Topological Insulators
and Topological Band Theory
E
E
k=La
k=Lb
k=La
k=Lb
The Quantum Spin Hall Effect
and 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
Characterized by energy gap: absence of low energy electronic excitations
Covalent Insulator
Atomic Insulator
e.g. intrinsic semiconductor
e.g. solid Ar
The vacuum
electron
4s
Egap ~ 10 eV
3p
Dirac
Vacuum
Egap = 2 mec2
~ 106 eV
Egap ~ 1 eV
Silicon
positron ~ hole
The Integer Quantum Hall State
2D Cyclotron Motion, Landau Levels
Egap  c
E
Energy gap, but NOT an insulator
Quantized Hall conductivity :
2
Jy
B
J y   xy Ex
Ex
e
 xy  n
h
Integer accurate to 10-9
Graphene
E
-  - - 
-  - 
-  -  - 
www.univie.ac.at
Novoselov et al. ‘05
Low energy electronic structure:
Two Massless Dirac Fermions
k
E  v | k |
Haldane Model (PRL 1988)
E   v 2 | k |2  m 2
Add a periodic magnetic field B(r)
•
•
•
Band theory still applies
Introduces energy gap
Leads to Integer quantum Hall state
 xy
e2

h
The band structure of the IQHE state looks just like an ordinary insulator.
Topological Band Theory
The distinction between a conventional insulator and the quantum Hall state
is a topological property of the manifold of occupied states
| (k ) : Brillouin zone (a torus)
Hilbert space
Classified by the Chern (or TKNN) topological invariant (Thouless et al, 1982)
n
Insulator
IQHE state
1
2
d
k  k u (k )  k u (k )  Integer

2 i BZ
: n=0
: xy = n e2/h
The TKNN invariant can only change
at a quantum phase transition where the
energy gap goes to zero
Analogy: Genus of a surface : g = # holes
g=0
g=1
Edge States
Gapless states must exist at the interface between different topological phases
IQHE state
n=1
Vacuum
n=0
y
n=1
n=0
x
Smooth transition : gap must
pass through zero
Edge states ~ skipping orbits
Gapless Chiral Fermions : E = v k
Band inversion – Dirac Equation
E
M>0
Egap
Egap
M<0
K’
Haldane Model
K
ky
Domain wall bound state y0
Jackiw, Rebbi (1976)
Su, Schrieffer, Heeger (1980)
Quantum Spin Hall Effect in Graphene
Kane and Mele PRL 2005
The intrinsic spin orbit interaction leads to a small (~10mK-1K) energy gap
Simplest model:
|Haldane|2
(conserves Sz)
 H
H 
 0
0   H Haldane

H   0
J↓
J↑


*
H Haldane

0
E
Bulk energy gap, but gapless edge states
Edge band structure
Spin Filtered edge states
vacuum
↓
↑
↑
↓
QSH Insulator
0
/a
k
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
Topological Insulator : A New B=0 Phase
There are 2 classes of 2D time reversal invariant band structures
Z2 topological invariant: n = 0,1
n is a property of bulk bandstructure, but can be understood by
considering the edge states
Edge States for 0<k</a
n=1 : Topological Insulator
n=0 : Conventional Insulator
E
E
Kramers degenerate at
time reversal
invariant momenta
k* = -k* + G
k*=0
k*=/a
k*=0
k*=/a
Quantum Spin Hall Insulator in HgTe quantum wells
Theory: Bernevig, Hughes and Zhang, Science 2006
d
HgxCd1-xTe
HgxCd1-xTe
HgTe
Predict inversion of conduction and valence
bands for d>6.3 nm → QSHI
Expt: Konig, Wiedmann, Brune, Roth, Buhmann, Molenkamp, Qi, Zhang Science 2007
d< 6.3 nm
normal band order
conventional insulator
Landauer Conductance G=2e2/h
↑
V
d> 6.3nm
inverted band order
QSH insulator
↓
I
G=2e2/h
Measured conductance 2e2/h independent of W for short samples (L<Lin)
↓
0
↑
3D Topological Insulators
There are 4 surface Dirac Points due to Kramers degeneracy
ky
L4
L1
L3
E
E
kx
OR
L2
2D Dirac Point
Surface Brillouin Zone
k=La
k=Lb
k=La
k=Lb
How do the Dirac points connect? Determined
by 4 bulk Z2 topological invariants n0 ; (n1n2n3)
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
EF
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
Bi2 Se3
ARPES Experiment : Y. Xia et al., Nature Phys. (2009).
Band Theory :
H. Zhang et. al, Nature Phys. (2009).
• n0;(n1,n2,n3) = 1;(000) : Band inversion at G
• Energy gap: D ~ .3 eV : A room temperature
topological insulator
• Simple surface state structure :
Control EF on surface by
exposing to NO2
Similar to graphene, except
only a single Dirac point
EF
Superconducting Proximity Effect
Fu, Kane PRL 08
s wave superconductor
Surface states acquire
superconducting gap D
due to Cooper pair tunneling
Topological insulator
BCS Superconductor :
-k↓
ck† c-† k   Dei
(s-wave, singlet pairing)
k↑
Superconducting surface states
† †
k -k
cc
 D surface e
i
(s-wave, singlet pairing)
Half an ordinary superconductor
Highly nontrivial ground state
-k
←
Dirac point
↑
↓
→
k
  h / 2e
Majorana Fermion at a vortex
Ordinary Superconductor :
Andreev bound states in vortex core:
E
D
0
-D
E ↑,↓
  2
 0
Bogoliubov Quasi Particle-Hole
redundancy :
 † E ,   - E ,
-E ↑,↓
Surface Superconductor :
Topological zero mode in core of h/2e vortex:
E
D
0
-D
E=0
Majorana fermion :  0   0
• Particle = Anti-Particle
• “Half a state”
• Two separated vortices define one zero energy
†
fermion state (occupied or empty)
Majorana Fermion
• Particle = Antiparticle :
  †
• Real part of Dirac fermion :  = †;  = 1i 2 “half” an ordinary fermion
• Mod 2 number conservation  Z2 Gauge symmetry :  → ± 
Potential Hosts :
Particle Physics :
•
Neutrino (maybe)
- Allows neutrinoless double b-decay.
- Sudbury Neutrino Observatory
Condensed matter physics : Possible due to pair condensation
•
•
•
•
Quasiparticles in fractional Quantum Hall effect at n=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
Kitaev, 2003
• 2 separated Majoranas = 1 fermion :  = 1i 2
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
Non-Abelian Statistics
y a  Uab y b
Manipulation of Majorana Fermions
Control phases of S-TI-S Junctions
f1
f2
Tri-Junction :
A storage register for Majoranas
Majorana
present

-
0
Create
Braid
Measure
A pair of Majorana bound
states can be created from
the vacuum in a well defined
state |0>.
A single Majorana can be
moved between junctions.
Allows braiding of multiple
Majoranas
Fuse a pair of Majoranas.
States |0,1> distinguished by
• presence of quasiparticle.
• supercurrent across line
junction
E
E
0
E
1
0
0
0
f-
0
0
0
0
f-
0
f-
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.