Hartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany

Transcription

1 Hartmut Buhmann Physikalisches Institut, EP3 Universität Würzburg Germany

2 Part I and II Insulators and Topological Insulators HgTe crystal structure Part III quantum wells Two-Dimensional TI Quantum Spin Hall Effect quantized conductivity non-locality spin polarization strained layers Three-Dimensional TI quantum Hall effect super currents

3 Material classification in solid state physics E conduction band E F valence band insulator semiconductor (insulator for T 0 K) halfmetal metal

4 Insulating States C.L.Kane and E.J.Mele, PRL 95, (2005) C.L.Kane and E.J.Mele, PRL 95, (2005) A.Bernevig and S.-C. Zhang, PRL 96, (2006) C.L.Kane and E.J. Mele, Science 314, 1692 (2006)

5 Insulating States C.L.Kane and E.J.Mele, PRL 95, (2005) C.L.Kane and E.J.Mele, PRL 95, (2005) A.Bernevig and S.-C. Zhang, PRL 96, (2006) C.L.Kane and E.J. Mele, Science 314, 1692 (2006)

6 The Usual 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 ~ 10 ev 3p Dirac Vacuum E gap = 2 m e c 2 ~ 10 6 ev Silicon positron ~ hole

7 Insulating States C.L.Kane and E.J.Mele, PRL 95, (2005) C.L.Kane and E.J.Mele, PRL 95, (2005) A.Bernevig and S.-C. Zhang, PRL 96, (2006) C.L.Kane and E.J. Mele, Science 314, 1692 (2006)

8 Quantum Hall Effect Nobel Prize K. von Klitzing 1985 xy B n e s 1 i h 2 e xy Hall resistance semiconductor 2DEG xx zero longitudinal resistance

9 The Integer Quantum Hall State 2D Cyclotron Motion, Landau Levels E E gap c Energy gap bulk insulator

10 Edge States

11 Edge States skipping orbits chiral edge modes / edge states

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

13 Topological Insulator 1d dispersion E -k F k F k The overlap of the edge channels destroys the QHE state! Back scattering becomes possible

14 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 1 2 n d u u 2i BZ k k k ( k) ( k) u(k) = Bloch wavefunction Classified by Chern (or TKNN) integer topological invariant (Thouless et al, 1982) (Thouless, Kohmoto, Nightingale & Den Nijs) Insulator : n = 0 IQHE state : n > 0 (integer) s xy = n e 2 /h

15 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

16 Geometry and Topology The transformation for one topological state to another is only possible by a phase transition

17 Edge States The TKNN invariant can only change with a phase transition where the energy gap goes to zero Vacuum Vacuum n = 0 n = 3 n = 0 Dn = # Chiral Edge Modes

18 Edge Channels xy B n e s 1 i h 2 e xx bulk insulating n: number of occupied edge modes per Landau level zero longitudinal resistance Quantized Hall conductivity : 2 s xy n e h Integer accurate to 10-9

19 Quantum Hall Effect 3 Edge Channel Picture D 0 D 0 - D 0 I +

20 The Insulating State Topologically Generalized C.L.Kane and E.J.Mele, PRL 95, (2005) C.L.Kane and E.J.Mele, PRL 95, (2005) A.Bernevig and S.-C. Zhang, PRL 96, (2006) C.L.Kane and E.J. Mele, Science 314, 1692 (2006)

21 A two-dimensional topological insulator B quantum Hall effect state with broken time reversal symmetry (TRS)

22 A two-dimensional topological insulator B What happens under time reversal?

23 A two-dimensional topological insulator B the time reversed state has opposite momentum and spin the effective magnetic field is reversed

24 A two-dimensional topological insulator B = 0 Quantum Spin Hall State helical edge states invariant under time reversal consisting of two counter propagating spin polarized edge channels and an insulating bulk the required magnetic field can be provided by the intrinsic spin orbit interaction

25 Protected Edge States suppressed backscattering: deflection into the bulk is not possible, because of the insulating bulk state

26 Protected Edge States suppressed backscattering: a deflection of 180 has to reverse momentum and spin spin reversion is possible in two different ways: acquired phase change: destructive interference Dj = 2

27 Stability of Helical Edge States Kramers Pairs, 1 k s k, s, 1 x 2e h 2 G QSHE 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

28

29 Topological Insulator : A New B=0 Phase There are 2 classes of 2D time reversal invariant band structures Z 2 topological invariant: = 0,1 Edge States for 0<k</a =0 : Conventional Insulator =1 : Topological Insulator Kramers degeneracy at points with time reversal invariant momenta k* = k* + G k*=0 k*=/a k*=0 k*=/a even number of bands crossing Fermi energy odd number of bands crossing Fermi energy

30 3D Topological Insulator There are 4 surface Dirac Points due to Kramers degeneracy k y L 4 L 3 k x L 1 L 2 2D Dirac Point Surface Brillouin Zone 0 = 0 : Weak Topological Insulator E E k=l a k=l b OR k=l a k=l b How do the Dirac points connect? Determined by 4 bulk Z 2 topological invariants 0 ; ( ) k y Related to layered 2D QSHI ; ( ) ~ Miller indices Fermi surface encloses even number of Dirac points k x 0 = 1 : Strong Topological Insulator Fermi circle encloses odd number of Dirac points Topological Metal : 1/4 graphene Berry s phase Robust to disorder: impossible to localize k y k x E F

31 3D Topological Insulator Bi 1-x Sb x Theory: Predict Bi 1-x Sb x is a topological insulator by exploiting inversion symmetry of pure Bi, Sb (Fu,Kane PRL 07) Experiment: ARPES (Hsieh et al. Nature 08) Bi 1-x Sb x is a Strong Topological Insulator 0 ;( 1, 2, 3 ) = 1;(111) 5 surface state bands cross E F between G and M Bi 2 Se 3 ARPES Experiment : Y. Xia et al., Nature Phys. (2009). Band Theory : H. Zhang et. al, Nature Phys. (2009). 0;(1,2,3) = 1;(000) : Band inversion at G Energy gap: D ~.3 ev : A room temperature topological insulator Simple surface state structure : Similar to graphene, except only a single Dirac point E F

32 QSHE in Graphene Graphene edge states gap C.L.Kane and E.J.Mele, PRL 95, (2005) Graphene spin-orbit coupling strength is too weak gap only about 30 ev. not accessible in experiments

33 QSHE in HgTe Helical edge states for inverted HgTe QW E H1 DEg = mev (typically) E1 0 k B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)

34 Effective Tight Binding Model preserves TRS TKKN invariant zero Hall conductivity quantized spin Hall conductivity n n n n n s 2 n

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 H ( k, k ) eff x y hk ( ) 0 * 0 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 H ( k, k ) eff x y hk ( ) 0 * 0 h ( k) h( k) A 2 M ( k) k Asin k sin x i kx sin kx isin kx M ( k) k 2 at the G-point k 0 A k M ( k) Ak x ik x x ik x M ( k) relativistic Dirac equation M 0 normal QW for M 0 inverted QW with tunable parameter M (band-gap)

37 Bandstructure HgTe B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) E 4.0nm 6.2 nm 7.0 nm k E1 H1 normal gap H1 E1 inverted gap

38 Bandstructure HgTe edge states for M > 0 M < 0 B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006) states with linear dispersion

39 Quantum Spin Hall Effect M > 0 M < 0 normal insulator QSHE bulk entire sample insulating bulk insulating

40 Mass domain wall inverted band structure E H1 E1 0 k states localized on the domain wall which disperse along the x-direction y M>0 M<0 y M>0 x M<0 0 M x M>0 helical edge states

41 Summary

42

43 HgTe band structure 1000 semi-metal or topological metal 500 E (mev) D 8 6 E g k (0.01 ) 7 band inversion D.J. Chadi et al. PRB, 3058 (1972) 6 8 E G E G 300 mev

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

45 HgTe Semi-metal with bulk band inversion 1000 E (mev) k (0.01 ) How to open the bulk band gap? 1. quantum confinement 2D TI 2. strain 3D TI

46

47 Quantum Well Structures E E g2 E g1 quantized energy levels in z-direction z (growth direction) x y quasi free electron gas in the xy-plane

48 MBE Quantum Well Growth by Molecular Beam Epitaxy QW

49 MBE Cluster at Würzburg

50 MBE Integrated in UHV transport system with 6 chambers Riber 3200 MBE Horizontal system 8 cell ports Maximum substrate dimension 2 wavers Pumped by a kryo pump, three nitrogen coldshields and a Tisublimation pump

51 HgTe Quantum Wells MBE-Growth 6.0 Bandgap vs. lattice constant (at room temperature in zinc blende structure) Bandgap energy (ev) lattice constant a [Å] 0 CT-CREW 1999

52 HgTe Quantum Wells free electron gas in the QW by donor doping of the barriers

53 HgTe Quantum Well Structures layer structure Carrier densities: n s = 1 x x cm -2 Carrier mobilities: = 1 x x 10 6 cm 2 /Vs gate insulator cap layer doping layer barrier quantum well barrier doping layer buffer substrate Au 100 nm Si N /SiO nm HgCdTe x = nm HgCdTe x = nm HgCdTe with I 10 nm HgCdTe x = nm HgTe 10 nm HgCdTe x = nm HgCdTe with I 10 nm HgCdTe x = nm CdTe CdZnTe(001) symmetric or asymmetric doping

54 E / mev n to p Transitions H1 n max = 1.35 x cm H2 p max = 3.2 x cm E ,0 0,1 0,2 0,3 0,4 0,5 k / nm -1 gate insulator cap layer doping layer barrier quantum well barrier doping layer buffer substrate Au 100 nm Si3N 4/SiO2 25 nm CdTe 10 nm HgCdTe x = nm HgCdTe with I 10 nm HgCdTe x = nm HgTe 10 nm HgCdTe x = nm HgCdTe with I 10 nm HgCdTe x = nm CdTe CdZnTe(001) R xy / Q nm QW T = 1.5 K 0V V Gate B / T -2V +5V +5,0 V +4,5 V +4,0 V +3,5 V +3,0 V +2,5 V +2,0 V +1,5 V +1,0 V +0,5 V 0 V -2,0 V

55 E (mev) E (mev) HgTe-Quantum Wells QW Barrier 1000 HgTe Hg 0.32 Cd 0.68 Te VBO k (0.01 ) 7 7 k (0.01 )

56 HgTe-Quantum Wells Typ-III QW d E (mev) k (0.01 ) G 6 HgCdTe HgTe G 8 HH1 E1 LH1 inverted band structure HgCdTe

57 Bandstructure G 6 HgTe HgCdTe G 6 HgTe HgCdTe G 8 G 8 B.A Bernevig et al., Science 314, 1757 (2006) Zero Gap for 6.3 nm QW structures B. Büttner et al., Nature Physics 7, 418 (2011)

58 E [mev] E [mev] Dirac Band Structure 80 E2 d c 40 E1 0 H1-40 H2 H3-80 H d [nm] E1 H1 in-plane direction (k x,k y )=(1,0) (k x,k y )=(1,1) -100 H k [nm -1 ]

59 xx [ xx [ Zero Gap B = 1 T 80k 70k 7 6 B = 5 T 3k s xy [e²/h] k 50k 40k 30k 20k s xy [e²/h] k 1k 0 10k -1-0,5 0,0 0,5 1,0 V G - V Dirac [V] V G - V Dirac [V] 0 B. Büttner et al., Nature Physics 7, 418 (2011)

60 LL-Dispersion Kane model effective model LLchart Parameters: M ~ 0 ( 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 (2011)

61

62 x G 2e h 2 x

63 Experiment e e k k

64 Small Samples L W (L x W) m 2.0 x 1.0 m 1.0 x 1.0 m 1.0 x 0.5 m

65 QSHE Size Dependence 10 6 (1 x 1) m 2 non-inverted R xx / (2 x 1) m 2 (1 x 1) m 2 G = 2 e 2 /h 10 3 (1 x 0.5) m (V Gate - V thr ) / V König et al., Science 318, 766 (2007)

66 m 2 m 6 5 4

67 Conductance Quantization in 4-terminal geometry!? 20 V R xx / k G = 2 e 2 /h I (V Gate - V thr ) / V

68 Small Samples

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

70 Multi-Terminal Probe T t t I e G h I e G h 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 2

71 QSHE in inverted HgTe-QWs G = 2/3 e 2 /h V I V R xx (k I 5 4 G = 2 e 2 /h 10 5 (1 x 0.5) m (V Gate - V thr ) (V) (2 x 1) m 2 R R 2t 4t 3

72 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 (2009)

73 R (k) Non-locality I: 1-3 V nl : LB: 8.6 k 12 V 10 8 G=3 e 2 /h 4-terminal V* (V) A. Roth, HB et al., Science 325,295 (2009)

74 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 (2009)

75 R nl (k) Non-locality V LB: 4.3 k ,0-5,5-5,0-4,5-4,0-3,5-3,0 V (V) A. Roth, HB et al., Science 325,295 (2009)

76 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!

77 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

78 Additional Scattering Potential Transition from scattering strength 2 e 2 /h 3/2 e 2 /h A. Roth, HB et al., Science 325,295 (2009)

79 Additional Scattering Potential R 14,23 / k V gate / V A. Roth, HB et al., Science 325,295 (2009)

80 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 (2009)

81 V Potential Fluctuations different gate sweep direction V x LB: 22.1 k R nl / k 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 (2006)

82 metal insulator QW HgTe Quantum Well Structures layer structure impurity states discharging charging gate insulator Au 100 nm Si N /SiO cap layer doping layer barrier quantum well barrier doping layer buffer substrate 25 nm HgCdTe x = nm HgCdTe x = nm HgCdTe with I 10 nm HgCdTe x = nm HgTe 10 nm HgCdTe x = nm HgCdTe with I 10 nm HgCdTe x = nm CdTe CdZnTe(001) hysteresis effects: J. Hinz et al., Semicond. Sci. Technol. 21 (2006)

83

84 Spin Polarizer 2D topological insulator attached to four metallic contacts/leads voltage contacts electron drain electron source

85 Spin Polarizer 2D topological insulator attached to four metallic contacts/leads 50% 1 : 4 50% electron drain electron source

86 Spin Polarizer 2D topological insulator attached to four metallic contacts/leads electron drain 100% 100 % 50% I = 0 electron drain 100% electron source

87 Spin Injector 2D topological insulator attached to four metallic contacts/leads 100% spin injection 100 % even with backscattering!

88 Spin Detection 100% D spin detection 100 %

89 Spin Manipulation D spin injection and spin manipulation by SO induced spin rotation (Rashba effect)

90 QSHE SHE

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

92 Rashba Effect structural inversion asymmetry (SIA) Y.A. Bychkov and E.I. Rashba, JETP Lett. 39, 78 (1984); J. Phys. C 17, 6039 (1984): Rashba-Term: H R R s k s k x y y x B eff p E 2 2 k E Ei k * 2m (for electrons) 2 2 k E Ei k * 2m 3 (for holes)

93 Origin of the Spin-Hall Effect intrinsic B eff p E z E x 2DEG J.Sinova et al., Phys. Rev. Lett. 92, (2004)

94 intrinsic SHE Rashba and Spin-Hall Effect Rashba effect B eff p E

95 Split Gate H-Bar

96 QSHE Spin-Detector 5 4 QSHI-type detector R 34,12 / k 3 2 injector sweep V injector-gate / V x 0.04 QSHE n-type injector SHE x QSHI-type detector detector sweep 0.2 p-type injector V detector-gate / V

97 E / mev E / mev n-regime Q2198 p-regime n = cm -2 n H1 dl = cm -2 E1 H k / nm -1 E F 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, (2004) Intrinsic Spin Hall Effekt Nature Physics 6, 448 (2010)

98 QSHE Spin-Injector 5 4 QSHI-type injector R 12,34 / k 3 2 detector sweep V detector-gate / V x 0.04 QSHE n-type detector SHE x QSHI-type detector injector sweep 0.2 p-type detector

99 C. Brüne et al., Nature Physics, 8, 485 (2012)