Post on 14-Apr-2018
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?