Topological insulators and the quantum anomalous Hall state. David Vanderbilt Rutgers University

Transcription

1 Topological insulators and the quantum anomalous Hall state David Vanderbilt Rutgers University

2 Outline Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z 2 ) 2D ( Quantum spin Hall ) insulator 3D topological insulators QAH strategies Heavy-atom adlayers on magnetic substrates Other ideas Summary

3 1D: BZ is really a loop Reciprocal space is really periodic Brillouin zone can be regarded as a loop E E π/a 0 k π/a k

4 2D, 3D: BZ is a closed manifold 2-D Crystal 3-D Crystal k z k y Brillouin zone k y Brillouin zone 0 k x 0 k x (3-torus)

5 Invisible information: Topology Without spin-orbit coupling Reds are topologically equivalent Greys are topologically equivalent Reds and greys are inequivalent

6 Topology of closed surfaces

7 Compare: Two contexts for topology Geometric topology Same if can be deformed without singularity (pinch, rip, etc.) Band structure topology Same if can be adiabatically perturbed without gap closure

8 Compare: Gauss-Bonnet Theorem Gaussian (geometrical) curvature Euler characteristic = 2(1-genus)

9 Compare: Chern theorem k y Brillouin zone 0 k x ( F Ω ) Berry curvature Chern number

10 Berry phase and curvature in the BZ u k k y φ k x 0 2π/a Bloch function

11 Chern Theorem Region B Region A Stokes applied to A: φ Stokes applied to B: Subtract: Chern theorem:

12 Chern theorem k y Brillouin zone 0 k x ( F Ω ) Berry curvature Chern number

13 Berry curvature in the Brillouin zone u k k y Ω k x 0 2π/a FS Anomalous Hall conductivity:

14 Ordinary Hall conductivity Measure σ xy in presence of B-field

15 Anomalous Hall conductivity (AHC) Ferromagnet Measure σ xy in absence of B-field

16 Berry curvature in the Brillouin zone u k k y Ω k x 0 2π/a FS Anomalous Hall conductivity:

17 Berry curvature in the Brillouin zone k y Ω Insulator k x u k 0 2π/a BZ = 2πC Quantum Anomalous Hall: Chern number or TKNN invariant

18 Quantum anomalous Hall effect Ferromagnetic insulator J E σ xy = e 2 /h Like integer quantum Hall, but no B ext

19 Outline Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z 2 ) 2D ( Quantum spin Hall ) insulator 3D topological insulators QAH strategies Heavy-atom adlayers on magnetic substrates Other ideas Summary

20 Quantum Hall effect B

21 Quantum Hall effect Density of states E F Energy σ xy σ xy = 2 σ xx

22 Hall effects: The big picture Induced by B-field Ferromagnetic sample Metal Ordinary Hall (1879) Anomalous Hall (1881) Topological insulator Quantum Hall (1980) Quantum Anomalous Hall?

23 QAH insulator k y Ω Insulator k x u k 0 2π/a BZ = 2πC Quantum Anomalous Hall: Chern number

24 Proof of principle: QAH insulators

25 Flux tubes in Haldane model (Real materials: spin-orbit interaction gives similar effects)

26 String Berry phases for normal band φ (k x ) k y k x k x

27 String Berry phases in QAH band φ (k x ) k y C = 1 k x k x

28 Bulk-boundary correspondence Topological phase A Topological phase B Something special at boundary

29 Bulk-boundary correspondence Country A Drive on left Hong Kong Country B Drive on right China Something special at boundary?

30 Bulk-boundary correspondence

31 Quantum Hall effect Semiclassical picture: Skipping orbits (edge states) Quantum picture: Chiral edge channels σ xy = e 2 /h exactly!

32 Quantum anomalous Hall effect QAH insulator: Chiral edge channels FM insulator (2D crystal) No B ext σ xy = e 2 /h exactly!

33 Edge states: 2D QAH insulator J Edge current k E E F C = +1 Conservation of charge chiral surface state

34 QAH insulators QAH insulator = Chern insulator Quantized Hall conductance even in the absence of macroscopic magnetic fields Quite possibly at room temperature Usefulness: Precision measurement? Dissipationless wires for microelectronics? Magnetoelectric coupling?

35 Can QAH insulators be found? Requirements Spontaneously broken TR (FM or FiM) Insulator Strong spin-orbit coupling (heavy atoms) Prefer gap > 0.2 ev (Q Hall at T room ) Proposals Magnetically doped TR-invariant TI s Magnetic adatoms on graphene 2D adlayer on a magnetic insulator

36 Magnetic doping: Claim for QAH Observed below ~1K

37 Hall effects: The big picture Induced by B-field Ferromagnetic sample Metal Ordinary Hall (1879) Anomalous Hall (1881) Topological insulator Quantum Quantum Hall Anomalous? Hall (1980) (2013)

38 Outline Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z 2 ) 2D ( Quantum spin Hall ) insulator 3D topological insulators QAH strategies Heavy-atom adlayers on magnetic substrates Other ideas Summary

39 2D Z 2 topological insulator (QSH) QSH = Quantum spin Hall k Z 2 insulator (conserved TR)

40 Z 2 Topological Insulator ( Quantum spin Hall ) Spin up, C = +1 Spin down, C = 1

41 Z 2 Topological Insulator ( Quantum spin Hall ) Spin up, C = +1 Spin down, C = 1 Then turn on spin-orbit coupling (SOC): Obeys T symmetry Total C = 0 Z 2 invariant is odd

42 Meaning of Z and Z 2 Z = group of integers under addition Z 2 = {0,1} under addition (mod 2) Or equivalently, Z 2 = { +, } under multiplication

43 2D Z 2 topological insulator (QSH) QSH = Quantum spin Hall k Z 2 insulator (conserved TR)

44 Edge states: 2D QAH insulator down up up k π/a π/a Z = N up N down = Invariant

45 Edge states: 2D TR-invariant insulator k π/a 0 π/a Z 2 = N cross (mod 2) = Invariant

46 3D TR-invariant topological insulator k π/a 0 π/a

47 3D TR-invariant topological insulator Figures from Hasan and Kane, RMP, 2010 (Adapted from Xia et al., 2008; Hsieh, Xia, Qian, Wray, et al., 2009a; and Xia, Qian, Hsieh, Wray, et al., 2009)

48 Bi 2 Se 3 and Bi 2 Te 3

49 Spin chirality of surface states

50 Outline Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z 2 ) 2D ( Quantum spin Hall ) insulator 3D topological insulators QAH strategies Heavy-atom adlayers on magnetic substrates Other ideas Summary

51 Our strategy

52 Our strategy Heavy atoms -Large spin-orbit Au, Hg, Tl, Pb, Bi Magnetic insulator -Breaks time reversal -FM or A-type AFM Advantages: -Spins align automatically -No doping -Large gap insulators -Large spin-orbit M Magnetic insulator MnTe, MnSe, EuS Disadvantages: -Preparing surfaces is difficult

53 Attempt I: One ML heavy atoms l l l 6 layers MnTe 1 ML heavy atom l Directly on Mn Polar surface Pb Mn Te Spins l (Bottom: 1 ML iodine) I

54 1 ML Tl on MnTe Surface Band Structure Energy (ev) E F K Γ M Γ K'

55 1 ML Tl on MnTe Surface Band Structure Zoomed In Energy (ev) E F C = 0 C = 1 C = 0 K Γ M Γ K'

56 Three observations Non-zero Chern numbers are common Provided E hop, E SO and E mag are at similar scale Bands are generically isolated in 2D No symmetry-induced degeneracies No accidental degeneracies However, if E hop is too large, there is no global gap

57 Attempt II: 1/3 ML heavy atoms 1/3 ML Bi on MnSe E E F (ev) C = 0 Γ M K Γ K'

58 Attempt II: 1/3 ML heavy atoms Result: l l l Bands tend to be flatter Global band gaps are easier to find But Chern numbers are typically all zero 1 ML 1/3 ML?

59 Attempt III: 2/3 ML honeycomb Top view Side view

60 2/3 ML of Pb on MnTe C=0 C=-1 C=-1 K Γ M Γ K' E F is in gap of 36 mev with C=-1 This is a QAH insulator! Even larger minimum direct gap (>0.2eV above)

61 Search for Chern Insulators Strained -2%

62 Our champion to date Bi/Br on MnSe: 142 mev gap E E F (ev) C = 0 C = +1 Γ M K Γ K'

63 Status l l First principles proof of principle Gaps can be as large as 0.14 ev Surfaces studied in current work are not experimentally realistic Probably unstable l Theory / experimental collaboration needed to find practical examples

64 Outline Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z 2 ) 2D ( Quantum spin Hall ) insulator 3D topological insulators QAH strategies Heavy-atom adlayers on magnetic substrates Other ideas Summary

65 Another idea c) GdN EuO SrO C = -1 Gap = 130 mev Kevin Garrity and D.V. Chern insulators from a magnetic rocksalt interface arxiv:

66 Another idea LaAlO 3 host Mn d(x 2 -y 2 ) Mn d(z 2 ) La 2 MnIrO 6 layer LaAlO 3 host C = 2 (unrelaxed) H. Zhang, H. Huang, K. Haule, and D.V. QAH phase in (001) double-perovskite monolayers via intersite spin-orbit coupling arxiv:

67 Summary Berry curvature and topology 2D quantum anomalous Hall (QAH) insulator TR-invariant insulators (Z 2 ) 2D ( Quantum spin Hall ) insulator 3D topological insulators QAH strategies Heavy-atom adlayers on magnetic substrates Other ideas