TOPOLOGICAL BANDS IN GRAPHENE SUPERLATTICES

Transcription

1 TOPOLOGICAL BANDS IN GRAPHENE SUPERLATTICES 1) Berry curvature in superlattice bands 2) Energy scales for Moire superlattices 3) Spin-Hall effect in graphene Leonid Levitov ISSP U Tokyo

2 MIT Manchester Justin Song Song, Shytov, LL PRL 111, (2013) Song, Samutpraphoot, LL arxiv: (2014) Gorbachev, Song et al arxiv: (2014) Lensky, Song, Samuthrapoot, LL, arxiv: (2014) 2

3 Stacked Van der Waals heterostructures Stacked atomically thin layers: van der Waals crystals, atomic precision, axes alignment Image from: Geim & Grigorieva, Nature 499, 419 (2013) 3

4 Gap opening for G/hBN 4

5 Gap opening for G/hBN Activated behavior: gap ~ K 5

6 Gap opening for G/hBN Activated behavior: gap ~ K 6

7 Valley index Nonlocal measurements: (2014) Gorbachev, Song, et. al. Science 7

8 Topological currents Electrons in crystals have charge, energy, momentum and Berry's curvature Semiclassical eqs of motion: 8

9 Topological currents Electrons in crystals have charge, energy, momentum and Berry's curvature Semiclassical eqs of motion: Hall currents at B=0 9

10 Graphene-based topological materials Quantized transport, Topological bands, Anomalous Hall effects Chern invariant 1 C= Ω (k ) k 2π Ω(k )= k A k, A k =i ψ(k) k ψ(k ) 10

11 Graphene-based topological materials Quantized transport, Topological bands, Anomalous Hall effects Chern invariant 1 C= Ω (k ) k 2π Ω(k )= k A k, A k =i ψ(k) k ψ(k ) Pristine graphene: massless Dirac fermions, Berry phase yet no Berry curvature 11

12 Massive (gapped) Dirac particles A/B sublattice asymmetry a gap-opening perturbation Berry curvature hot spots above and below the gap T-reversal symmetry: Ω( k )= Ω( k ) Ω( k ) 0 Valley Chern invariant (for closed bands) 1 C= Ω (k ) 2π k 12

13 Massive (gapped) Dirac particles A/B sublattice asymmetry a gap-opening perturbation Berry curvature hot spots above and below the gap T-reversal symmetry: Ω( k )= Ω( k ) Ω( k ) 0 Valley Chern invariant (for closed bands) 1 C= Ω (k ) 2π k D. Xiao, W. Yao, and Q. Niu, PRL 99, (2007) 13

14 Bloch bands in G/hBN superlattices Song, Shytov, LL, PRL 111, (2013) Song, Samutpraphoot, LL, PNAS (2015) Moiré wavelength C-C spacing can as large as 14nm 100 times 14

15 Bloch bands in G/hBN superlattices Song, Shytov, LL, PRL 111, (2013) Song, Samutpraphoot, LL, PNAS (2015) Moiré wavelength can as large as 14nm 100 times C-C spacing Focus on one valley, K or K' 15

16 The variety of G/hBN superlattices: San-Jose et al. arxiv: , Jung et al arxiv: , Song, Shytov LL PRL (2013), Kindermann PRB (2012) Sachs, et. al. PRB (2011) Incommensurate (moire) chirality/mass sign changing Dean et.al. Nature 497, 213 (2013) Ponomarenko et al Nature 497, 594 (2013) 16

17 The variety of G/hBN superlattices: San-Jose et al. arxiv: , Jung et al arxiv: , Song, Shytov LL PRL (2013), Kindermann PRB (2012) Sachs, et. al. PRB (2011) Incommensurate (moire) Commensurate stacking chirality/mass sign global A/B asymmetry changing global gap Dean et.al. Nature 497, 213 (2013) Ponomarenko et al Nature 497, 594 (2013) Woods, et.al. Nature Phys 10, 451 (2014) 17

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19 Low-energy Hamiltonian San-Jose et al. arxiv: , Jung et al arxiv: , Song, Shytov LL PRL (2013), Kindermann PRB (2012) Sachs, et. al. PRB (2011) Constant global gap at DP Spatially varying gap, Bragg scattering Focus on one valley Song, Samutpraphoot, LL arxiv (2014) 19

20 Incommensurate/Moire case 20

21 Commensurate case 21

22 Band topology tunable by crystal axes alignment Topological bands C=1 Trivial bands C=0 22

23 Future (RVHE) Valley Current (VHE) 1) Measure Chern numbers (separately gated regions for VHE injection and detection) 2) Waveguides for valley currents 3) Valley population accumulation (optical probes) 23

24 Interactions in G/hBN superlattices Incommensurability from lattice mismatch and twist angle impacts energy scales of superlattice structures 24

25 Interactions in G/hBN superlattices Incommensurability from lattice mismatch and twist angle impacts energy scales of superlattice structures Oscillating gap Interactions 25

26 Non-interacting theory (i) Oscillating gives a first order gap that vanishes, since 26

27 Non-interacting theory (i) Oscillating gives a first order gap that vanishes, since (ii) A large gap at edge of Superlattice Brillouin Zone, 27

28 Non-interacting theory (i) Oscillating gives a first order gap that vanishes, since (ii) A large gap at edge of Superlattice Brillouin Zone, (iii) At 3rd order perturbation theory for obtain small gap (100 mk) at Dirac point: 28

29 Interacting theory (i) Interactions enhance both velocity and mass terms ( ) (ii) Scalar term Identity not enhanced due to Ward follows from gauge invariance Can obtain giant enhancements to 3rd order gap at Dirac point,, as large as three orders of magnitude. 29

30 A two-stage RG flow RG Equations: Stage 1: Stage 2: 30

31 RG Flow of interaction enhanced couplings to Moiré potential 31

32 Interaction enhanced gap features (I) Sensitivity to (controlled by twist angle) and screening (controlled by gates) 32

33 Interaction enhanced gap features (I) Sensitivity to (controlled by twist angle) and screening (controlled by gates) (ii) Gap can be as large as room temperature 33

34 Interaction enhanced gap features (I) Sensitivity to (controlled by twist angle) and screening (controlled by gates) (ii) Gap can be as large as room temperature (iii) Gap is sensitive to cut-off length scale and can be smeared out by charge puddles 34

35 Interaction enhanced gap features (I) Sensitivity to (controlled by twist angle) and screening (controlled by gates) (ii) Gap can be as large as room temperature (iii) Gap is sensitive to cut-off length scale and can be smeared out by charge puddles (iv) For self-terminated RG, get gap that scales (one loop large-n) 35

36 Spintronics in graphene? 36

37 Spin manipulation in graphene Slow spin relaxation due to weak SO in graphene 37

38 Spin manipulation in graphene Slow spin relaxation due to weak SO in graphene BUT: short spin lifetimes w ferromagnets 38

39 Spin manipulation in graphene Slow spin relaxation due to weak SO in graphene BUT: short spin lifetimes w ferromagnets 39

40 Spin manipulation in graphene Slow spin relaxation due to weak SO in graphene BUT: short spin lifetimes w ferromagnets BUT: impossible to make (edge disorder) 40

41 Spin manipulation in graphene Slow spin relaxation due to weak SO in graphene BUT: short spin lifetimes w ferromagnets NEW WAYS TO GENERATE & DETECT SPIN/VALLEY CURRENTS? BUT: impossible to make (edge disorder) 41

42 Spin-Hall effect without spin-orbit -Large value, persists at room T and low B -Stems from Dirac spectrum SH 0.1 B Zeeman-split bands ( ) (k ) vk / 2 Finite density of electrons and holes 42

43 SHE mechanism Opposite Lorentz force on the up-spin and down-spin Spin current in a transverse direction 43

44 SHE mechanism Opposite Lorentz force on the up-spin and down-spin Spin current in a transverse direction d xy SH SHE coefficient xy xy SH d xx the Zeeman splitting 44

45 SHE mechanism Opposite Lorentz force on the up-spin and down-spin Spin current in a transverse direction d xy SH SHE coefficient xy xy SH d xx the Zeeman splitting Need to understand Hall resistivity 45

46 Steep xy, giant SHE Abanin et al. PRL 2011 Quasiclassical result: xy (k ) B xy ne carrier density Steepening large d xy d Diverges at the Dirac point Singularity smeared by disorder and interactions giant SHE at the Dirac point 46

47 Predict SHE coefficient -Grows with B and inverse T -Saturates at DP width T ~ SH 1 / 3 -Large enhancement in clean samples SH ebv02 C, T 2 T ebv02, T 3 47

48 Nonlocal measurement V35 Rnl I 26 I SD Vnl 300 K Good agreement w theory: Abanin et al, Science Peak at the Dirac point -Growth as a function of 1/T, B -Magnitude 48

49 Future -Predict large spin accumulation: ns cm times larger than GaAs -Generate/detect spin currents using local magnetic fields -Spin injection into graphene and other materials Abanin et al. PRL (2011), Science (2012) 49