1. Univ Toronto, Nov 4, 2009 Topological Insulators J. G. Checkelsky, Y.S. Hor, D. Qu, Q. Zhang, R. J. Cava, N.P.O. Princeton University 1.Introduction 2.Angle resolved photoemission (Hasan) 3.Tentative transport signatures 4.Giant fingerprint signal 5.Insulator and Superconductor

2. cond. band valence band cond. band Top Bottom Conventional insulatorTopological insulator k s s Surface states are helical (spin locked to k) Large spin-orbit interactn Surface state has Dirac dispersion kxkx kyky Fu, Kane ’06 Zhang et al. ’06 Moore Balents ‘06 Xi, Hughes, Zhang ‘09 crystal s Surface states may cross gap A new class of insulators cond. band valence band cond. band valence band

3. 1. Time-reversal invariance prevents gap formation at crossing cond. band ? Violates TRI cond. band 2. Suppression of 2k F scattering Spinor product kills matrix element Large surface conductance? 2D Fermi surface Under time reversal (k↑)  (-k↓) s k Protection of helical states Kane, Mele, PRL ‘05 valence band

4. Helicity and large spin-orbit coupling Spin-orbit interaction and surface E field  effectv B = v  E in rest frame spin locked to B Rashba-like Hamiltonian Like LH and RH neutrinos in different universes v E B v E B spin aligned with B in rest frame of moving electron s s k k Helical, massless Dirac states with opposite chirality on opp. surfaces of crystal

5. A twist of the mass (gap) Doped polyacetylene (Su, Schrieffer, Heeger ‘79)  x  e /2 Domain wall (soliton) traps ½ charge Mobius strip 1.Gap-twist produces domain wall 2.Domain wall traps fractional charge 3.Topological (immune to disorder) Mobius strip like H HH

6. m(x) Jackiw Rebbi, PRD ‘76 Goldstone Wilczek, PRL ‘81 Callan Harvey, Nuc Phys B ‘85 Fradkin, Nuc Phys B ‘87 D. Kaplan, Phys Lett B ’92 Dirac modes on domain walls of mass field Chiral zero-energy mode Domain-wall fermion vacuum Topological insulator Chiral surface states? Dirac fermions as domain wall excitation z k QFT with background mass-twist field x   Callan-Harvey: Domain walls exchange chiral current to solve anomaly problm

7. Bi Sb Bi 1-x Sb x Fu Kane prediction of topological insulator z k Mass twist Mass twist traps surface Weyl fermions Fu, Kane, PRB ‘06 ARPES confirmation Hsieh, Hasan, Cava et al. Nature ‘08 Confirm 5 surf states in BiSb

8. 20 eV photons + - Angle-resolved photoemission spectroscopy (ARPES) velocity selector  E k || Intensity E quasiparticle peak

9. Hsieh, Hasan, Cava et al. Nature 2008 ARPES of surface states in Bi 1-x Sb x

10. ARPES results on Bi 2 Se 3 (Hasan group) Se defect chemistry difficult to control for small DOS Xia, Hasan et al. Nature Phys ‘09 Large gap ~ 300meV As grown, Fermi level in conduction band

11. Photoemission evidence for Topological Insulators Why spin polarized?  Rashba term on surface What prevents a gap?  Time Reversal Symmetry What is expected from transport? No 2 k F scattering SdH Surface QHE (like graphene except ¼) Weak anti-localization Hsieh, Hasan et al., Nature ‘09

12. Bi 2 Se 3 : Typical Transport Roughly spherical Fermi surface (period changes by ~ 30%) Metallic electron pocket with mobility ~ 500-5000 cm 2 /Vs Carrier density ~ 10 19 e - /cm 3

13. Quantum oscillations of Nernst in metallic Bi 2 Se 3 Major problem confrontg transport investigation As-grown xtals are always excellent conductors,  lies in conduction band (Se vacancies).  (1 K) ~ 0.1-0.5 m  cm, n ~ 1 x 10 18 cm -3 m* ~ 0.2, k F ~ 0.1 Å -1

14. Fall into the gap Decrease electron density Solution: Tune  by Ca doping cond. band valence band  electron doped hole doped target Hor et al., PRB ‘09 Checkelsky et al., arXiv/09

15. Resistivity vs. Temperature : In and out of the gap Onset of non- metallic behavior ~ 130 K SdH oscillations seen in both n-type and p-type samples Non-metallic samples show no discernable SdH Checkelsky et al., arXiv:0909.1840

17. Magnetoresistance of gapped Bi2Se3 Logarithmic anomaly Conductance fluctuations Giant, quasi-periodic, retraceable conductance fluctuations Checkelsky et al., arXiv:0909.1840

18. Magneto-fingerprints Giant amplitude (200-500 X too large) Retraceable (fingerprints) Spin degrees Involved in fluctuations Fluctuations retraceable Checkelsky et al., arXiv:0909.1840

19. Angular Dependence of R(H) profile Cont. For δG, 29% spin term For ln H, 39% spin term (~200 e 2 /h total) Theory predicts both to be ~ 1/2π (Lee & Ramakrishnan), (Hikami, Larkin, Nagaoka)

20. Conductance -- sum over Feynman paths Universal conductance fluctuations (UCF)  G = e 2 /h Universal Conductance Fluctuations in a coherent volume defined by thermal length L T = hD/kT At 1 K, L T ~ 1  m For large samples size L, H LTLT Stone, Lee, Fukuyama (PRB 1987) LTLT L = 2 mm “Central-limit theorem” UCF should be unobservable in a 2-mm crystal! Quantum diffusion our xtal

21. Taking typical 2D L T = 1 µm at 1 K, For systems size L > L T, consider (L/L T ) d systems of size L T, UCF suppressed as For AB oscillation, assuming 60 nm rings, N -1/2 ~ 10 -8 Size Scales

22. Quasi-periodic fluctuations vs T Fluctuation falls off quickly with temperature For UCF, expect slow power law decay ~T -1/4 or T -1/2 AB, AAS effect exponential in L T /P  Doesn’t match!

23. Non-Metallic Samples in High Field Fluctuation does not change character significantly in enhanced field

24. Next Approach: Micro Samples

25. Micro Samples Cont. Sample is gate-able SdH signal not seen in 10 nm thick metallic sample Exploring Callan-Harvey effect in a cleaved crystal x  

26. (b) (c) (a)

27. Desperately seeking Majorana bound state SC1 SC2  (x) Majorana bound state  = 0  =  Open  at  by Proximity effect Surface topological states Fu and Kane, PRL 08 bound state wf electron creation oper. Neutral fermion that is its own anti-particle Majorana oper. Gap “twist” traps Majorana

28. Cu Doping: Intercalation between Layers Hor et al., arXiv 0909.2890 Intercalation of Cu between layers Confirmed by c-axis lattice parameter increase and STM data Crystal quality checked by X- ray diffraction and electron diffraction

29. Diamagnetic Response at low T Typical M for type II: -1000 A / m From M(H), κ ~ 50 χ ~ -0.2 Impurity phases not SC above 1.8 K (Cu 2 Se, CuBi 3 Se 5, Cu 1.6 Bi 4.8 Se 8....) Small deviations from Se stoichiometry suppress SC

30. Cu Doping: Transport Properties Not complete resistive transition Up to 80% transition has been seen Carrier density relatively high

31. Upper Critical Field H C2 H C2 anisotropy moderate ξ c = 52 Å, ξ ab = 140 Å H C2 estimate by extrapolation Similar shape for H||ab

32. Ca Doping: Conclusions Ca doping can bring samples from n- type to p-type Non-metallic samples at threshold between the two reveal new transport properties G ~ ln(H) at low H δG ~ e 2 /h, quasiperiodic Hard to fit with mesoscopic interpretation No LL quantization seen up to 32 T Metallic nanoscale samples show no LL

33. Summary Doping of Bi 2 Se 3 creates surprising effects Ca doping: Quantum Corrections to Transport become strong Cu Doping: Superconductivity Next stage: 1.nm-thick gated, cleaved crystals 2.Proximity effect and Josephson current expt

34. END