How do the Dirac points connect? Determined

by 4 bulk 2 topological invariants n0 ; (n1n2n3)

kx

ky

Moore & Balents PRB 07

Roy, cond-mat 06

Fu, Kane & Mele PRL 07

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 n0;(n1,n2,n3) = 1;(111)

• 5 surface state bands cross EF

between G and M

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

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

• n0;(n1,n2,n3) = 1;(000) : Band inversion at G

• Energy gap: D ~ .3 eV : A room temperature

topological insulator

• Simple surface state structure :

Similar to graphene, except

only a single Dirac point

EF

Control EF on surface by

exposing to NO2

Surface Quantum Hall Effect

21

2x y

en

hs

+

2

2xy

e

hs

2

2xy

e

hs

B

n=1 chiral edge state

Orbital QHE :

M↑ M↓

E=0 Landau Level for Dirac fermions. “Fractional” IQHE

Anomalous QHE : Induce a surface gap by depositing magnetic material

Chiral Edge State at Domain Wall : DM ↔ -DM

†

0( v )

zMH i s s D - - +

Mass due to Exchange field

Egap = 2|DM|

EF

0

1

2

-2

-1

TI

2

sg n ( )2

xy M

e

hs D

2

2

e

h+

2

2

e

h-

Topological Magnetoelectric Effect

Consider a solid cylinder of TI with a magnetically gapped surface

21

2x y

eJ E n E M

hs

+

Magnetoelectric Polarizability

M E

EJ

M

21

2

en

h

+

The fractional part of the magnetoelectric polarizability is determined

by the bulk, and independent of the surface (provided there is a gap)

Analogous to the electric polarization, P, in 1D.

Qi, Hughes, Zhang ‟08; Essin, Moore, Vanderbilt „09

d=1 : Polarization P

d=3 : Magnetoelectric

poliarizability

[ ]2

T rB Z

e

A

2

2

2[ ]

4 3T r

B Z

ed

h + A A A A A

formula “uncertainty quantum”

e

2/e h

(extra end electron)

(extra surface

quantum Hall layer)

L D E B

DL

E B

P E

topological “q term”

2

2

e

h q

0 2T R s ym . : o r m o d q

Topological Superconductivity

†

†

1

2B d G

H H

k

Intrinsic anti-unitary particle – hole symmetry

1

B d G B d GH H

- -

†

E E E E

- -

0

*

0

B d G

H

H

H

D

D -

Bogoliubov de Gennes

Hamiltonian

BCS mean field theory : † † † † * D

Bloch - BdG Hamiltonians satisfy

Topological classification problem similar to time reversal symmetry

*x

Particle – hole redundancy

21 +

0 1

1 0x

=

E

k

E

k

1( ) ( )

B d G B d GH H

- - -k k

same state

Bulk-Boundary correspondence : Discrete end state spectrum

0

D

-D

E

E

-E0

D

-D

E=0

n=0 “trivial” n=1 “topological”

†

0 0E E

G G

Majorana fermion

bound state

1D 2 Topological Superconductor : n = 0,1

END

†

E E-G G

EG

(Kitaev, 2000)

Majorana Fermion : Particle = Antiparticle

Real part of a Dirac fermion :

“Half a state”

Two Majorana fermions define a single two level system

Zero mode

†

1 1 2

† †

2 1 2

;

( ) ;

i

i i

+ +

- - - , 2i j i j

†

0 1 2 02H i 0empty

occupied

21

i

†

Periodic Table of Topological Insulators and Superconductors

Anti-Unitary Symmetries :

- Time Reversal :

- Particle - Hole :

Unitary (chiral) symmetry :

1( ) ( ) 1

2 ; H H

- - - k k

1( ) ( ) 1

2 ; H H

- + - k k

1( ) ( ) ; H H

- - k k

Real

K-theory

Complex

K-theory

Bott Periodicity

Altland-

Zirnbauer

Random

Matrix

Classes

Schnyder, Ryu,

Furusaki, Ludwig 2008

Kitaev, 2008

Hasan and Kane, Rev Mod Phys 82, 3045 (2010).

Moore, Nature 464, 194 (2010).

Qi and Zhang, Phys. Today 63, 33 (2010).

Ryu, Schnyder, Furusaki and Ludwig, New J. Phys. 12, 065010 (2010).

Qi and Zhang, Rev Mod Phys, to appear, arXiv:1008.2026.

Moore and Hasan, Annual Review of Condensed Matter, 2, 44 (2010).

Further Reading:

Major accomplishments :

Topological band theory of insulators and superconductors

is well understood:

- Topological Invariants and bulk-boundary correspondence

- Robustness to disorder and weak interactions

- Electromagnetic and/or gravitational response

Rapid materials progress:

- Several materials have been identified and characterized

experimentally.

- Even more materials have been predicted, based on band

structure calculations.

- Detailed characterization of topological insulators via transport,

optics and spectroscopy is developing.

Grand Challenges

• Perfect existing and new materials

• Design and implement heterostructure devices

• Find Majorana

• Classify and characterize many body topological phases

• Find applications for technology

Perfect New and Existing Materials

Real 3D topological insulator materials

are not such great insulators. Electrical

conductance is dominated by the bulk.

Challenge for materials theory in conjunction

with experiments.

Success Story : Bi2Te2Se

Xiong, et al (Princeton) „11

Electrical resistivity in Bi2Se3

(Checkelsky et al „09)

Topological insulator devices

• Topological Insulator – Trivial Insulator

- protect the surface states

- control the surface state Fermi energy

(modulation doping)

• Topological Insulator – Magnetic Insulator

- achieve magnetically gapped surface states

- anomalous quantum Hall effect

- topological magnetoelectric effect

• Topological Insulator – Superconductor

- achieve proximity induced superconductivity

in the surface states.

Requires control interfaces between materials.

Challenge for materials theory and experiment

M. ↑

T.I.

S.C.

T.I.

I.

T.I.

Potential Hosts :

Particle Physics : Neutrino (maybe)

- Allows neutrinoless double b-decay.

Condensed matter physics : Possible due to pair condensation

- n=5/2 Fractional quantum Hall effect

- Topological superconductivity

Topological Quantum Computation

Find Majorana

† †0

Ettore Majorana 1906 – 1938?

Observation of a Majorana fermion is among

the great challenges of physics today

1937 : Majorana publishes his modification

of the Dirac equation that allowes spin ½

particles to be their own antiparticle.

1938 : Majorana mysteriously disappears at sea

What is the best way to achieve topological superconductivity?

SC

M

TI

SC

TI

Majorana bound state at a vortex (0D) 1D Chiral Majorana mode at a interface with

a magnetic material

• Exotic superconductors (superfluids)- Surface of 4He

- p+ip superconductor (eg Sr2RuO4)

- CuxBi2Se3 ?

• Ordinary superconductor heterostructures- superconductor – topological insulator

- superconductor – semiconductor (eg InAs wire)

SC InAs

Majorana

end state

What are the most feasible experimental

signatures of Majorana modes ?

Classify and Characterize Interacting Topological States

Topological Insulators are like the

Integer Quantum Hall effect. The

single particle energy gap is correctly

described by non interacting band

theory.

Interacting systems exhibit a much

richer collection of fractional quantum

Hall states. Understanding these was

one of the greatest triumphs of many

body physics.

What is a fractional topological insulator ?

Classify possible states

Characterize quasiparticle excitations and surface states.

Need to develop new techniques:

- Parton construction?

- B-F theory?

- Entanglement spectrum?

What is the generalization of the bulk – boundary

correspondence for interacting systems ?