Hoffman Lab Microscopes

Transcription

1 Hoffman Lab Microscopes Scanning Tunneling Microscope Force Microscope Ultra-high vacuum STM SmB 6 Ca-YBCO Sb(111) Pr x Ca 1-x Fe 2 As 2 Bi 2 Se 3 K x Sr 1-x Fe 2 As 2 Bi-2212 Bi-2201 Ba(Fe 1-x Co x ) 2 As 2 NbSe 2 nm Li 0.9 Mo 6 O 17 NdFeAsO 1-x F x VO 2 Nd 2 Fe 14 B

2 Nanoscale Imaging of Topological Materials: Sb and SmB 6 Jenny Hoffman Experiments: Samples: Sb: Dillon Gardner, Young Lee SmB 6 : Dae-Jeong Kim, Zach Fisk Mike Yee Anjan Yang He Jason Zhu Soumyanarayanan (Harvard) Funding: Theory: Hsin Lin, Arun Bansil Jay Sau, Anton Akhmerov, Bert Halperin

3 Outline 3 Topological Insulators Scanning Tunneling Microscopy Nanoscale Band Structure Topological: Sb Insulator: SmB 6

4 My kids asked me 4 What in the world is a topological insulator??

5 A topological insulator is a material with 5 conducting surface insulating bulk But what makes it topological?

6 Band Theory 6 E Single atom: Crystal: 3s 2p 2s 1s

7 Band Theory: Metal 7 E Single atom: Crystal: 3s 2p 2s Fermi level 1s

8 Band Theory: Insulator 8 E Single atom: Crystal: 3s 2p 2s Fermi level 1s

9 Spin-Orbit Coupling Band Inversion Energy 9 ε F s Band gap: 2m e c 2 ~ 10 6 ev p 3/2 Vacuum Material Vacuum

10 Spin-Orbit Coupling Band Inversion Energy 10 Metallic surface states ε F p 3/2 Band gap: 2m e c 2 ~ 10 6 ev s Vacuum Material Vacuum topologically protected spin-polarized surface states Why spin-polarized?

11 Band Theory: Insulator 11 E Single atom: Crystal: Add momentum information: 3s 2p Fermi level E 2s Brillouin zone edge k 1s

12 12 Spin-Polarized Surface State Hamiltonian ε bulk k

13 13 Spin-Polarized Surface State Hamiltonian 0. Free electrons in 2 dimensions: H = E 0 + k2 2m 1. Rashba: Surface breaks symmetry H R = Ez 2. Moving electron with velocity v sees E-field as B v E ε bulk surface state k

14 14 Spin-Polarized Surface State Hamiltonian 0. Free electrons in 2 dimensions: H = E 0 + k2 2m 1. Rashba: Surface breaks symmetry H R = Ez 2. Moving electron with velocity v sees E-field as B v E 3. Spin-orbit coupling splits bands: H SO v E σ ε bulk surface state k momentum spin

15 15 Spin-Polarized Surface State Hamiltonian 0. Free electrons in 2 dimensions: H = E 0 + k2 2m 1. Rashba: Surface breaks symmetry H R = Ez 2. Moving electron with velocity v sees E-field as B v E 3. Spin-orbit coupling splits bands: H SO v E σ momentum spin 4. k p theory: expand in small k around Γ point H = E 0 + k2 2m + v(k xσ y k y σ x ) ε bulk surface state k

16 Spin-Polarized Surface State Hamiltonian 0. Free electrons in 2 dimensions: H = E 0 + k2 2m 1. Rashba: Surface breaks symmetry H R = Ez ε bulk surface state 2. Moving electron with velocity v sees E-field as B v E 3. Spin-orbit coupling splits bands: H SO v E σ k momentum spin 4. k p theory: expand in small k around Γ point H = E D 0 + k2 + (v v(k xσ y y σ x ) 2m 0 + αk 2 )(k x σ y k y σ x ) new term to bend band down 16

17 Spin-Polarized Surface State Hamiltonian 0. Free electrons in 2 dimensions: H = E 0 + k2 2m 1. Rashba: Surface breaks symmetry H R = Ez ε bulk surface state 2. Moving electron with velocity v sees E-field as B v E 3. Spin-orbit coupling splits bands: H SO v E σ M K momentum spin 4. k p theory: expand in small k around Γ point H = E D 0 + k2 + (v v(k xσ y y σ x ) 2m 0 + αk 2 )(k x σ y k y σ x ) + 1 λ(k k 3 ) crystal symmetry 17

18 Spin-Polarized Surface State Hamiltonian 0. Free electrons in 2 dimensions: H = E 0 + k2 2m 1. Rashba: Surface breaks symmetry H R = Ez 2. Moving electron with velocity v sees E-field as B v E 3. Spin-orbit coupling splits bands: H SO v E σ momentum spin 4. k p theory: expand in small k around Γ point H = E D v(k xσ y y σ x ) 0 + k2 + (v 2m 0 + αk 2 )(k x σ y k y σ x ) + 1 λ(k k 3 ) M ε bulk But what are they good for? surface state Liang Fu s 5-parameter Hamiltonian crystal symmetry [PRL 101, (2009)] K 18

19 Applications of Topological Insulators 1. Spintronics How can we realize these applications? 2. Topological quantum computing Majorana fermion = self-antiparticle (non-abelian anyon) Topological-superconductor interface t TI proximity ½ triplet: ( + )/ 2 SC braid the world-lines information is encoded topologically Fu & Kane, PRL 100, (2008) 19

20 Metrics for topological spintronics devices 1. Enhance spin-momentum locking Metric: Spin-Orbit Coupling (v 0 ) 2. Reduce scattering Metric: Mean Free Path (l f ) H. Beidenkopf, Nat. Phys Reduce vulnerability to external B & magnetic impurities: Metric: g-factor Need a probe which measures: nanoscale, B-dependent, filled & empty states 20

21 Outline 21 Topological Insulators Scanning Tunneling Microscopy Nanoscale Band Structure Topological: Sb Insulator: SmB 6

22 Scanning Tunneling Microscopy Sample Tip E E Tip GW E F (sample) E F (tip) Sample vacuum V Sample LDOS under tip TIP DOS 22

23 Scanning Tunneling Microscopy Sample ev I( V ) LDOS( E) de E F Tip E E E F (tip) E F (sample) Tip GW Sample vacuum V Sample LDOS under tip TIP DOS 23

24 Types of STM Measurements di/dv Spectrum Local Density of States (x, y, E) x E y di/dv Map Topography Constant current mode: di dv 24

25 Momentum Information from STM? Clean Samples LLs Impurities QPI Wildöer, PRB 55, R16013 (1997) Crommie, Nature 363, 524 (1993) Dirac dispersion ε N eħv F NB Measure q ε infer k(ε) 25

26 Single Layer Graphene 26 Landau levels Quasiparticle interference Luican+Andrei, PRB 83, (2012) Zhang+Crommie, Nat. Phys. 5, 722 (2009) v LL = m/s 40% Discrepancy! v QPI = m/s

27 Outline 27 Topological Insulators Scanning Tunneling Microscopy Nanoscale Band Structure Topological: Sb Insulator: SmB 6

28 Sb(111): Topography A 100 mv, 1 GΩ C B A C B 3 bilayers c = 11.3 Å 100 mv, 1/8 GΩ A a = 4.31 Å 31

29 32 Sb(111): Spectroscopy and Band Structure Surface states: H = E D + k2 2m + (v 0 + αk 2 )(k x σ y k y σ x ) λ(k k 3 ) Theory extrapolate inner cone Experimental Data K ε T Γ M ε S ε S ε T

30 33 Sb(111): Spectroscopy and Band Structure Surface states: H = E D + k2 2m + (v 0 + αk 2 )(k x σ y k y σ x ) λ(k k 3 ) Theory extrapolate inner cone K Data ε T Γ M ε T ε S ε S

31 Spectroscopy in Magnetic Field 300 mv, 0.5 GΩ 300 mv, 0.5 GΩ 100 mv, 0.2 GΩ 34

32 Landau Levels 35 Dirac Fermions: ε = ħv F k ε N = ε D + eħv F NB ε 3 ε 2 ε 1 ε ε T k QPI LL Rashba: 2 split parabolas ω c = eb mc ; ε 0 = 1 2 ħω c + gμ B B ε ε S ε n ± = ħω c n ± 2nv 0 2 mħω c + ε 0 2 k

33 Landau Levels 36 Let s focus on Dirac-like first Dirac Fermions: ε = ħv F k ε N = ε D + eħv F NB ε 3 ε 2 ε 1 ε ε T k QPI LL Rashba: 2 split parabolas ω c = eb mc ; ε 0 = 1 2 ħω c + gμ B B ε ε S ε n ± = ħω c n ± 2nv 0 2 mħω c + ε 0 2 k

34 Landau Levels: Data 37 N= N > 6 Hold that thought ε N = ε D + eħv LL NB v LL = m/s

35 Quasiparticle Interference Imaging 38 Sb defects unperturbed H 2 O ME (experimentalist) perturbation interference patterns

36 2-dim band structure: topographic map for e - scattering between any degenerate k points gives a standing wave with wavevector q q k y k x Contours of Constant Energy

37 41 QPI in Sb(111) Surface states: H = E D + k2 2m + (v 0 + αk 2 )(k x σ y k y σ x ) λ(k k 3 ) ε 3 K ε T Γ M ε S q

38 Quasiparticle Interference on Sb(111) 42 Sb(111) Topography di/dv (density of states)

39 Quasiparticle Interference on Sb(111) 44 QPI disperses ~linearly Dominant modes along Γ M and Γ K Fourier transform Γ-K Γ-M Will this agree with Landau levels??

40 LL & QPI: Dispersion Comparison v QPI = m/s v LL = m/s agree within 3% arxiv: STM Advantages Filled and empty states Nontrivial band structure sub-mev energy resolution B-field dependence Nanoscale spatial resolution 45

41 Outline 46 Topological Insulators Scanning Tunneling Microscopy Nanoscale Band Structure Topological: Sb Insulator: SmB 6

42 47 Metrics for topological devices 1. Enhance spin-momentum locking Metric: Spin-Orbit Coupling (v 0 ) 2. Reduce scattering Metric: Mean Free Path (l f ) H. Beidenkopf, Nat. Phys Reduce vulnerability to external B & magnetic impurities: Metric: g-factor

43 Reconstruct multi-component band structure 49 ε T ε S QPI LL K Γ M

44 Reconstruct multi-component band structure 50 ε T ε S QPI LL ε > ε S K Γ M

45 Reconstruct multi-component band structure ε T ε S QPI LL ε > ε S H = E D + k2 2m + (v 0 + αk 2 )(k x σ y k y σ x ) λ(k k 3 ) E D = 210 mv; m = 0.1m e ; v 0 = 0.51 ev Å; α = 110eV Å 3 ; λ = 230 ev Å 3 51

46 52 Metrics for topological devices 1. Enhance spin-momentum locking Metric: Spin-Orbit Coupling (v 0 =0.5 ev Å) 2. Reduce scattering Metric: Mean Free Path (l f ) H. Beidenkopf, Nat. Phys Reduce vulnerability to external B & magnetic impurities: Metric: g-factor

47 53 Sb(111) LL Sharpness & Lifetime Broadening LLs are sharpest at ε F in agreement with Fermi liquid theory: τ e e (ε)~1/ε 2 Sb(111) mean free path l ~ ħ Γ(ε F ) v F Sample 1: λ F ~ 65 nm Sample 2: λ F ~ 59 nm (same for 7.5T & 9T)

48 Sb(111) LL Sharpness & Lifetime Broadening LLs are sharpest at ε F in agreement with Fermi liquid theory: τ e e (ε)~1/ε 2 Comparable to distance between strong surface defects Sb(111) mean free path l ~ ħ Γ(ε F ) v F Sample 1: λ F ~ 65 nm Sample 2: λ F ~ 59 nm (same for 7.5T & 9T) c.f. Bi 2 Se 3 Bi 2 Te 3 24 nm H. Beidenkopf, Nat. Phys

49 55 Metrics for topological devices 1. Enhance spin-momentum locking Metric: Spin-Orbit Coupling (v 0 =0.5 ev Å) 2. Reduce scattering Metric: Mean Free Path (l f ~60 nm) H. Beidenkopf, Nat. Phys Reduce vulnerability to external B & magnetic impurities: Metric: g-factor

50 Quantify g-factor from low-energy LLs 57 Need to look at low N to get the g-factor Dirac Fermions: ε = ħv F k ε N = ε D + eħv F NB ε 3 ε 2 ε 1 ε ε T k QPI LL Rashba: 2 split parabolas ε ε S ε 0 = 1 2 ħω c + gμ B B ε n ± = ħω c n ± 2nv 0 2 mħω c + ε 0 2 k (where ω c = eb mc )

51 Landau level identification in Sb(111) 58 n=3 + n=2 ε + 0 = 1 2 ħω c + gμ B B n=0 n=1 + g = 12.8

52 59 Metrics for topological devices 1. Enhance spin-momentum locking Metric: Spin-Orbit Coupling (v 0 =0.5 ev Å) 2. Reduce scattering Metric: Mean Free Path (l f ~60 nm) H. Beidenkopf, Nat. Phys Reduce vulnerability to external B & magnetic impurities: Metric: g-factor (g = 12.8)

53 60 Sb(111) Robust Surface States in Semimetal Sb Material (Group) LL Range (N) Bulk Overlap (N) ε T Bi 2 Se 3 (RIKEN) PRB 82, (2010) Sb 2 Te 3 (MBE, CAS) PRL 108, (2012) Bi 2 Te 3 (BC) PRL 109, (2012) Pb 1-x Sn x Se (BC) Science 341, 1496 (2013) Bi 2 Te 2 Se (RIKEN) ACS Nano 7, 4105 (2013) 0 : 22 (9 T) ~ 18 (VB) -4 : 8 (7 T) ~ -3 (VB), ~6 (CB) ~ 0 : 14 (7 T) ~ N = :18 (3-7.5T) ~ N = 7-18 ~ 0 : 17 (11 T) ~ N = 0 Sb (Harvard) 1 : 27 (4 9 T) Full LL ε S suggests to shift focus from topological insulators to topological semimetals

54 61 Topological Semimetal vs. Insulator Sb: bulk semimetal Flat topography, electronic modulations primarily around impurities Bi 0.92 Sb 0.08 : bulk insulator Large chemical potential fluctuations Roushan, Nature 460, 1106 (2009) 100 mv, 1 GΩ

55 Sb vs. canonical Bi 2 Se 3 Sb Bi 2 Se 3 ε T ε S Zhu, Elfimov, Damascelli, PRL 107, (2011) Zhu, Elfimov, Damascelli, PRL 110, (2013) Sb is an excellent platform for exploring topological proximity effects 62

56 Outline 65 Topological Insulators Scanning Tunneling Microscopy Nanoscale Band Structure Topological: Sb Insulator: SmB 6

57 Topological Kondo Insulator 66 f band hybridization conduction band Hybridization gap, Δ f band 0K T k High T

58 Topological Kondo Insulator 67 topological Kondo insulator metal conduction band Hybridization gap, Δ E F topological surface states f band 0K T k High T

59 SmB 6 as possible TKI Resistivity resistance saturates Wolgast, PRB 88, (2013) Kondo insulator transition? LDA Zhang, PRX 3, (2013) Lu, PRL 110, (2013) 68

60 Hybridization gap in SmB 6 Transport: Δ = 4.6 mev Menth, PRL 22, 295 (1969) Δ = 11.2 mev Flachbart, PRB 64, (2001) Δ = 3.47 mev Wolgast, PRB 88, (2013) Reflectivity & transmissivity: Δ = 4.7 mev Travaglini, PRB 29, 893 (1984) Δ = 19 mev Gorshunov, PRB 59, 1808 (1999) Raman spectroscopy: Δ = 36 mev Nyhus, PRB 52, R14308 (1995) Planar tunneling / point contact spectroscopy: Δ = 2.7 mev Güntherodt, PRL 49, 1030 (1982) Δ = 14 mev Amsler, PRB 57, 8747 (1998) Δ = 22 mev Flachbart,PRB 64, (2001) Δ = 18 mev Zhang, PRX 3, (2013) ARPES: (seven contradictory papers in last few months!) Δ ranges from < 5 mev (entirely below E F ) to > 20 mev (spanning E F ) 69

61 SmB 6 ARPES Miyazaki, PRB 86, (2012) Xu, PRB 88, (2013) Jiang, Nat Comm 4, 3010 (2013) Neupane, Nat Comm 4, 2991 (2013) Frantzeskakis, PRX 3, (2013) Zhu + Damascelli, PRL 111, (2013) Suga, JPSJ 83, (2014) But where is the hybridization gap?? 70

62 SmB 6 ARPES 6 H 7/2 (mev) H 5/2 (mev) -15 (X) to -20 (Γ) Δ (mev) Spans E F? ~ 15 YES In-gap state? -4 to -8 mev, weakly dispersing ~ 20 YES 2 dispersing bands > 18 YES NO f band energies < 5 mev 2 dispersing bands, circular dichroism -4 mev, non-dispersing NO cannot resolve E F, 2 ev dispersing Reference Miyazaki, PRB 86, (2012) Xu (H. Ding), PRB 88, (2013) Jiang (D.L. Feng), Nat Com 4, 3010 (2013) Neupane (Z. Hasan), Nat Com 4, 2991 (2013) Frantzeskakis (M. Golden), PRX 3, (2013) Zhu (A. Damascelli), PRL 111, (2013) Suga, JPSJ 83, (2014) SmB 6 is not even a Kondo insulator!?!?! Need reliable, spatially resolved, empty + filled state measure of Δ 71

63 SmB 6 single crystal Å grown by Al flux method cleaved in cryogenic UHV exposes (001) plane surface B:Sm ratio > 6:1 insert into STM

64 SmB 6 atomic surface morphologies Sm 1 x 1 Sm 2 x 1 B filamentary disordered c.f. ARPES: 6 H 7/2 ~ -160 to H 5/2 ~ -20 to -14 band bending at this polar 1 x 1 surface 73

65 1x1 surfaces: Polarity-driven surface states? 4.13 Å Zhu & Damascelli, PRL 111, (2013) 74

66 2x1 non-polar: SmB 6 hybridization gap? Can we just read Δ off from the spectrum? Δ 20 mv?? Not so fast 75

67 Tunneling into Kondo impurity: Fano resonance STM tip tunneling into Co d orbital tunneling into conduction band Co on Au(111) Not a gap: energy range of destructive interference Madhavan, Science 280, 567 (1998) Interference between two tunneling channels gives Fano resonance: di dv V q + ε ε 2 q = ratio between tunneling channels ε = (ev ε 0 )/w ; ε 0 = bare resonance; w = resonance width 76

68 Tunneling into a Kondo Lattice Dirty Kondo lattice: Fano is the limiting case Yang, PRB 79, (2009) Wolfle, Dubi & Balatsky, PRL 105, (2010) Clean Kondo lattice: analytic model, structureless conduction band Maltseva, Dzero & Coleman, PRL 103, (2009) Clean Kondo lattice: computational model, realistic conduction band Figgins & Morr, PRL 104, (2010) 77

69 ARPES filled states SmB 6 bands from ARPES our model LDA for empty states Sm 5d conduction band D 1 = 1.6 ev ; D 2 = 3 ev Jiang, Nat Comm 4, 3010 (2013) Antonov, PRB 66, (2002) 78

70 79 Figgins vs. Maltseva Models Raw di/dv data Figgins Raw data Maltseva Both models: γ = k B T (T = 8 K) t f t c = v ~155 mev in both models

71 Recapturing the band structure from di/dv Raw di/dv data Figgins Figgins decomposition v = 155mV t f t c = di/dv is dominated by the bare conduction band gap in STM di/dv is the true hybridization gap Δ 2v2 D 13 mev 80

72 Kondo gap closes at T > 50K 81

73 82 Comparison to Point Contact Spectroscopy STM (vacuum tunneling) Planar tunneling Pb contact, unknown size Amsler, PRB 57, 8747 (1998) Ag contact, ~ 100 mm prominent peak is on opposite side Zhang, PRX 3, (2013)

74 Electron vs. hole Hole-like conduction band Electron-like conduction band In 3-dim, outside shell will give more prominent peak than inside shell Expect conduction band peak on negative side 83

75 84 Electron-like conduction band Electron vs. Hole Hole-like conduction band

76 Tunneling ratio: t f /t c 85 N f N cf our data N c All 3 models show di/dv 0 in gap (< 10% of background) c.f. non-zero di/dv in data

77 6 H 7/2 (mev) H 5/2 (mev) -15 (X) to -20 (Γ) Δ (mev) Comparison to ARPES Spans E F? ~ 15 YES In-gap state? 2x1? Reference -4 to -8 mev, weakly dispersing ~ 20 YES 2 dispersing bands > 18 YES NO f band energies 2 dispersing bands, circular dichroism -4 mev, non-dispersing < 5 mev NO cannot resolve E F, 2 ev dispersing yes (LEED) yes (folding) yes (folding) Miyazaki, PRB 86, (2012) Xu (H. Ding), PRB 88, (2013) Jiang (D.L. Feng), Nat Com 4, 3010 (2013)? Neupane (Z. Hasan), Nat Com 4, 2991 (2013) no (LEED) no (LEED) Frantzeskakis (Golden), PRX 3, (2013) Zhu (Damascelli), PRL 111, (2013)? Suga, JPSJ 83, (2014) partial 2x1 surface gives the -8 mev state: this is the f band itself, not an in-gap state!! complete 1x1 surface causes band-bending, looks like Kondo metal? 86

78 1. Band bending on varying surface morphologies Need to decompose di/dv into DOS & interference 2. Hybridization gap spans E F Kondo insulator SmB 6 conclusions Sm 1 x 1 Sm 2 x Residual in-gap spectral weight Next step: QPI to look for Dirac cone arxiv:

79 Nanoscale Band Structure o Quasiparticle Interference o Landau Quantization Reconciled!! Conclusions arxiv: Topological surface states in Sb: o mean free path l = 65 nm o spin-orbit coupling v 0 = 0.51 ev Å o g-factor g = 12.8 SmB 6 : Topological Kondo Insulator? o Hybridization gap spans E F Kondo insulator o In-gap spectral weight topological surface state? arxiv: