A microwave realization of artificial graphene Matthieu Bellec - - PowerPoint PPT Presentation

▶
a microwave realization of artificial graphene
SMART_READER_LITE
LIVE PREVIEW

A microwave realization of artificial graphene Matthieu Bellec - - PowerPoint PPT Presentation

Feb 25, 2023 119 likes •762 views

A microwave realization of artificial graphene Matthieu Bellec Laboratoire de Physique de la Matire Condense, CNRS & University of Nice-Sophia Antipolis, Nice, France Journe de la physique nioise Sophia Antipolis, June 20th, 2014 A

slide-1
SLIDE 1

A microwave realization of artificial graphene

Matthieu Bellec

Laboratoire de Physique de la Matière Condensée, CNRS & University of Nice-Sophia Antipolis, Nice, France Journée de la physique niçoise Sophia Antipolis, June 20th, 2014

slide-2
SLIDE 2

A microwave realization of artificial graphene

Co-workers

Ulrich Kuhl LPMC Fabrice Mortessagne LPMC Gilles Montambaux LPS, Orsay

2 / 27

slide-3
SLIDE 3

Amazing graphene

Outlook, Nature 483, S29 (2012)

3 / 27

slide-4
SLIDE 4

Amazing graphene

  • Outlook, Nature 483, S29 (2012)

3 / 27

slide-5
SLIDE 5
  • Tight-binding Hamiltonian in regular honeycomb lattice
slide-6
SLIDE 6
  • Tight-binding Hamiltonian in regular honeycomb lattice
  • Bloch states representation

|Ψki = 1 p N ∑

j

(lA|fA

j i + lB|fB j i)eik·Rj

  • Effective Bloch Hamiltonian in (A, B) basis

HB

eff = t1

✓ f(k) f ⇤(k) ◆

with f(k) = 1 + eik·a1 + eik·a2

  • Dispersion relation : e(k) = ±t1|f(k)|
slide-7
SLIDE 7

Dispersion relation

5 / 27

slide-8
SLIDE 8

Dispersion relation

Low energy expension ) Dirac Hamiltonian for massless particle with vf ' c/300

5 / 27

slide-9
SLIDE 9

Dispersion relation

Low energy expension ) Dirac Hamiltonian for massless particle with vf ' c/300

  • Applications : very high charge carrier mobility

5 / 27

slide-10
SLIDE 10

Dispersion relation

Low energy expension ) Dirac Hamiltonian for massless particle with vf ' c/300

  • Applications : very high charge carrier mobility
  • Fundamental interests e.g. Klein tunneling

5 / 27

slide-11
SLIDE 11

Artificial graphene

E.S. Reich, Nature 497, 422 (2013)

6 / 27

slide-12
SLIDE 12

Artificial graphene

''The objective of creating these articial graphene-like lattices is to produce new systems that have properties that graphene does not have.'' A. Castro Neto

E.S. Reich, Nature 497, 422 (2013)

6 / 27

slide-13
SLIDE 13

Typical experimental set-up with microwave lattices

slide-14
SLIDE 14

Formal analogy between the Schrödinger and the Helmholtz equations Free particle

[∆ + V( ~ r)]y( ~ r) = Ey( ~ r)

Varying potential V Microwave cavity

[∆ + (1 e( ~ r))k2]y( ~ r) = k2y( ~ r)

Varying permittivity e

slide-15
SLIDE 15

Formal analogy between the Schrödinger and the Helmholtz equations Free particle

[∆ + V( ~ r)]y( ~ r) = Ey( ~ r)

Varying potential V Microwave cavity

[∆ + (1 e( ~ r))k2]y( ~ r) = k2y( ~ r)

Varying permittivity e TM modes : y(

~ r) = Ez( ~ r) ! energy everywhere (continuum state)

TE modes : y(

~ r) = Bz( ~ r) ! energy confined inside (bound state)

Attractive implementation to perform quantum analogue measurements

slide-16
SLIDE 16

1 Microwaves in a honeycomb lattice

A flexible experimental artificial graphene

slide-17
SLIDE 17

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

slide-18
SLIDE 18

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

slide-19
SLIDE 19

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

slide-20
SLIDE 20

The experimental set-up in reality...

slide-21
SLIDE 21

Experimental setup

10 / 27

slide-22
SLIDE 22

Experimental setup

10 / 27

slide-23
SLIDE 23

Experimental setup

10 / 27

slide-24
SLIDE 24

An artificial atom

  • Dielectric cylinder : n = 6

11 / 27

slide-25
SLIDE 25

An artificial atom

  • Dielectric cylinder : n = 6
  • We measure the reflected signal S11(n). At n = n0 ! 1 |S11(n0)|2 ' 2s

Γ |Ψ0(r1)|2 Γ1 : lifetime (Γ ⇠ 10 MHz) s : antenna coupling (weak and constant)

11 / 27

slide-26
SLIDE 26

An artificial atom

  • Dielectric cylinder : n = 6
  • We measure the reflected signal S11(n). At n = n0 ! 1 |S11(n0)|2 ' 2s

Γ |Ψ0(r1)|2 Γ1 : lifetime (Γ ⇠ 10 MHz) s : antenna coupling (weak and constant)

  • Most of the energy is confined in the disc (J0) and spreads evanescently (K0)

11 / 27

slide-27
SLIDE 27

Coupling between two discs

  • The frequency splitting ∆n(d) gives the coupling strength t1(d) = ∆n(d)/2

12 / 27

slide-28
SLIDE 28

Coupling between two discs

  • The frequency splitting ∆n(d) gives the coupling strength t1(d) = ∆n(d)/2
  • |t1(d)| = a|K0(gd/2)|2 + d

12 / 27

slide-29
SLIDE 29

Experimental (local) density of states

13 / 27

slide-30
SLIDE 30

Experimental (local) density of states

We have a direct acces to the LDOS

g(r1, n) = |S11(n)|2 h|S11|2in j0

11(n) ⇠

s Γ h|S11|2in ∑

n

|Ψn(r1)|2d(n nn)

13 / 27

slide-31
SLIDE 31

Experimental (local) density of states

We have a direct acces to the LDOS

g(r1, n) = |S11(n)|2 h|S11|2in j0

11(n) ⇠

s Γ h|S11|2in ∑

n

|Ψn(r1)|2d(n nn)

13 / 27

slide-32
SLIDE 32

Experimental (local) density of states

We have a direct acces to the LDOS

g(r1, n) = |S11(n)|2 h|S11|2in j0

11(n) ⇠

s Γ h|S11|2in ∑

n

|Ψn(r1)|2d(n nn)

13 / 27

slide-33
SLIDE 33

Experimental (local) density of states

  • One can get the wavefunction associated to the eigenfrequency nn

13 / 27

slide-34
SLIDE 34

Experimental (local) density of states

  • One can get the wavefunction associated to the eigenfrequency nn

13 / 27

slide-35
SLIDE 35

Experimental (local) density of states

  • By averaging g(ri, n) over all the antenna positions ri, we obtain the DOS

14 / 27

slide-36
SLIDE 36

Experimental (local) density of states

  • By averaging g(ri, n) over all the antenna positions ri, we obtain the DOS

14 / 27

slide-37
SLIDE 37

Experimental (local) density of states

  • By averaging g(ri, n) over all the antenna positions ri, we obtain the DOS
  • Tight-binding compatible. Main features have been taken into account

Dirac shift, band asymmetry ! next n.n. couplings.

14 / 27

slide-38
SLIDE 38

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

slide-39
SLIDE 39

Graphene under strain – Motivations

  • Mechanical response & electronic properties ! tunable electronic properties

16 / 27

slide-40
SLIDE 40

Graphene under strain – Motivations

  • Mechanical response & electronic properties ! tunable electronic properties
  • Induce a robust, clean bulk spectral gap in graphene

16 / 27

slide-41
SLIDE 41

Graphene under strain – Motivations

  • Mechanical response & electronic properties ! tunable electronic properties
  • Induce a robust, clean bulk spectral gap in graphene
  • 20% deformations required to open a gap

) Uni-axial strain ineffective to achieve bulk gapped graphene.

Peirera et al., Phys. Rev. B 80 045401 (2009)

16 / 27

slide-42
SLIDE 42

Graphene under strain – Motivations

  • Mechanical response & electronic properties ! tunable electronic properties
  • Induce a robust, clean bulk spectral gap in graphene
  • 20% deformations required to open a gap

) Uni-axial strain ineffective to achieve bulk gapped graphene.

Peirera et al., Phys. Rev. B 80 045401 (2009)

  • Strain in artificial systems : a = a

t ∂t ∂a

with a site separation and t coupling term

amicrowave ' 2agraphene

16 / 27

slide-43
SLIDE 43
  • TB Hamiltonian in uni-axial strained honeycomb lattice
slide-44
SLIDE 44
  • TB Hamiltonian in uni-axial strained honeycomb lattice
  • Anisotropy parameter : b = t0/t
  • Bloch Hamiltonian

HB

eff = t|f(k)|

✓ eif(k) eif(k) ◆

with f(k) = b + eik·a1 + eik·a2

  • Dispersion relation : e(k) = ±t|f(k)|
  • Eigenstates : Ψk,±(r) =

1 p 2 ✓ eif(k) ±1 ◆ eik·r

  • Berry phase : g = 1

2

I

dk rkf(k)

slide-45
SLIDE 45

Topological phase transition

Gapless phase Gapped phase

  • Berry phase ±p vanishes to 0 ) Topological phase transition
  • Dirac points move, merge (at b = bc = 2) and annihilate
  • G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009)

18 / 27

slide-46
SLIDE 46

Topological phase transition

Gapless phase Gapped phase

  • Berry phase ±p vanishes to 0 ) Topological phase transition
  • Dirac points move, merge (at b = bc = 2) and annihilate
  • Phase transition from gapless to gapped phase (Lifshitz transition)
  • G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009)

18 / 27

slide-47
SLIDE 47

Topological phase transition

  • Berry phase ±p vanishes to 0 ) Topological phase transition
  • Dirac points move, merge (at b = bc = 2) and annihilate
  • Phase transition from gapless to gapped phase (Lifshitz transition)
  • G. Montambaux et al., Eur. Phys. J. B 72, 509 (2009)

18 / 27

slide-48
SLIDE 48

Observation of the topological phase transition

  • Transition at b = bc and bandgap opening

19 / 27

slide-49
SLIDE 49

Observation of the topological phase transition

25

  • Transition at b = bc and bandgap opening

19 / 27

slide-50
SLIDE 50

Observation of the topological phase transition

  • Transition at b = bc and bandgap opening
  • Presence of edge states at Dirac frequency

19 / 27

slide-51
SLIDE 51

Observation of the topological phase transition

  • Transition at b = bc and bandgap opening
  • Presence of edge states at Dirac frequency

19 / 27

slide-52
SLIDE 52

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

slide-53
SLIDE 53

Edges in the honeycomb lattice

  • Edge states generally appear.

Absence of armchair edge states is an exception !

  • How to figure it out ?

! Zak phase

  • S. Ryu and Y. Hatsugai, PRL (2002)
  • P. Delplace et al., PRB (2011)

21 / 27

slide-54
SLIDE 54

Zigzag and bearded edge states

Experimental wavefunction intensities at nD

  • b allows to control and manipulate edge states
  • Localization length x depends on b

22 / 27

slide-55
SLIDE 55

Zigzag and bearded edge states

23 / 27

slide-56
SLIDE 56

Zigzag and bearded edge states

Diagram of existence obtained via a tight-binding approach

23 / 27

slide-57
SLIDE 57

Observation of armchair edge states at Dirac frequency

  • No edge states along anisotropy (horiz.) axis ! Zak phase
  • States live only on one sublattice (A for b < 1 and B for b > 1)
  • Edge states apparition doesn’t depend on the phase transition
  • The localisation length x decreases with b > 1

24 / 27

slide-58
SLIDE 58

Armchair edge states

Diagram of existence obtained via a tight-binding approach

25 / 27

slide-59
SLIDE 59

1 Microwave artificial graphene

Flexible experiment TB compatible Access to the DOS & wavefunctions

slide-60
SLIDE 60

1 Microwave artificial graphene

Flexible experiment TB compatible Access to the DOS & wavefunctions

2 Honeycomb lattices under strain

Observation of a topological phase transition Manipulation of edge states via strain

slide-61
SLIDE 61

1 Microwave artificial graphene

Flexible experiment TB compatible Access to the DOS & wavefunctions

2 Honeycomb lattices under strain

Observation of a topological phase transition Manipulation of edge states via strain

More info : Phys. Rev. Lett. 110, 033902 (2013) – Phys. Rev. B 88, 115437 (2013) ArXiv : 0000.0000 (2014) available soon :-) mail : bellec@unice.fr – web : www.unice.fr/mbellec

slide-62
SLIDE 62

Future work

  • Inhomogeneous strain

! pseudo-magnetic field, Landau

levels, etc.

  • Quantum search algorithm

(collab. Univ. Nottingham)

  • Quasicrystals (collab. INLN)
  • Selective enhancement of topologically

induced interface states (collab. Univ. Lancaster)

27 / 27

slide-63
SLIDE 63

Future work

  • Inhomogeneous strain

! pseudo-magnetic field, Landau

levels, etc.

  • Quantum search algorithm

(collab. Univ. Nottingham)

  • Quasicrystals (collab. INLN)
  • Selective enhancement of topologically

induced interface states (collab. Univ. Lancaster)

Thanks for your attention !

27 / 27