1. Topological Insulators
Charles L. Kane, University of Pennsylvania E
k=La
k=Lb E
k=La
k=Lb
Trieste, July 4, 2013

2. Thanks! Mentors and Role Models :
Matthew Fisher
Steve Girvin
Dung Hai Lee
Patrick Lee
Tom Lubensky
Allan MacDonald
Eugene Mele
Naoto Nagaosa
Nick Read
Tom Rosenbaum
Collaborators on Topological Insulators:
Zahid Hasan
Matthew Fisher
Liang Jiang
Max Metlitski
Eugene Mele
Joel Moore
Tony Pantev
John Preskill
Andrew Rappe
Students and Post Docs :
Sugata Chowdhury
Liang Fu
Jeffrey Teo
Ben Wieder
Steve Young
Saad Zaheer
Fan Zhang
Financial Support :

3. Thanks! Mentors and Role Models :
Matthew Fisher
Steve Girvin
Dung Hai Lee
Patrick Lee
Tom Lubensky
Allan MacDonald
Eugene Mele
Naoto Nagaosa
Nick Read
Tom Rosenbaum
Collaborators on Topological Insulators:
Zahid Hasan
Matthew Fisher
Liang Jiang
Max Metlitski
Eugene Mele
Joel Moore
Tony Pantev
John Preskill
Andrew Rappe
Students and Post Docs :
Sugata Chowdhury
Liang Fu
Jeffrey Teo
Ben Wieder
Steve Young
Saad Zaheer
Fan Zhang
Financial Support :

4. Topological Insulators
Introduction : Topological Band Theory
Quantum Spin Hall Effect in 2D
Topological insulators in 3D
Proximity effects and heterostructure devices

5. The Insulating State
Characterized by energy gap: absence of low energy electronic excitations
Covalent Insulator
Atomic Insulator
The vacuum
e.g. intrinsic semiconductor
e.g. solid Ar
electron 4s
Dirac
Vacuum
Egap ~ 10 eV
Egap = 2 mec2
~ 106 eV 3p
Egap ~ 1 eV
positron ~ hole
Silicon

6. The Integer Quantum Hall State
2D Cyclotron Motion,
Landau Levels
sxy = e2/h E
IQHE with zero net magnetic field
Haldane model : Graphene with a periodic magnetic field B(r) (Haldane PRL 1988)
Band structure k + -
B(r) = 0
Zero gap,
Dirac point
B(r) ≠ 0
Energy gap
sxy = e2/h Eg

7. Topological Band Theory
The distinction between a conventional insulator and the quantum Hall state
is a topological property of the band structure
Classified by the TKNN (or Chern) topological invariant (Thouless et al, 1984)
Insulator : n = 0
IQHE state: sxy = n e2/h
Bulk - Boundary correspondence:
Conducting states occur at interfaces where n changes
Edge States at a domain wall
Gapless Chiral Dirac Fermions
QHE state
n=1
Vacuum
n=0 E Eg
Haldane
Model K’ K k

8. Fall 2004 : Graphene Intrinsic spin orbit interaction : 2lSO
Gene Mele : Did you see the paper by Geim and Novoselov et al.?
We should run with graphene.
CLK : I have an idea: Graphene actually has an energy gap!
Graphene: 2D Dirac fermions with 2 valleys (tz = ±1) and 2 spins (sz = ±1) E p
Intrinsic spin orbit interaction : K’ K
Respects all symmetries, so it must* be present.
Two copies of integer quantum Hall state
2lSO

9. Quantum Spin Hall Effect in Graphene
Simplest model:
|Haldane|2
(conserves Sz) J↑ J↓ E
Bulk energy gap, but gapless edge states
p/a E ↑ ↓ ↓ ↑ V
Two terminal Conductance: G = 2e2/h
Edge states form a unique 1D conductor
Half an ordinary 1D electron gas
Protected by time reversal symmetry even when spin conservation is violated.

10. time reversal invariant momenta
Time Reversal Invariant 2 Topological Insulator
2D Time reversal invariant band structures are characterized by a
Z2 topological invariant, n = 0,1
n=0 : Conventional Insulator
n=1 : Topological Insulator E
k*=0
k*=p/a
Edge States E
k*=0
k*=p/a
Kramers degenerate at
time reversal invariant momenta
k* = -k* + G
Stability to Interactions and Disorder: E
Kramers parity
switching:
T2=+1 → T2=-1 F ↑ ↑↓ p
Edge states are impossible to localize and remain in presence of interactions

11. Quantum Spin Hall Insulator in HgTe quantum wells
Theory: Bernevig, Hughes and Zhang, Science 2006
HgTe
HgxCd1-xTe d
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
d> 6.3nm
inverted band order
QSH insulator
G=2e2/h
Laurens Molenkamp
Measured conductance 2e2/h independent of W for short samples (L<Lin)

12. Summer 2006: Race to three dimensions
Early July :
Liang Fu : I think there are multiple Z2 invariants in 3D.
CLK : I think there’s just one, which specifies an even
or odd number of surface Dirac points.
July 12 : Joel Moore and Leon Balents
“Topological invariants of time-reversal invariant band structures”
- Homotopy theory for 4 Z2 invariants in 3D
- Coined the name “Topological Insulators”
July 20 : Rahul Roy
“Three dimensional topological invariants for time reversal invariant
Hamiltonians and the three dimensional quantum spin Hall effect”
July 26 : Fu, Kane, Mele
“Topological Insulators in Three Dimensions”
Characterized surface states
Strong vs weak TI’s
Explicit model

13. 3D Topological Insulators
3D insulators are characterized by 4 Z2 topological invariants : (n0 ; n1n2n3 )
n0 = 1 : Strong Topological Insulator
Surface Fermi circle encloses odd number of Dirac points on all faces ky EF
Topological Metal on the surface :
1/4 graphene
Spin and momentum correlated.
Protected by time reversal symmetry
Robust to disorder:
impossible to localize or passivate. kx
Described by 2D Massless Dirac Hamiltonian
n0 = 0 : Weak Topological Insulator
(n1n2n3 )
Layered 2D topological insulator :
surface states on the sides, but less robust
surface states

14. semiconductor Egap ~ 30 meV
Bi1-xSbx
Pure Bismuth
semimetal
Alloy : .09<x<.18
semiconductor Egap ~ 30 meV
Pure Antimony
semimetal EF La EF Ls La
Egap La Ls Ls E EF T L T L T L k
Inversion symmetry 
Predict: Bi1-xSbx is a strong topological insulator: (1 ; 111).
Other predicted strong TI’s : strained HgTe, strained PbxSn1-xTe

15. Bi1-xSbx Bi2 Se3 ARPES on 3D Topological Insulators EF 5 surface state
Hsieh et al. Nature ’08
5 surface state
bands cross EF
between G and M
Zahid Hasan
Bi2 Se3
ARPES Experiment + Band theory : Y. Xia et al., Nature Phys. ‘09
Band Theory : H. Zhang et. al, Nature Phys. ‘09
Energy gap: D ~ .3 eV :
A room temperature topological
insulator
Simple surface state structure :
A textbook Dirac cone EF

16. Surface Quantum Hall Effect
Fu, Kane ’07;
Qi, Hughes, Zhang ’08,
Essen, Moore, Vanderbilt ‘09
Broken time reversal symmetry allows a surface gap
Orbital QHE: E=0 Landau Level for Dirac fermions 2 1 B
n=1 chiral
edge state -1 -2
“fractional” integer quantum Hall effect impossible in 2D
Allowed on surface of TI because a surface has no boundary.
Anomalous QHE: Induce a surface gap by depositing magnetic material
Gap due to
Exchange field EF M↑ M↓
Chiral Edge State
at Domain Wall TI

17. Superconducting Proximity Effect
A route to engineering topological superconductivity
Fu, Kane ‘08
s wave superconductor
Topological insulator
Similar to 2D spinless px+ipy topological superconductor, except T invariant
Supports Majorana fermion bound states D -D E SC
h/2e TI
S.C. M
QSHI

18. In Search of Majorana B SC
Kitaev (2006) : Zero energy Majorana modes associated
with topological superconductivity offer
A topologically protected memory for quantum information
Prospects for a (rudimentary) topological quantum computer
Ettore Majorana 1906 – 1938?
Superconductor – semiconductor nanowire structure
Theory: Sau, Lutchyn, Tewari, Das Sarma ’09
Oreg, von Oppen, Alicea, Fisher ’10
Expt : Mourik, …, Kouwenhoven, Nature ‘12 SC
InSb
nanowire
Majorana
end state B
Zero bias peak in tunneling conductance observed. Attributed to 0D Majorana end state.

19. Conclusion
Topological Insulators have taught us that new phases with useful and/or interesting and/or
beautiful properties can be found if you know what is possible and you know where to look.
The road ahead:
Perfect the materials and phases that we have found. Many
materials challenges remain.
There has been much recent progress on what is (in principle) possible :
- States with fractionalized or non-Abelian excitations.
- Symmetry protected topological states (beyond free fermions)
- Exotic surface terminations of more conventional states.
There has been less progress on “where to look”
- May require new theoretical or computational methods
- There is much room for good ideas