Edge states in strained honeycomb lattices : a microwave experiment

Transcription

1 Edge states in strained honeycomb lattices : a microwave experiment Matthieu Bellec 1, Ulrich Kuhl 1, Gilles Montambaux 2 and Fabrice Mortessagne 1 1 Laboratoire de Physique de la Matière Condensée, CNRS & University of Nice-Sophia Antipolis, Nice, France 2 Laboratoire de Physique du Solide, CNRS & University of Paris XI, Orsay, France Workshop on Waves in Complex Media, Grenoble, France December 11, 2013

2 Amazing graphene Outlook, Nature 483, S29 (2012) 1 / 20

3 Amazing graphene Electronic properties Outlook, Nature 483, S29 (2012) 1 / 20

4 Tight-binding Hamiltonian in regular honeycomb lattice

5 Tight-binding Hamiltonian in regular honeycomb lattice Bloch states representation Ψ k = 1 N (λ A φ A j + λ B φ B j )e ik R j j Effective Bloch Hamiltonian in (A, B) basis ( ) 0 f (k) H B eff = t 1 f with f (k) = 1 + e (k) 0 ik a 1 + e ik a 2 Dispersion relation : ɛ(k) = ±t 1 f (k)

6 Dispersion relation Low energy expension (i.e. at the vicinity of the Dirac point) : Linear dispersion relation Dirac Hamiltonian for massless particle : H = v f σ q with v f = 3 ta 2 h c 300 Linear density of states 3 / 20

7 Artificial graphene E.S. Reich, Nature 497, 422 (2013) 4 / 20

8 Typical experimental set-up with microwave lattices

9 Formal analogy between the Schrödinger and the Helmholtz equations Free particle [ + V( r)]ψ( r) = Eψ( r) Varying potential V Microwave cavity [ + (1 ɛ( r))k 2 ]ψ( r) = k 2 ψ( r) Varying permittivity ɛ

10 Formal analogy between the Schrödinger and the Helmholtz equations Free particle [ + V( r)]ψ( r) = Eψ( r) Varying potential V Microwave cavity [ + (1 ɛ( r))k 2 ]ψ( r) = k 2 ψ( r) Varying permittivity ɛ Attractive implementation to perform quantum analogue measurements TM modes : ψ( r) = E z ( r) energy everywhere (continuum state) TE modes : ψ( r) = B z ( r) energy confined inside (bound state)

11 Formal analogy between the Schrödinger and the Helmholtz equations Free particle [ + V( r)]ψ( r) = Eψ( r) Varying potential V Microwave cavity [ + (1 ɛ( r))k 2 ]ψ( r) = k 2 ψ( r) Varying permittivity ɛ Attractive implementation to perform quantum analogue measurements TM modes : ψ( r) = E z ( r) energy everywhere (continuum state) TE modes : ψ( r) = B z ( r) energy confined inside (bound state)

12 1 Microwaves in a honeycomb lattice A flexible experimental artificial graphene

13 1 Microwaves in a honeycomb lattice A flexible experimental artificial graphene 2 Topological phase transition in strained graphene Lifshitz transition from gapless to gapped phase

14 1 Microwaves in a honeycomb lattice A flexible experimental artificial graphene 2 Topological phase transition in strained graphene Lifshitz transition from gapless to gapped phase 3 Edge states in graphene ribbon Zigzag, bearded and armchair edges under strain

15 1 Microwaves in a honeycomb lattice A flexible experimental artificial graphene 2 Topological phase transition in strained graphene Lifshitz transition from gapless to gapped phase 3 Edge states in graphene ribbon Zigzag, bearded and armchair edges under strain

16 Experimental setup 6 / 20

17 Experimental setup 6 / 20

18 Experimental setup 6 / 20

19 An artificial atom 5 mm 8 mm Dielectric cylinder : n = 6 7 / 20

20 An artificial atom 5 mm 8 mm Dielectric cylinder : n = 6 We measure the reflected signal S 11 (ν). At ν = ν 0 1 S 11 (ν 0 ) 2 2σ Γ Ψ 0(r 1 ) 2 Γ 1 : lifetime (Γ 10 MHz) // σ : antenna coupling (weak and constant) 7 / 20

21 An artificial atom 5 mm 8 mm Dielectric cylinder : n = 6 We measure the reflected signal S 11 (ν). At ν = ν 0 1 S 11 (ν 0 ) 2 2σ Γ Ψ 0(r 1 ) 2 Γ 1 : lifetime (Γ 10 MHz) // σ : antenna coupling (weak and constant) Most of the energy is confined in the disc (J 0 ) and spreads evanescently (K 0 ) 7 / 20

22 Coupling between two discs d = 11 mm d = 13 mm d The frequency splitting ν(d) gives the coupling strength t 1 (d) = ν(d)/2 8 / 20

23 Coupling between two discs d The frequency splitting ν(d) gives the coupling strength t 1 (d) = ν(d)/2 t 1 (d) = α K 0 (γd/2) 2 + δ 8 / 20

24 Experimental (local) density of states We have a direct acces to the LDOS g(r 1, ν) = S 11(ν) 2 ϕ σ S (ν) ν Γ S 11 2 ν Ψ n (r 1 ) 2 δ(ν ν n ) n 9 / 20

25 Experimental (local) density of states We have a direct acces to the LDOS g(r 1, ν) = S 11(ν) 2 ϕ σ S (ν) ν Γ S 11 2 ν Ψ n (r 1 ) 2 δ(ν ν n ) n 9 / 20

26 Experimental (local) density of states We have a direct acces to the LDOS g(r 1, ν) = S 11(ν) 2 ϕ σ S (ν) ν Γ S 11 2 ν Ψ n (r 1 ) 2 δ(ν ν n ) n 9 / 20

27 Experimental (local) density of states One can get the wavefunction associated to the eigenfrequency ν n 9 / 20

28 Experimental (local) density of states By averaging g(r i, ν) over all the antenna positions r i, we obtain the DOS 10 / 20

29 Experimental (local) density of states ~ DOS By averaging g(r i, ν) over all the antenna positions r i, we obtain the DOS 10 / 20

30 Experimental (local) density of states TB model ~ DOS By averaging g(r i, ν) over all the antenna positions r i, we obtain the DOS Tight-binding compatible. Main features have been taken into account Dirac shift, band asymmetry next n.n. couplings. 10 / 20

31 1 Microwaves in a honeycomb lattice A flexible experimental artificial graphene 2 Topological phase transition in strained graphene Lifshitz transition from gapless to gapped phase 3 Edge states in graphene ribbon Zigzag, bearded and armchair edges under strain

32 TB Hamiltonian in uni-axial strained honeycomb lattice

33 TB Hamiltonian in uni-axial strained honeycomb lattice Anisotropy parameter : β = t /t Bloch Hamiltonian ( ) H B eff = t 0 e iφ(k) f (k) e iφ(k) 0 with f (k) = 1 + t t e ik a 1 + t t e ik a 2 Eigenstates : u k,± = 1 ( ) e iφ(k) 2 ±1 Berry phase : γ = 1 dk k φ(k) 2

34 Topological phase transition Two independant Dirac points opposite Berry phases (±π ) K and K0 with different topologies characterized by G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009) 12 / 20

35 Topological phase transition Two independant Dirac points opposite Berry phases (±π ) Berry phase K and K0 with different topologies characterized by ±π vanishes to 0 Topological phase transition G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009) 12 / 20

36 Topological phase transition Gapless phase Gapped phase Two independant Dirac points K and K with different topologies characterized by opposite Berry phases (±π) Berry phase ±π vanishes to 0 Topological phase transition Dirac points move, merge (at β = β c = 2) and annihilate G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009) 12 / 20

37 Topological phase transition Gapless phase Gapped phase Two independant Dirac points K and K with different topologies characterized by opposite Berry phases (±π) Berry phase ±π vanishes to 0 Topological phase transition Dirac points move, merge (at β = β c = 2) and annihilate Phase transition from gapless to gapped phase (Lifshitz transition) G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009) 12 / 20

38 Topological phase transition Two independant Dirac points K and K with different topologies characterized by opposite Berry phases (±π) Berry phase ±π vanishes to 0 Topological phase transition Dirac points move, merge (at β = β c = 2) and annihilate Phase transition from gapless to gapped phase (Lifshitz transition) G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009) 12 / 20

39 Observation of the topological phase transition Transition at β c = 1.8 ( = 2!) Bandgap opening 13 / 20

40 Observation of the topological phase transition 25 0 Transition at β c = 1.8 ( = 2!) Bandgap opening 13 / 20

41 Observation of the topological phase transition Transition at β c = 1.8 ( = 2!) Bandgap opening Presence of edge states at Dirac frequency 13 / 20

42 1 Microwaves in a honeycomb lattice A flexible experimental artificial graphene 2 Topological phase transition in strained graphene Lifshitz transition from gapless to gapped phase 3 Edge states in graphene ribbon Zigzag, bearded and armchair edges under strain

43 Edges in HC lattices zigzag armchair bearded 14 / 20

44 Edges in HC lattices zigzag armchair bearded Edge states generally appear. Absence of armchair edge states is an exception! 14 / 20

45 Edges in HC lattices zigzag armchair bearded Edge states generally appear. Absence of armchair edge states is an exception! How to figure it out? Zak phase S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, (2002) P. Delplace et al., Phys. Rev. B 84, (2011) 14 / 20

46 Edges in HC lattices armchair 14 / 20

47 Observation of edge states at Dirac frequency No edge states along compression direction y States live only on one sublattice Edge states apparition doesn t depend on the transition β c = 1.8 The localisation length ξ decreases with β 15 / 20

48 Armchair edge states in strained HC lattice

49 Armchair edge states in strained HC lattice From TB model at B n+1, we get : βa n e iky + A n+1 + A n+2 = 0 For β > 1 {A n } 0. Only B sites are excited. From TB model at A n+1, we get : B n+1 + B n + βb n+2 e iky = 0 Presence of edge states for : k y < arccos β 2 (3 β2 )

50 Armchair edge states in strained HC lattice Edge states - B sites No edge states at β = 1 Presence of ES for β = 1. Only ES for β > β c = 2 No edge states 17 / 20

51 Armchair edge states in strained HC lattice Edge states - B sites No edge states at β = 1 Presence of ES for β = 1. Only ES for β > β c = 2 No edge states ES - A sites 17 / 20

52 Armchair edge states in strained HC lattice No edge states at β = 1 Presence of ES for β = 1. Only ES for β > β c = 2 Edge states on A site for β < 1 17 / 20

53 Armchair edge states in strained HC lattice Localisation length ξ max min No edge states at β = 1 Presence of ES for β = 1. Only ES for β > β c = 2 Edge states on A site for β < 1 Edge states weakly localized close to the red line (ξ ) 17 / 20

54 Armchair edge states in strained HC lattice Localisation length ξ Measured and calculated intensities A n 2 (β = 0.4) B n 2 (β = 1.8, 2.5 and 3.5) max 1 β = 3.5 min Bn n 17 / 20

55 Armchair edge states in strained HC lattice Localisation length ξ Measured and calculated intensities A n 2 (β = 0.4) B n 2 (β = 1.8, 2.5 and 3.5) max 1 β = 2.5 min Bn n 17 / 20

56 Armchair edge states in strained HC lattice Localisation length ξ Measured and calculated intensities A n 2 (β = 0.4) B n 2 (β = 1.8, 2.5 and 3.5) max 1 β = 1.8 min Bn n 17 / 20

57 Armchair edge states in strained HC lattice Localisation length ξ Measured and calculated intensities A n 2 (β = 0.4) B n 2 (β = 1.8, 2.5 and 3.5) max 1 β = 0.4 min An n 17 / 20

58 Zigzag and bearded edge states in strained HC lattice zigzag bearded 18 / 20

59 Zigzag and bearded edge states in strained HC lattice Zigzag Bearded 19 / 20

60 Zigzag and bearded edge states in strained HC lattice Zigzag Bearded 19 / 20

61 Zigzag and bearded edge states in strained HC lattice max min 19 / 20

62 Zigzag and bearded edge states in strained HC lattice max min 19 / 20

63 Zigzag and bearded edge states in strained HC lattice max min 19 / 20

64 Zigzag and bearded edge states in strained HC lattice max min 19 / 20

65 1 Microwave artificial graphene Flexible experiment TB compatible Access to the DOS & wavefunctions

66 1 Microwave artificial graphene Flexible experiment TB compatible Access to the DOS & wavefunctions 2 Topological phase transition in strained HC lattices Observation of the gapped phase Observation of armchair edge states Study of various edge states

67 1 Microwave artificial graphene Flexible experiment TB compatible Access to the DOS & wavefunctions 2 Topological phase transition in strained HC lattices Observation of the gapped phase Observation of armchair edge states Study of various edge states Thanks for your attention! More info : Phys. Rev. Lett. 110, (2013) Phys. Rev. B 88, (2013) mail : bellec@unice.fr web :