Introductory lecture on topological insulators

Reza Asgari

Workshop on graphene and topological insulators , IPM. 19-20 Oct. 2011

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Outlines

-Introduction New phases of materials, Insulators -Theory quantum Hall effect, edge modes, topological invariance -Conclusion

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-In classical world we have solid, liquid and gas phases

-In quantum world we have metals, insulators, magnetisms , superconductors, etc : spontenious symmetry breaking

Broken rotational symmetry Broken gauge symmetry

Phases of matter

Qi and Zhang PRB 2008

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New phases of matter in quantum electron systems

-Quantum Hall effect

-Super fluidity and superconductors

-Localization, disorder and quantum magnetism

Nobel prize ’85,’98

Nobel prize ’72,’73,’87,’96’03

Nobel prize ’70,’77,’94,’07

- Quantum phase transition, condensation Nobel prize ’82,’01

Hasan and kane RMP, 2011

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New phases of matter in quantum electron systems

-Quantum Hall effect

- Super fluidity and superconductors

-Localization, disorder and quantum magnetism

QHE without magnetic field? Majorana fermions?

Sc without BCS-like paradigm? New routes to HTCS?

Spin-liquid, fractionalization

-Quantum phase transition, condensation Q- phase transition, New universality classes

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Insulators

How many insulators do we know?

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Insulators

Band insulator

Peierls insulator

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Insulators

Mott Insulator: large Coulomb repulsion electron can not move

Anderson localization: Impurity scattering

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Insulators

What else?

TI is a band insulator characterized by a topological number and has gapless excitations at its boundaries.

Topological Insulators !

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Topological Insulators

TI electronic phases: Kane & Mele 2005 Fu, Kane & Mele 2007 Moore & Balents 2007 Roy 2009

TI predictions: Berneving, Hughes & Zhang 2006 Fu & Kane 2007

TI observations: König et al, 2007 Hsieh et al 2008

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All electronic states with an energy gap topologically equivalent to the vacuum?

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Hall effect ( Edwin Hall 1879)

Metals feature no gap: current can flow

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Hall effect at high magnetic field

Electrons + high magnetic field Discrete energy levels

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Insulators vs IQHE

12( )m c mε ω= +

2

0 ( )x xy yeE j N Eh

σ≠ → = =Hall current

Von Klitzing, Dorda & Pepper, Phys. Rev. Lett. 45, 494 (1980)

NOT insulator!

Hasan and kane RMP, 2011

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Bulk insulators: conduction through the edges

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What is the difference between a QHE states and an ordinary insulator?

Thouless, Kohmoto, Nightingale & den Nijs, Phys. Rev. Lett. 49, 405 (1982)

Topology

( ){ ( )} { ( )}m m mH k u k E u k= Periodic conditions Sphere

2

1

1 1 | |2

N

k k k km

c d k u ui Nπ =

−= ∇ × < ∇ >∑∫First Chern number Topological invariant

2

xyeNch

σ = 01

cc=

=

Ordinary insulator

IQHE 2Z

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Quantum Hall effect without magnetic field

Haldane model (1988)

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Edge sates and the bulk boundary correspondence

( ) ( )F zH q q m yν σ σ= +

( ) 0m y >

( ) 0m y <

Jackiw, and Rebbi, Phys. Rev. D 13, 3398 (1976)

0

( ') ' / 1( , )

1

y

Fx

x

m y d yiq x

q x y e eν

ψ−∫

∝

R Ln N N∆ = − Bulk boundary correspondence

0)( F

x

x vdq

qdE=

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Kramers’ theorem

In QHE can only occur when time reversal symmetry is broken.

Spin-orbit interaction allows a different topological class when time reversal symmetry is preserved

THEOREM: All eigenstates of a time reversal invariant Hamiltonian commute with TR operator are at least twofold degenerate

Topological Insulator. 2ZFu and kane PRL, 2009

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quantum spin Hall effect

Bernevig, and Zhang, Phys. Rev. Lett. 96 106802 (2006).

sx x xy yJ J Eσ↑ ↓− =

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2D topological insulator (HgTe/CdTe)

Bernevig, Hughes, and Zhang, Science 314, 1757 (2006)

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Band structures (HgTe/CdTe) Without SOI With SOI

Effective edge Hamiltonian zedge yH A k σ=

5

3.6

5.5 10 /

A eV AA m sν

≈

= = ×

Bernevig, Hughes, and Zhang, Science 314, 1757 (2006)

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Exp (HgTe/CdTe)

K�Önig, Wiedmann, Brne, Roth, Buhmann, Molenkamp, Qi and Zhang, Science 318, 766 (2007)

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3D topological insulator

Kane & Mele Phys. Rev. Lett. 98, 106803 (2007)

• Insulator in bulk • Odd number of Dirac cone at surface • Z2 topological insulator

Ordinary insulator Trivial insulator with even # of Dirac cones

Z2 topological insulator

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3D topological insulator

Zhang , Cheng, Chen, Jia, Ma, He, Wang, Zhang,Dai, Fang, Xie, and Xue Phys. Rev. Lett., 103, 266803 (2007)

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3D TI ( Bi(1-x)Sbx )

Hsieh, Qian, Wray, Xia,Hor, Cava and M. Z. Hasan, Nature 452, 970 (2008)

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Hsieh, Xia, Qian, Wray, Dil, Meier,Osterwalder, Patthey, Checkelsky, Ong, Fedorov, Lin,

Bansil, Grauer, Hor, Cava and Hasan, Nature 460, 1101 (2009).

3D TI ( Bi2Sb3 )

28 Zhang , Liu, Qi, Dai, Fang, and Zhang, Nature Phys, 5, 438 (2009)

3D TI ( different alloys )

29 S. c. Zhang , KITP (2009)

Theory

30 S. c. Zhang , KITP (2009)

Theory

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Effective model Hamiltonian

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Conclusion

1. Many topological insulators ( Strong SOI) are found . 2. Gapless spectrum in the surface ( Dirac like or Majorana (SC)). 3. Modes on the surface are stable against perturbations and disorders. 4. Transport properties? 5. In the presence of any strain? 6. ……

Thanks for your attention

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