Topological Insulators
A new state of matter with three dimensional
topological electronic order
L. Andrew Wray
Lawrence Berkeley National LabPrinceton University

Surface States (Topological Order in 3D)“Search & Discovery”: PHYSICS TODAY 2009 (April)

REVIEWS------------------------------------------------

MZH & C.L. Kane, Rev. of Mod. Phys. 82, 3045 (2010)
MZH & J.E. Moore, Ann. Rev. of Cond-Mat. Phys. (2011)
X.L. Qi & S.C.Zhang, RMP (in press) 2011

History of Z2 Topological Insulators
2005: Theoretical prediction of the Z2 TI phase (C.L. Kane andE.J. Mele PRL 2005)
2006-2007: Achievement of a 2D TI phase in HgTe (B.A.
Bernevig, T.L. Hughes, S.-C. Zhang, SCIENCE 2006, M. König etal. SCIENCE 2007)
2007-2009: First discovery of a 3D TI (Bi1-xSbx alloy, L. Fu et al.
PRL 2007, D. Hsieh et al. NATURE 2008, SCIENCE 2009)
2008: Discovery of the M2X3 TI class (Y. Xia, arXiv 2008, H.-J.
Zhang et al. NATURE 2009, D. Hsieh et al. NATURE 2009)2010: Symmetry breaking: Observation of unconventional
superconductivity in CuxBi2Se3 , magnetism in MnxBi2-xTe3(Wray et al. Nat. Phys. 2010, Hor et al. PRB 2010)2010-2011: Many new ternary TIs, building of TI interfaces andnanodevices

Lecture Outline:
1. “Experimentally discovering” topological insulators
2. Understanding topological order
3. New material properties, new possibilities
4. Discussion

Bismuth Selenide
e- doped as Cu0.12Bi2Se3

Y. Xia, arXiv 2008, D. Hsieh et al. NATURE 2009, L.A. Wray et al. Nat. Phys. 2010

Helical Dirac fermions
One to One Spin-Linear Momentum Locking+K-K-K+K

Hsieh et.al., SCIENCE 09, NATURE 09

Spin-texture  Absence of Backscattering
STM (Roushan et.al.)

Spin-ARPES (Hsieh et.al.)Xu, Moore et. (06)Spin-Independent

Spin-Dependent
Roushan et.al.,NATURE 09

Photoemission on a TI

Wray PRB 2011

Macroscopic Effects

L
Conductivity by wire size:σB ~ A/LσS ~ r/L
A=πr2

Critical crystal size for σB~σS in Bi2Te2Se is ~1X1X0.1mm!!

More than just a surface state
Typical Rashba surface state

Disallowed band structure!

For visual clarity, 3D parity symmetry has been broken so that eachband is singly degenerate away from the Kramers points

Tying a knot

Note: this does not closely approximate a real band structure

Energy ((eV)

How to see the topological connection: band bendingtight binding model
Momentum (A-1)

Band bending creates new topological surface states!

Wray et al., Nat. Phys. 2010, arXiv 2011

Energy ((eV)

Partner exchange and symmetry inversion

Non-TI “gap”

Momentum (A-1)

Wray, Nature Physics 2010
Wray, arXiv:1105.4794 2011

Weak TI++Strong TI+kZ=π++ kX=π

kZ=0 +Energy (eVeV)-----+++kZ=π++ kX=π

kZ=0 +---Momentum
L. Fu, PRL 2007

Inducing the topological statelattice strain and spin orbit coupling

Xu et al., Science 2011

Topological Invariants Define Surface States

Fully gapped
Bi2Se3  Bi2Se3“interface”

Topological Insulator

Quantum Hall Effectν (Chern Number): Thouless et.al.,{νο } (Chern Parity invariants) Z2

3D Topological Insulators

Protected Surface States = New 2DEG

X
Nature 08 (subm. 2007)

2D Topological InsulatorsScience 07 (subm. 2007)

Edge States (1D) by TRS

Chiral Edge States (1D)Topological Insulator in 2D: Quantum Hall State
Thouless et.al, (‘82), (Berry Phase ‘84)nh/e2Hall conductance:

σxy = ne2/hn = Chern no. ( Edge states)
Chern : Quantum version (Hilbert space)of Gauss-Bonnet formula

Topological Property

1nn=

TKNN invariant:Topological Quantum Number

2π i ∫BZ

d 2k ⋅ ∇k u (k ) × ∇k u (k )Electron-occupied Bulk bands

Finite n  topologically “protected” edge-states

Quantum Hall Effect (insulator) : 2D Topological insulator w/ LL
Haldane model (QAH) : 2D Topological insulator w/o LL
time-reversal invariant!
+2e2/h

1 invariant  QSHE
Quantum spin Hall effectKane & Mele (05a), Kane & Mele (05b) [σspinHall Not quantized]
Bernevig, Hughes, Zhang (06), Sheng, Haldane et.al., (06)Expt: Molenkamp group HgCdTe-QWells, Science (2007)

4 Invariants  3D TI
Distinct Topological state in 3D Topo Insulator
Moore & Balents(07), Fu & Kane(07), Fu, Kane &Mele(07), Roy (2009)

Expt: MZH group Bi-based Semiconductors, KITP Proc.(2007)
Nature 2008 [Submitted in 2007]
3D TIs -> Superconductors and Magnets (Tc)
Many others afterwards, ~ 800 papers on arXiv

QHE phases

Transport

σxy = ne2/hTopological quantum number

Topo Insulatorsνο = Θ/πΘ=π
Θ=π (odd)
Θ=2π
Θ=2π (even)

How to experimentally “measure” thetopological quantum numbers (νi) ?
4 TQNs  16 distinct insulators{ν0, ν1 ν2 ν3 }No quantized transport
via :{νi}Topological quantum number

Topological “Order Parameters”?

Spin-sensitiveMomentum-resolved
Edge vs. Bulk

So what can they do?

Magnetoelectric Effects(not “Electromagnetic”)

Spin helical states meeting defects

Biswas PRB 2010; arXiv 2010

Local Magnetic Monopoles

Dyons

Magnetic order creates effective “axions”

X.-L. Qi, Science 2009

Possibilities for Surface Magnetism

CuxBi2Se3 (Tc ~ 3.8K) : Hor et.al., PRL 2010

A Majorana Platform
CuxBi2Se3

Topological Surface States:
Superconductivity in doped topological insulators
Wray et.al., Nature Physics (2010)

Majorana FermionsFu, PRL 2008; Wray, Nature Physics 2010; PRB 2011

Non-Abelian Statistics

Topological Superconductor (TSC)?
Kitaev/Ludwig D3 class of TSC (proposed by Fu & Berg 09)

m/µµ from ARPES
ARPES Expts

If ODD parity  TSC[ analog of SF He-3(B) ]Wray et.al., Nature Physics (2010)

Slide from J. Moore

Topological insulators are:Simple•Exact non-interacting models (DFT, k.p)
•Most complexity reduces to 1D (much nicer than cuprates!!!)•Surface is robust against non-magnetic scattering

Complicated•Theory is difficult to learn, and few people know it•Many surface instabilities, particularly because they occupy the same orbitals as bulk
•Lots of new phases and new physics to explore (Majorana Fermions, Dyons, magnetoelectriceffect, unusual surface transport, unusual interface physics)•Lots of different compounds! (Tl chalcogenides, Heuslers, M2X3, etc)
•Many “simple” issues are actually complicated:-2nd order backscattering is allowed from Anderson impurities-Self energy is poorly understood