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
A microwave realization of artificial graphene Co-workers Ulrich Kuhl Gilles Montambaux Fabrice Mortessagne LPMC LPMC LPS, Orsay 2 / 27
Amazing graphene Outlook, Nature 483 , S29 (2012) 3 / 27
���������� ����������� Amazing graphene Outlook, Nature 483 , S29 (2012) 3 / 27
� � Tight-binding Hamiltonian in regular honeycomb lattice
� � Tight-binding Hamiltonian in regular honeycomb lattice • Bloch states representation 1 j i ) e i k · Rj N ∑ ( l A | f A j i + l B | f B | Ψ k i = p j • E ff ective Bloch Hamiltonian in ( A , B ) basis ✓ ◆ f ( k ) 0 with f ( k ) = 1 + e � i k · a1 + e � i k · a2 H B eff = � t 1 f ⇤ ( k ) 0 • Dispersion relation : e ( k ) = ± t 1 | f ( k ) |
Dispersion relation 5 / 27
Dispersion relation Low energy expension ) Dirac Hamiltonian for massless particle with v f ' c /300 5 / 27
Dispersion relation Low energy expension ) Dirac Hamiltonian for massless particle with v f ' c /300 • Applications : very high charge carrier mobility 5 / 27
Dispersion relation Low energy expension ) Dirac Hamiltonian for massless particle with v f ' c /300 • Applications : very high charge carrier mobility • Fundamental interests e.g. Klein tunneling 5 / 27
Artificial graphene E.S. Reich, Nature 497 , 422 (2013) 6 / 27
Artificial graphene ''The objective of creating these arti � cial 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
Typical experimental set-up with microwave lattices
Formal analogy between the Schrödinger and the Helmholtz equations Free particle Microwave cavity [ � ∆ + V ( ~ r )] y ( ~ r ) = E y ( ~ r ) r )) k 2 ] y ( r ) = k 2 y ( [ � ∆ + ( 1 � e ( ~ ~ ~ r ) Varying potential V Varying permittivity e
Formal analogy between the Schrödinger and the Helmholtz equations Free particle Microwave cavity [ � ∆ + V ( ~ r )] y ( ~ r ) = E y ( ~ r ) r )) k 2 ] y ( r ) = k 2 y ( [ � ∆ + ( 1 � e ( ~ ~ ~ r ) Varying potential V Varying permittivity e TM modes : y ( ~ r ) = E z ( ~ r ) ! energy everywhere (continuum state) TE modes : y ( ~ r ) = B z ( ~ r ) ! energy confined inside (bound state) Attractive implementation to perform quantum analogue measurements
1 Microwaves in a honeycomb lattice A flexible experimental artificial graphene
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
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
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
The experimental set-up in reality...
Experimental setup 10 / 27
Experimental setup 10 / 27
Experimental setup 10 / 27
���� ���� An artificial atom • Dielectric cylinder : n = 6 11 / 27
���� ���� An artificial atom • Dielectric cylinder : n = 6 • We measure the reflected signal S 11 ( n ) . At n = n 0 ! 1 � | S 11 ( n 0 ) | 2 ' 2 s Γ | Ψ 0 ( r 1 ) | 2 Γ � 1 : lifetime ( Γ ⇠ 10 MHz) s : antenna coupling (weak and constant) 11 / 27
���� ���� An artificial atom • Dielectric cylinder : n = 6 • We measure the reflected signal S 11 ( n ) . At n = n 0 ! 1 � | S 11 ( n 0 ) | 2 ' 2 s Γ | Ψ 0 ( r 1 ) | 2 Γ � 1 : lifetime ( Γ ⇠ 10 MHz) s : antenna coupling (weak and constant) • Most of the energy is confined in the disc ( J 0 ) and spreads evanescently ( K 0 ) 11 / 27
� � ��������� � ��������� Coupling between two discs • The frequency splitting ∆ n ( d ) gives the coupling strength t 1 ( d ) = ∆ n ( d ) /2 12 / 27
� Coupling between two discs • The frequency splitting ∆ n ( d ) gives the coupling strength t 1 ( d ) = ∆ n ( d ) /2 • | t 1 ( d ) | = a | K 0 ( g d /2 ) | 2 + d 12 / 27
Experimental (local) density of states 13 / 27
Experimental (local) density of states We have a direct acces to the LDOS g ( r 1 , n ) = | S 11 ( n ) | 2 s j 0 Γ h | S 11 | 2 i n ∑ | Ψ n ( r 1 ) | 2 d ( n � n n ) 11 ( n ) ⇠ � h | S 11 | 2 i n n 13 / 27
Experimental (local) density of states We have a direct acces to the LDOS g ( r 1 , n ) = | S 11 ( n ) | 2 s j 0 Γ h | S 11 | 2 i n ∑ | Ψ n ( r 1 ) | 2 d ( n � n n ) 11 ( n ) ⇠ � h | S 11 | 2 i n n 13 / 27
Experimental (local) density of states We have a direct acces to the LDOS g ( r 1 , n ) = | S 11 ( n ) | 2 s j 0 Γ h | S 11 | 2 i n ∑ | Ψ n ( r 1 ) | 2 d ( n � n n ) 11 ( n ) ⇠ � h | S 11 | 2 i n n 13 / 27
� � � � ��� � � Experimental (local) density of states One can get the wavefunction associated to the eigenfrequency n n 13 / 27
� ��� � � Experimental (local) density of states One can get the wavefunction associated to the eigenfrequency n n 13 / 27
Experimental (local) density of states • By averaging g ( r i , n ) over all the antenna positions r i , we obtain the DOS 14 / 27
������ Experimental (local) density of states • By averaging g ( r i , n ) over all the antenna positions r i , we obtain the DOS 14 / 27
������ �������� Experimental (local) density of states • By averaging g ( r i , n ) 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 . 14 / 27
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
Graphene under strain – Motivations • Mechanical response & electronic properties ! tunable electronic properties 16 / 27
Graphene under strain – Motivations • Mechanical response & electronic properties ! tunable electronic properties • Induce a robust, clean bulk spectral gap in graphene 16 / 27
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 ine ff ective to achieve bulk gapped graphene. Peirera et al., Phys. Rev. B 80 045401 (2009) 16 / 27
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 ine ff ective to achieve bulk gapped graphene. Peirera et al., Phys. Rev. B 80 045401 (2009) • Strain in artificial systems : a = a ∂ t with a site separation and t coupling term t ∂ a a microwave ' 2 a graphene 16 / 27
� �� � � � TB Hamiltonian in uni-axial strained honeycomb lattice
� � � � �� TB Hamiltonian in uni-axial strained honeycomb lattice • Anisotropy parameter : b = t 0 / t • Bloch Hamiltonian ✓ e � i f ( k ) ◆ 0 H B eff = � t | f ( k ) | e i f ( k ) 0 with f ( k ) = b + e � i k · a 1 + e � i k · a 2 • Dispersion relation : e ( k ) = ± t | f ( k ) | ✓ ◆ 1 e � i f ( k ) e i k · r p • Eigenstates : Ψ k , ± ( r ) = ± 1 2 • Berry phase : g = 1 I d k r k f ( k ) 2
Topological phase transition Gapless phase Gapped phase 0 • Berry phase ± p vanishes to 0 ) Topological phase transition • Dirac points move, merge (at b = b c = 2 ) and annihilate G. Montambaux et al., Eur. Phys. J. B 72 , 509 (2009) 18 / 27
Topological phase transition Gapless phase Gapped phase 0 • Berry phase ± p vanishes to 0 ) Topological phase transition • Dirac points move, merge (at b = b c = 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
Topological phase transition • Berry phase ± p vanishes to 0 ) Topological phase transition • Dirac points move, merge (at b = b c = 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
Observation of the topological phase transition • Transition at b = b c and bandgap opening 19 / 27
Observation of the topological phase transition 25 0 • Transition at b = b c and bandgap opening 19 / 27
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