Topological insulators

Transcription

1 Topological insulators Jaroslav Fabian Institute for Theoretical Physics University of Regensburg Stara Lesna, DFG SFB 689

2 what are topological insulators? gapped bulk states + conducting (gapless) edge (surface) states due to topology M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82, 345 (21)

3 :outline: mesoscopics for pedestrians topological insulators edge states in HgTe quantum wells magnetism of HgTe edge states B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, (212)

4 mesoscopics for pedestrians bulk states quantum point contacts current states

5 How much current per spin flows in a quantum channel?.. conductance quantum.. resistance quantum

6 conductance quantization: n-channel transport.. conductance quantization

7 conductance quantization B. J. van Wees, Phys. Rev. Lett. 6, 848 (1988). D. A. Wharam et al. J. Phys. C 21, L29 (1988). e n=1 (2 spins) GaAs/AlGaAs QPC lead resistance subtracted B. J. van Wees, Phys. Rev. Lett. 6, 848 (1988) picture from C. W. J. Beenakker and H. van Houten, Solid State Physics, 44, 1 (1991).

8 integer quantum Hall effect: topological edge states bulk states bulk Landau levels edge states edge states no backscattering

9 integer quantum Hall effect: topological edge states voltage probe

10 integer quantum Hall effect K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (198) Si MOSFET Original MOSFET von Klitzing s web site SdH IQHE GaAs/AlGaAs Cl picture from

11 topological nature of the integer quantum Hall effect 1 st Chern number Blount-Berry curvature Blount-Berry phase Bloch state D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 45 (1982) Q. Niu, D.J. Thouless, and Y.-S. Wu, Phys. Rev. B 31, 3372 (1985)

12 topological insulators another look at the quantum Hall edge states quantum Hall edge states come in spin pairs: time reversal symmetry is broken skipping orbits bulk orbits (Landau levels)

13 topological insulators edge states with time reversal preserved: spin-orbit coupling spin-orbit edge states come in spin pairs, but move opposite: time reversal symmetry is preserved edge states no backscattering!!! the edge states are topologically protected against TR scattering

14 emergence of spin-orbit fields space-inversion symmetry breaking

15 Time reversal points spin degeneracy preserved

16 Z 2 invariance stable to continuous change of band parameters even number of crossings odd number of crossings

17 (non-exotic) materials classes of topological insulators Graphene Kane and Mele, 25, spin quantum Hall effect 2d topological insulators Zhang and co., 26, HgTe quantum wells (see later) 3d topological insulators (Zhang and Co) BiSe, BiTe, BiSb yet to be experimentally confirmed first 3d experimental TI other special materials structures

18 Electronic structure of CdTe and HgTe CdTe HgTe normal band ordering 1.6 ev gap narrow-gap semiconductor inverted band structure negative band gap -.3 ev

19 HgTe/CdTe quantum wells CdTe CdTe CdTe CdTe HgTe HgTe 2d topological insulator

20 HgTe/CdTe quantum wells: TI states B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (26). CdTe HgTe 2d topological insulator trivial interface states CdTe B. A. Volkov and O. A. Pankratov, JETL Lett. 42, 178 (1985) M. I. Dyakonov and A. V. Khaetskii, JETP Lett. 33, 11 (1981).

21 experimental evidence of TI states mesoscopic transport tunable gate and width Konig et al. (Molenkapm group, Wurzburg), Science 318, 766 (27)

22 experimental evidence of TI states mesoscopic transport (d) R14,23 (Ω) G =.3 e 2 /h G = 2 e 2 /h V g -V thr (V) Konig et al. (Molenkapm group, Wurzburg), Science 318, 766 (27)

23 magnetism of the TI edge states in HgTe how the edge states evolve with B-field? CdTe CdTe HgTe B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (26).

24 magnetism of the TI edge states in HgTe how the edge states evolve with B-field? B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (26).

25 magnetism of the TI edge states in HgTe B = B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, (212) E [mev] (a) Fig. (b) Fig. (c) k [1 6 1/m]] k [1 6 1/m] (b) v k < (c) v k > bulk states v k < v k > y [nm] ρ [1 13 1/m 2 ] ρ [1 13 1/m 2 ] E [mev] (a) Fig. (b) k [1 6 1/m] Fig. (c) k [1 6 1/m] (b) (c) v k < v k < v k > v k > TI states ρ [1 14 1/m 2 ] y [nm] 1 ρ [1 14 1/m 2 ]

26 magnetism of the TI edge states in HgTe B =1 T B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, (212) E [mev] (a) Fig. (b) Fig. (c) (b) v k < v k < (c) v k > v k > QH edge states ρ [1 14 1/m 2 ] ρ [1 14 1/m 2 ] E [mev] (a) Fig. (b) Fig. (c) (b) v k < v k < (c) v k > v k > QH edge states ρ [1 14 1/m 2 ] ρ [1 14 1/m 2 ] k [1 9 1/m] y [nm] k [1 9 1/m] y [nm]

27 magnetism of the TI edge states in HgTe how the edge states evolve with B-field? B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, (212) 1 E [mev] 5-5 SQH QH B [T]

28 magnetism of the TI edge states in HgTe bulk magnetic susceptibility B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, (212) χ [1 17 J/(Tm) 2 ] T = 1 K T = 1 K T = 1 K μ = 2 mev /B [1/T] χ [1 17 J/(Tm) 2 ] T = 1 K T = 1 K T = 1 K μ = 2 mev /B [1/T]

29 Conclusion topological insulators are a new playground for (not only*) solid state physicists *X. Qi, E. Witten, and S. Zhang, Axion topological field theory of topological insulators, arxiv: