E
k=a k=b
E
k=a k=b
Topological Insulatorsand Topological Band Theory
The Quantum Spin Hall Effectand Topological Band Theory
I. Introduction - Topological band theory
II. Two Dimensions : Quantum Spin Hall Insulator - Time reversal symmetry & Edge States - Experiment: Transport in HgCdTe quantum wells
III. Three Dimensions : Topological Insulator - Topological Insulator & Surface States - Experiment: Photoemission on BixSb1-x and Bi2Se3
IV. Superconducting proximity effect - Majorana fermion bound states - A platform for topological quantum computing?
Thanks to Gene Mele, Liang Fu, Jeffrey Teo, Zahid Hasan + group (expt)
The Insulating State
Covalent Insulator
Characterized by energy gap: absence of low energy electronic excitations
The vacuumAtomic Insulator
e.g. solid Ar
Dirac Vacuum
Egap ~ 10 eV
e.g. intrinsic semiconductor
Egap ~ 1 eV3p
4s
Silicon
Egap = 2 mec2 ~ 106 eV
electron
positron ~ hole
The Integer Quantum Hall State
2D Cyclotron Motion, Landau Levels
gap cE E
Quantized Hall conductivity :
ExB
Jy
y xy xJ E2
xy hne
Integer accurate to 10-9
Energy gap, but NOT an insulator
Graphene
2
xy
e
h
E
k
Low energy electronic structure:
Two Massless Dirac Fermions
Haldane Model (PRL 1988)
Add a periodic magnetic field B(r)
• Band theory still applies• Introduces energy gap
• Leads to Integer quantum Hall state
v | |E k
2 2 2v | |E m k
The band structure of the IQHE state looks just like an ordinary insulator.
Novoselov et al. ‘05
www.univie.ac.at
Topological Band Theory
g=0 g=1
21( ) ( ) Integer
2 BZn d u u
i 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
Analogy: Genus of a surface : g = # holes
| ( ) : ( )k
Brillouin zone a torus Hilbert space
Insulator : n = 0IQHE state : xy = n e2/h
The TKNN invariant can only changeat a quantum phase transition where theenergy gap goes to zero
Classified by the Chern (or TKNN) topological invariant (Thouless et al, 1982)
Edge StatesGapless states must exist at the interface between different topological phases
IQHE staten=1
Egap
Domain wall bound state 0
Vacuumn=0
Edge states ~ skipping orbits
n=1 n=0
Band inversion – Dirac Equation
x
y
M<0
M>0
Gapless Chiral Fermions : E = v k
E
Haldane Modelky
Egap
KK’
Smooth transition : gap must pass through zero
Jackiw, Rebbi (1976)Su, Schrieffer, Heeger (1980)
Quantum Spin Hall Effect in Graphene
The intrinsic spin orbit interaction leads to a small (~10mK-1K) energy gap
Simplest model:|Haldane|2
(conserves Sz)
Haldane*Haldane
0 0
0 0
H HH
H H
Edge states form a unique 1D electronic conductor• HALF an ordinary 1D electron gas
• Protected by Time Reversal Symmetry
• Elastic Backscattering is forbidden. No 1D Anderson localization
Kane and Mele PRL 2005
J↑ J↓
E
Bulk energy gap, but gapless edge states Edge band structure
↑↓
0 /a k
Spin Filtered edge states
↓ ↑QSH Insulator
vacuum
Topological Insulator : A New B=0 Phase There are 2 classes of 2D time reversal invariant band structures
Z2 topological invariant: = 0,1
is a property of bulk bandstructure, but can be understood byconsidering the edge states
=0 : Conventional Insulator =1 : Topological Insulator
Kramers degenerate at
time reversal
invariant momenta
k* = k* + G
E
k*=0 k*=/a
E
k*=0 k*=/a
Edge States for 0<k</a
Expt: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007
Measured conductance 2e2/h independent of W for short samples (L<Lin)
d< 6.3 nmnormal band orderconventional insulator
d> 6.3nminverted band orderQSH insulator
Quantum Spin Hall Insulator in HgTe quantum wells
Theory: Bernevig, Hughes and Zhang, Science 2006
HgTe
HgxCd1-xTe
HgxCd1-xTed
Predict inversion of conduction and valence bands for d>6.3 nm → QSHI
G=2e2/h
↑↓
↑ ↓V 0I
Landauer Conductance G=2e2/h
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
0 = 1 : Strong Topological Insulator
Fermi circle encloses odd number of Dirac points
Topological Metal :
1/4 graphene
Robust to disorder: impossible to localize
0 = 0 : Weak Topological Insulator
Fermi circle encloses even number of Dirac pointsRelated to layered 2D QSHI
OR
4
1 2
3
EF
How do the Dirac points connect? Determined by 4 bulk Z2 topological invariants 0 ; (123)
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 0;(1,2,3) = 1;(111)
• 5 surface state bands cross EF
between and M
ARPES Experiment : Y. Xia et al., Nature Phys. (2009).Band Theory : H. Zhang et. al, Nature Phys. (2009).Bi2 Se3
• 0;(1,2,3) = 1;(000) : Band inversion at
• Energy gap: ~ .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 byexposing to NO2
Superconducting Proximity Effect
s wave superconductor
Topological insulator
Fu, Kane PRL 08
BCS Superconductor :† † ik kc c e
k↑
-k↓
Superconducting surface states
† †surface
ik kc c e
Surface states acquiresuperconducting gap due to Cooper pair tunneling
Half an ordinary superconductorHighly nontrivial ground state
-k
k
↑
←
↓
→
Dirac point
(s-wave, singlet pairing)
(s-wave, singlet pairing)
Majorana fermion : • Particle = Anti-Particle
• “Half a state”
• Two separated vortices define one zero energy
fermion state (occupied or empty)
Majorana Fermion at a vortex
Ordinary Superconductor :
Andreev bound states in vortex core:
0
E
E ↑,↓
-E ↑,↓
Bogoliubov Quasi Particle-Hole redundancy :
†, ,E E
Surface Superconductor :
Topological zero mode in core of h/2e vortex:
0
E †0 0
E=0
0 2
/ 2h e
Majorana Fermion• Particle = Antiparticle : †
• Real part of Dirac fermion : = †; = i“half” an ordinary fermion
• Mod 2 number conservation Z2 Gauge symmetry : → ±
Potential Hosts :
Particle Physics :
• Neutrino (maybe)
- Allows neutrinoless double -decay. - Sudbury Neutrino Observatory
Condensed matter physics : Possible due to pair condensation
• Quasiparticles in fractional Quantum Hall effect at =5/2• h/4e vortices in p-wave superconductor Sr2RuO4
• s-wave superconductor/ Topological Insulator• among others....
Current Status : NOT OBSERVED
† † 0
Majorana Fermions and Topological Quantum Computation
• 2 separated Majoranas = 1 fermion : = i 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 bU
Non-Abelian Statistics
Manipulation of Majorana FermionsControl phases of S-TI-S Junctions
12
0
Majorana present
Tri-Junction : A storage register for Majoranas
CreateA pair of Majorana boundstates can be created from the vacuum in a well definedstate |0>.
BraidA single Majorana can bemoved between junctions.Allows braiding of multipleMajoranas
MeasureFuse a pair of Majoranas.
States |0,1> distinguished by• presence of quasiparticle.• supercurrent across line junction
E
00
1E
E
0
0
0
0
0
0
0
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 and Bi2Te3
• Superconductor/Topological Insulator structures host Majorana Fermions - A Platform for Topological Quantum Computation
• Experimental Challenges - 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
- Effects of disorder on surface states and critical phenomena - Protocols for manipulating and measureing Majorana fermions.
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