Hartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany

Transcription

1 Hartmut Buhmann Physikalisches Institut, EP3 Universität Würzburg Germany

2 Outline Insulators and Topological Insulators HgTe quantum well structures Two-Dimensional TI Quantum Spin Hall Effect experimental observations: quantized conductivity non-locality spin polarization Three-Dimensional TI Quantum Hall Effect

3 The Insulating States Topologically Generalized C.L.Kane and E.J.Mele, PRL 95, (25) C.L.Kane and E.J.Mele, PRL 95, (25) A.Bernevig and S.-C. Zhang, PRL 96, 1682 (26) C.L.Kane and E.J. Mele, Science 314, 1692 (26)

4 The Usual Boring Insulating State Characterized by energy gap: absence of low energy electronic excitations Covalent Insulator e.g. intrinsic semiconductor Atomic Insulator e.g. solid Ar The vacuum electron E gap ~ 1 ev 4s E gap ~ 1 ev 3p Dirac Vacuum E gap = 2 m e c 2 ~ 1 6 ev Silicon positron ~ hole

5 The Insulating State Topologically Generalized C.L.Kane and E.J.Mele, PRL 95, (25) C.L.Kane and E.J.Mele, PRL 95, (25) A.Bernevig and S.-C. Zhang, PRL 96, 1682 (26) C.L.Kane and E.J. Mele, Science 314, 1692 (26)

6 The Integer Quantum Hall State 2D Cyclotron Motion, Landau Levels E E gap c Energy gap, but NOT an insulator

7 Edge States Quantized Hall conductivity : bulk insulating J E y xy x 2 xy n e h Integer accurate to 1-9

8 Perfect Conductor magnetic field 1d dispersion E k F k F k The QHE state spatially separates the right and left moving states (spinless 1D liquid) suppressed backscattering first example of a new Topological Insulator

9 Topological Insulator

10 Topological Band Theory The distinction between a conventional insulator and the quantum Hall state is a topological property of the manifold of occupied states H( k) : Brillouin zone (torus) Bloch Hamiltonians with energy gap Classified by Chern (or TKNN) integer topological invariant (Thouless et al, 1982) 1 2 n d u u 2i BZ k k k ( k) ( k) u(k) = Bloch wavefunction Insulator : n = IQHE state : xy = n e 2 /h

11 Geometry and Topology Gauss and Bonnet topological 1 2 S KdA 2(1 g) geometric Example: Mug to Torus S: surface K -1 : product of the two radii of curvatures da: area element g: number of handles

12 Edge States The TKNN invariant can only change at a phase transition where the energy gap goes to zero Vacuum Vacuum n = n = 3 n = Dn = # Chiral Edge Modes This approach can actually be generalized to a spinfull QHE at zero magnetic field: the Quantum Spin Hall Effect

13 The Insulating State Topologically Generalized C.L.Kane and E.J.Mele, PRL 95, (25) C.L.Kane and E.J.Mele, PRL 95, (25) A.Bernevig and S.-C. Zhang, PRL 96, 1682 (26) C.L.Kane and E.J. Mele, Science 314, 1692 (26)

14 A two-dimensional topological insulator B quantum Hall effect topological invariant broken time reversal symmetry (TRS)

15 A two-dimensional topological insulator B

16 A two-dimensional topological insulator B helical edge states invariant under time reversal

17 Protected Edge States suppressed backscattering:

18 Protected Edge States suppressed backscattering: spin rotation: Dj = 2 Interference is destructive

19 Quantum Spin Hall Effect Stability of Helical Edge States: x 2e h G QSHE 2 x backscattering is only possible by time reversal symmetry breaking processes (for example external magnetic fields) or if more states are present k s k, s, 2 1, x x x x

20 Topological Insulator : A New B= Phase

21 Topological Insulator : A New B= Phase There are 2 classes of 2D time reversal invariant band structures Z 2 topological invariant: n =,1 Edge States for <k</a n= : Conventional Insulator n=1 : Topological Insulator Kramers degenerate at time reversal invariant momenta k* = k* + G k*= k*=/a k*= k*=/a even number of bands crossing Fermi energy odd number of bands crossing Fermi energy

22 QSHE in Graphene Graphene edge states gap C.L.Kane and E.J.Mele, PRL 95, (25) Graphene spin-orbit coupling strength is too weak gap only about 3 mev. not accessible in experiments

23 QSHE in HgTe Helical edge states for inverted HgTe QW E H1 E1 k B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (26)

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25 HgTe band structure semi-metal 1 5 E (mev) -5 D 8 6 E g k (.1 ) 7 semiconductor D.J. Chadi et al. PRB, 358 (1972) fundamental energy gap 6 8 E E 3 mev

26 CdTe and HgTe CdTe HgTe H = H 1 + H D + H mv + H so

27 E (mev) E (mev) HgTe-Quantum Wells QW Barrier 1 HgTe Hg.32 Cd.68 Te VBO k (.1 ) 7 7 k (.1 ) VBO = 57 mev

28 HgTe-Quantum Wells Typ-III QW d 6 HgCdTe HH1 E1 HgTe HgCdTe 8 inverted normal band structure

29 Bandstructure 6 HgTe HgCdTe 6 Zero Gap for 6.3 nm QW structures HgTe HgCdTe 8 8

30 E [mev] E [mev] Dirac Band Structure 8 E2 d c 4 E1 H1-4 H2 H3-8 H d [nm] E1 H1 in-plane direction (k x,k y )=(1,) (k x,k y )=(1,1) -1 H k [nm -1 ]

31 xx [ xx [ Zero Gap B = 1 T 8k 7k 7 6 B = 5 T 3k xy [e²/h] k 5k 4k 3k 2k xy [e²/h] k 1k 1k -1 -,5,,5 1, V G - V Dirac [V] V G - V Dirac [V] B. Büttner et al., Nature Physics 7, 418 (211)

32 LL-Dispersion Kane model effective model LLchart Parameters: M ~ (-.35 mev) D = mev nm² G = mev nm² A = mev nm E(VG) conversion taken from appropriate Kane model B. Büttner et al., Nature Physics 7, 418 (211)

33 Finite Mobility 1.4 xx 8k 6k xx,min =.36 e²/h 1.2 measurement theory 1. 4k 2k DV G xx in e 2 /h V G -V Dirac [V] T [K] B. Büttner et al., Nature Physics 7, 418 (211)

34 Bandstructure 6 HgTe HgCdTe 6 HgTe HgCdTe 8 8 B.A Bernevig et al., Science 314, 1757 (26)

35 Effective Tight Binding Model square lattice with 4 orbitals per site s,, s,, ( p ip ),, ( p ip ), x y x y nearest neighbor hopping integral basis: E1, H1, E1, H1 hk ( ) Heff ( kx, ky) * h ( k)

36 Effective Tight Binding Model square lattice with 4 orbitals per site s,, s,, ( p ip ),, ( p ip ), x y x y nearest neighbor hopping integral basis: E1, H1, E1, H1 hk ( ) Heff ( kx, ky) * h ( k) at the -point k A k h( k) A relativistic Dirac equation 2 M ( k) k Asin k sin x i kx sin kx isin kx M ( k) k 2 M ( k) Ak x ikx x ikx M ( k) HgTe M normalqw for M inverted QW with tunable parameter M (band-gap)

37 Mass domain wall inverted band structure E H1 E1 k states localized on the domain wall which disperse along the x-direction y M> M< y M> x M< M x M> helical edge states

38 Quantum Spin Hall Effect M > M < normal insulator QSHE bulk entire sample insulating bulk insulating

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40 x G 2e h 2 x

41 Experiment e e k k

42 Small Samples L W (L x W) mm 2. x 1. mm 1. x 1. mm 1. x.5 mm

43 QSHE Size Dependence 1 6 (1 x 1) mm 2 non-inverted R xx / (2 x 1) mm 2 (1 x 1) mm 2 G = 2 e 2 /h 1 3 (1 x.5) mm (V Gate - V thr ) / V König et al., Science 318, 766 (27)

44 R 2-8, 9-11 (k) QSHE Temperature Dependence Q2367 Microhallbar 1 x 1 mm 2 I(2-8), U(9-11); B= T; T= variabel; T = 1.8 K T = 2.5 K T = 4.2 K T = 6.4 K T = 9.8 K T = 19 K T = 28 K T = 48 K U G (V)

45 R (kohm) QSHE Excitation Voltage Dependence mV 14mV 12mV 1mV 8mV 6mV 4mV 2mV U g (V)

46 Conductance Quantization in 4-terminal geometry!? 2 18 V 16 R xx / k G = 2 e 2 /h I (V Gate - V thr ) / V

47 Small Samples

48 Role of Ohmic Contacts n-type metallic contact areas and leads areas where scattering takes place spin relaxation

49 Multi-Terminal Probe T t t I e G h I e G h m m m m generally ) ( e h n R t 3 exp 4 2 t t R R h e G t 2,exp 4 2 Landauer-Büttiker Formalism normal conducting contacts no QSHE ji j ij i ii i i T R M h e I m m 2

50 QSHE in inverted HgTe-QWs 4 35 G = 2/3 e 2 /h V I 4 R xx (k V 3 I G = 2 e 2 /h (2 x 1) mm 2 (1 x.5) mm (V Gate - V thr ) (V) R R 2t 4t 3

51 Non-locality mm 2 mm 6 5 4

52 I (na) Rnl (k) Non-locality I: 1-4 V nl :2-3 V V 3 I mk U Gate [ V ] A. Roth, HB et al., Science 325,295 (29)

53 R (k) Non-locality I: 1-3 V nl : LB: 8.6 k 12 V 1 8 G=3 e 2 /h 4-terminal V* (V) A. Roth, HB et al., Science 325,295 (29)

54 Non-locality 1 2 I: 1-4 V nl :2-3 V LB: 6.4 k 5 G=4 e 2 /h R 14,23 / k V gate / V A. Roth, HB et al., Science 325,295 (29)

55 R nl (k) Non-locality V LB: 4.3 k , -5,5-5, -4,5-4, -3,5-3, V (V) A. Roth, HB et al., Science 325,295 (29)

56 Back Scattering potential fluctuations introduce areas of normal metallic (n- or p-) conductance in which back scattering becomes possible QSHE The potential landscape is modified by gate (density) sweeps!

57 Back Scattering area of raised or lowered chem. Pot. the probalility for back scatterring increases strength length additional ohmic contact transition from a two to a three terminal device

58 Additional Scattering Potential Transition from scattering strength 2 e 2 /h 3/2 e 2 /h A. Roth, HB et al., Science 325,295 (29)

59 Additional Scattering Potential R 14,23 / k V gate / V A. Roth, HB et al., Science 325,295 (29)

60 Rnl (kw) Potential Fluctuations V LB: 4.3 k one additional contact: between 2 and k 4 3 V x A. Roth, HB et al., Science 325,295 (29)

61 V Potential Fluctuations different gate sweep direction V x LB: 22.1 k R nl / k 15 1 LB: 12.9 k V g / V Hysteresis effects due to charging of trap states at the SC-insulator interface J. Hinz, HB et al., Semicond. Sci. Technol. 21 (26) 51 56

62 metal insulator QW HgTe Quantum Well Structures layer structure impurity states discharging charging gate insulator Au 1 nm Si N /SiO cap layer doping layer barrier quantum well barrier doping layer buffer substrate 25 nm HgCdTe x =.7 1 nm HgCdTe x =.7 9 nm HgCdTe with I 1 nm HgCdTe x = nm HgTe 1 nm HgCdTe x =.7 9 nm HgCdTe with I 1 nm HgCdTe x =.7 25 nm CdTe CdZnTe(1) hysteresis effects: J. Hinz et al., Semicond. Sci. Technol. 21 (26) 51 56

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64 Spin Polarizer

65 Spin Polarizer 5% 1 : 4 5%

66 Spin Polarizer 1% 1 % 5% 1%

67 Spin Injector 1% spin injection 1 % even with backscattering!

68 Spin Detection 1% Dm spin detector 1 %

69 QSHE SHE QSHE Spin Hall Effect as creator and analyzer of spin polarized electrons SHE -1

70 Split Gate H-Bar

71 QSHE Spin-Detector x n-type injector.8.6 5x QSHI-type detector detector sweep QSHE.2 p-type injector SHE V detector-gate / V 4 QSHI-type detector R 34,12 / k V injector-gate / V injector sweep

72 n-regime Q2198 p-regime E / mev n = cm -2 n H1 dl = cm -2 E1 H k / nm -1 E F E / mev p = cm -2 n dl = cm -2 H1 Dk E1 H k / nm -1 E F eex j s, y pf pf Dk 16m J.Sinova et al., Phys. Rev. Lett. 92, (24) Intrinsic Spin Hall Effekt Nature Physics 6, 448 (21)

73 QSHE Spin-Injector x n-type detector.8.6 5x QSHI-type detector injector sweep QSHE.2 p-type detector SHE QSHI-type injector R 12,34 / k V detector-gate / V detector sweep

74 n-type 1 detector sweep 1 V 8 8 V R nonloc / 6 4 R nonloc / U gate / V p-type U gate / V R nonloc / 6 4 R nonloc / U gate / V U gate / V

75 Spin-Current Switch I 2 I 1

76 Spin-Current Switch I 2 I 1

77 Spin-Current Switch I 2 I 1 QSHE

78 Spin-Current Switch I 2 I 1 ballistic injection

79 I [na] ballstic 5, 9/1 11/12 1/2 4,5 4, 3,5 B I A C D F I E 3, 8/7 6/5 4/3 2,5 2, 1,5 QSHE 1, -3, -2,5-2, -1,5-1, -,5, U (Gate ACDF) [V]

80 I [na] I [na] /1 11/12 1/ B I A C D F I E 1 5, 9 4,5-3, -2,5-2, -1,5-1, -,5, U (Gate ACDF) [V] 8/7 6/5 4/3 4, 3,5 3, 2,5 2, 1,5 1, -3, -2,5-2, -1,5-1, -,5, U (Gate ACDF) [V]

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82

83

84 ¼ Graphene

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86 E, mev Bulk HgTe 2 1 E F fully strained 7 nm HgTe on CdTe substrate ,3 -,2 -,1,,1,2,3 k, nm -1 band structure - gap openning - two Dirac cones on different surfaces Qy*1(rlu) x-ray diffraction (115 reflex) Q2424-map_1.xrdml Qx*1(rlu)

87 R xx in Ohm R xy in Ohm n= n=3 n= R xx (SdH) R xy (Hall) B in Tesla 4 2

88 R xx [k] xy [e 2 /h] B [T] R xy [k] B [T] Brüne et al., Phys. Rev. Lett. 16, (211)

89 R xx in Ohm DOS Density of states for two independent surfaces with slightly different density 1 8 n n cm 11-2 cm B in Tesla

90 Brüne et al., Phys. Rev. Lett. 16, (211)

91 Summary the QSH effect which consists of an insulating bulk and two counter propagating spin polarized edge channels (Kramers doublet) quantized edge channel transport the QSH effect can be used as an effective spin injector and spin detector with 1 % spin polarization properties Strained HgTe bulk exhibits two-dimensional surface states

92 Acknowledgements Quantum Transport Group (Würzburg, ) C. Brüne C. Ames P. Leubner Ex-QT: C.R. Becker T. Beringer M. Lebrecht J. Schneider S. Wiedmann N. Eikenberg R. Rommel F. Gerhard B. Krefft A. Roth B. Büttner R. Pfeuffer L. Maier M. Mühlbauer A. Friedel Stanford University S.-C. Zhang X.L. Qi J. Maciejko T. L. Hughes M. König C. Thienel H. Thierschmann R. Schaller J. Mutterer A. Astakhova CX Liu Lehrstuhl für Experimentelle Physik 3: L.W. Molenkamp Collaborations: Texas A&M University J. Sinova Univ. Würzburg Inst. f. Theoretische Physik E.M. Hankiewicz B. Trautzettel

93 Quantum Spin Hall Effekt Science 318, 766 (27) The Quantum Spin Hall Effect: Theory and Experiment J. Phys. Soc. Jap. Vol. 77, 317 (28) Nonlocal edge state transport in the quantum spin Hall state Science 325, 294 (29) Intrinsic Spin Hall Effekt Nature Physics 6, 448 (21) Quantum Hall Effect from the Topological Surface States of Strained Bulk HgTe Phys. Rev. Lett. 16, (211)