Introductory lecture on topological insulators. Reza Asgari

Transcription

1 Introductory lecture on topological insulators Reza Asgari Workshop on graphene and topological insulators, IPM Oct. 2011

2 Outlines -Introduction New phases of materials, Insulators -Theory quantum Hall effect, edge modes, topological invariance -Conclusion 2

3 Phases of matter -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 Qi and Zhang PRB 2008 Broken gauge symmetry 3

4 New phases of matter in quantum electron systems -Quantum Hall effect Nobel prize 85, 98 -Super fluidity and superconductors Nobel prize 72, 73, 87, Localization, disorder and quantum magnetism Nobel prize 70, 77, 94, 07 - Quantum phase transition, condensation Nobel prize 82, 01 Hasan and kane RMP,

5 New phases of matter in quantum electron systems -Quantum Hall effect QHE without magnetic field? Majorana fermions? - Super fluidity and superconductors Sc without BCS-like paradigm? New routes to HTCS? -Localization, disorder and quantum magnetism Spin-liquid, fractionalization -Quantum phase transition, condensation Q- phase transition, New universality classes 5

6 Insulators How many insulators do we know? 6

7 Insulators Band insulator Peierls insulator 7

8 Insulators Mott Insulator: large Coulomb repulsion electron can not move Anderson localization: Impurity scattering 8

9 Insulators What else? Topological Insulators! TI is a band insulator characterized by a topological number and has gapless excitations at its boundaries. 9

10 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

11 All electronic states with an energy gap topologically equivalent to the vacuum? 11

12 Hall effect ( Edwin Hall 1879) Metals feature no gap: current can flow 12

13 Hall effect at high magnetic field Electrons + high magnetic field Discrete energy levels 13

14 Insulators vs IQHE Hall current 1 εm = ωc( m + 2) 2 e E 0 j = σ ( = N ) E h x xy y Von Klitzing, Dorda & Pepper, Phys. Rev. Lett. 45, 494 (1980) Hasan and kane RMP, 2011 NOT insulator! 14

15 Bulk insulators: conduction through the edges 15

16 What is the difference between a QHE states and an ordinary insulator? Topology H ( k){ u ( k)} E { u ( k)} = Periodic conditions Sphere m m m First Chern number 1 1 = < > N 2 c d k k uk k uk 2π i N m = 1 Topological invariant 2 e c = 0 σ xy = Nc h c = 1 Ordinary insulator IQHE Z 2 Thouless, Kohmoto, Nightingale & den Nijs, Phys. Rev. Lett. 49, 405 (1982) 16

17 Quantum Hall effect without magnetic field Haldane model (1988) 17

18 Edge sates and the bulk boundary correspondence m( y ) > 0 m( y ) < 0 H ( q) = ν q σ + m( y) σ F z iqx x ψ q ( x, y) e e x y m( y') dy' / ν F de( q dq x x ) = v F 0 Jackiw, and Rebbi, Phys. Rev. D 13, 3398 (1976) n = N N Bulk boundary correspondence R L 18

19 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 Fu and kane PRL, 2009 Z 2 Topological Insulator. 19

20 quantum spin Hall effect J J = σ E s x x xy y Bernevig, and Zhang, Phys. Rev. Lett (2006). 20

21 2D topological insulator (HgTe/CdTe) Bernevig, Hughes, and Zhang, Science 314, 1757 (2006) 21

22 Band structures (HgTe/CdTe) Without SOI With SOI Effective edge Hamiltonian H edge = z Ak σ y A 3.6 ev A A ν = = m / s Bernevig, Hughes, and Zhang, Science 314, 1757 (2006) 22

23 Exp (HgTe/CdTe) KÖnig, Wiedmann, Brne, Roth, Buhmann, Molenkamp, Qi and Zhang, Science 318, 766 (2007) 23

24 3D topological insulator 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 Kane & Mele Phys. Rev. Lett. 98, (2007) 24

25 3D topological insulator Zhang, Cheng, Chen, Jia, Ma, He, Wang, Zhang,Dai, Fang, Xie, and Xue Phys. Rev. Lett., 103, (2007) 25

26 3D TI ( Bi(1-x)Sbx ) Hsieh, Qian, Wray, Xia,Hor, Cava and M. Z. Hasan, Nature 452, 970 (2008) 26

27 3D TI ( Bi2Sb3 ) Hsieh, Xia, Qian, Wray, Dil, Meier,Osterwalder, Patthey, Checkelsky, Ong, Fedorov, Lin, Bansil, Grauer, Hor, Cava and Hasan, Nature 460, 1101 (2009). 27

28 3D TI ( different alloys ) Zhang, Liu, Qi, Dai, Fang, and Zhang, Nature Phys, 5, 438 (2009) 28

29 Theory S. c. Zhang, KITP (2009) 29

30 Theory S. c. Zhang, KITP (2009) 30

31 Effective model Hamiltonian 31

32 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. 32

33 Thanks for your attention 33