Majorana Fermions in Topological Insulators

I. Introduction - Topological band theory and topological insulatorsII. Two Dimensions : Quantum Spin Hall Effect - Time reversal symmetry and edge states - Experiment: Transport in HgTe quantum WellsIII. Three Dimensions : Topological Insulators - Topological Insulator Surface States - Experiment: Photoemission on BixSb1-x

IV. Superconducting proximity effect and Majoranafermions

- Majorana fermion bound states: A platform for topological quantum computing? - Fractional Josephson effect in a S-QSHI-S junction

Thanks to Gene Mele, Liang Fu, Jeffrey Teo

The Insulating State

The Integer Quantum Hall State

IQHE with zero net magnetic fieldGraphene with a periodic magnetic field B(r) (Haldane PRL 1988)

Band structure

B(r) = 0Zero gap, Dirac point

B(r) ≠ 0Energy gapσxy = e2/h

Eg

k

Eg ~ 1 eV

e.g. Silicon

g cE != hE

2D Cyclotron Motion, Landau Levels

σxy = e2/h

+ − + − + − +

+ − + − +

+ − + − + − +

Topological Band Theory

21( ) ( )

2 BZ

n d u ui!

= " # $ #% k kk 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!r

a

Insulator : n = 0IQHE state : σxy = n e2/h

The TKNN invariant can only changewhen 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 = 0,1

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

Kramers degenerate attime 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 2n!

" #=

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 ! " !

"

# = # #

# = ±

Γ4 Γ3

Γ1 Γ2

4

2

1

( 1) ( )n i

i n

! "=

# = $%%Bulk Brillouin Zone

Quantum spin Hall Effect :J↑ J↓

E

Two dimensions : Quantum Spin Hall Insulator

Theory: Bernevig, Hughes and Zhang, Science ’06Experiement: Konig et al. Science ‘07

HgTe

HgxCd1-xTe

HgxCd1-xTed

d < 6.3 nmNormal band order

d > 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 ~ p Normal

Inverted

G ~ 2e2/h in QSHI

Three Dimensional Topological InsulatorsIn 3D there are 4 Z2 invariants: (ν0 ; ν1ν2ν3) characterizing the bulk. These determine how surface states connect.

Fu, Kane & Mele PRL 07Moore & Balents PRB 07Roy, cond-mat 06

Surface Brillouin Zone

Λ4

Λ1 Λ2

Λ32D Dirac

Point

E

k=Λa k=Λb

E

k=Λa k=Λb

ν0 = 1 : Strong Topological Insulator

Fermi surface encloses odd number of Dirac pointsTopological Metal • Berry’s phase π around Fermi surface • Robust to disorder (antilocalization)

ν0 = 0 : Weak Topological InsulatorFermi surface encloses even number of Dirac pointsNormal Metal • Berry’s phase 0, less robust. • Equivalent to layered 2D QSHI

OR

EF

Bi1-xSbxPure Bismuth

semimetalAlloy : .09<x<.18

semiconductor Egap ~ 30 meVPure Antimony

semimetal

E

k

EF

LsLa Ls

La

Ls

LaEF EF

Egap

T L T L T L

Theory: Predict Bi1-xSbx is a topological insulator by exploiting inversion symmetry of Bi, Sb (Fu,Kane PRL’07)

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

• 5 surface state bands cross EF between Γ and M • Proves that Bi1-x Sbx is a Strong Topological Insulator

Proximity Effects, Energy GapsMinimal surfacestate model:

†

0 ( v )H i! " µ != # $ #rr

µ

Magnetic †

M zV M! " !=

Gap for M>Mc(µ)Broken time reversal symmetry“half quantized” QHE σxy = e2/2h

Two Gapped Phases :

M ↑TI

Superconducting † † *

SV ! ! ! !

" # # "= $ + $

s wave SCTISimilar to spinless px+ipy SC

No broken time reversal

Bogoliubov Spectrum (µ=0) ( )2 2 2( ) | | | | vE k M k± = ! ± +

|Δ|>|M| : Superconducting phase|Δ|<|M| : Insulating phase|Δ|=|M| : Critical : 2D gapless Majorana

↑←↓→

“half” a 2DEG

S-TI-M interfaceMajorana Fermions

S.C.M ↑Gapless 1D chiral Majorana fermions bound to domain wall

Vortex in 2D SC :Zero energy Majorana bound stateat vortex

h/2e

S-QSHI-M junction

Zero energy Majorana bound stateat junction

k

E

S.C.M

QSHI

Kitaev 2003 : 2N Majoranas = N qubits: fault tolerant quantum memory Braiding : Quantum computation

0

Δ

−Δ

E

“half” a state

( ) ir e

!" = "

Manipulation of Majorana Fermions

S-TI-S Tri-Junction :

φ1φ2

0

Create, Transport and Fuse Majorana fermions along line junctions

Majorana bound state present at tri-junction

2π/3

−2π/30

phase φ

+

−

S-TI-S Line Junction : A 1D “wire” for Majorana fermions

φ = 0φ = π

0<φ<π

E

Gapless 1D MajoranaFermions for φ = π

k

Evidence for good contact between BiSb and Nb : minimal Shottky barrier

The challenges :• Find suitable topological insulator (Bi1-x Sbx ? Eg ~ 30 meV)• Find suitable superconductor which makes good interface ( Nb ? )• Optimize proximity induced gap and discrete Andreev bound states• Control the superconducting phases with Josephson junctions• Measure current difference when Majoranas are fused ………

S-QSHI-S Josephson Junction Fu, Kane cond-mat/08

Andreev bound states in the junctionB = 0 B ≠ 0

B = 0 : “Half” a perfectly transmitting superconducting quantum point contact

B ≠ 0 : “Fractional Josephson Effect” (Kitaev 2001; Kwon, Sengupta, Yakovenko, 2004)

• 4π periodicity of E(φ) protected by local conservation of fermion parity. • Two coupled Majoranas

“Telegraph noise” at φ ∼ π • Switch fermion parity by inelastic scattering of thermally activated quasiparticles • τ ~ exp( Δ0/T ) • Noise S(ω→0) ~ exp( Δ0/T )

B

e

o

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

• Experimental Challenges

- Spin dependent Transport Measurements - Transport and magneto-transport measurements on Bi1-xSbx

- Superconducting proximity effect : - BiSb-Nb ? HgCdTe-Nb ?? - Characterize S-TI-S junctions - Create the Majorana bound states - Detect the Majorana bound states

• Theoretical Challenges

- Effects of disorder, interactions on surface states, superconductivity and critical phenomena

- Other Materials?