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Dirac- Microwave Billiards , Photonic Crystals and Graphene

Dirac- Microwave Billiards , Photonic Crystals and Graphene. Warsaw 2013. Graphene , Schrödinger-microwave billiards and photonic crystals Band structure and relativistic H amiltonian Dirac-microwave billiards Spectral properties Periodic orbits Edge states

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Dirac- Microwave Billiards , Photonic Crystals and Graphene

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  1. Dirac-MicrowaveBilliards, Photonic Crystals andGraphene Warsaw 2013 • Graphene, Schrödinger-microwave billiards and photonic crystals • Band structure and relativistic Hamiltonian • Dirac-microwave billiards • Spectral properties • Periodic orbits • Edge states • Quantum phase transitions • Outlook Supportedby DFG within SFB 634 S. Bittner, C. Cuno, B. Dietz, T. Klaus, M. Masi, M. Miski-Oglu, A. R., F.Iachello, N. Pietralla, L. von Smekal, J. Wambach

  2. Graphene • “What makes graphene so attractive for research is that the spectrum closely resembles the Dirac spectrum for massless fermions.”M. Katsnelson, Materials Today, 2007 • Two triangular sublattices of carbon atoms • Near each corner of the first hexagonal Brillouin zone the electron energy Ehas a conical dependence on the quasimomentum • but low • Experimental realization of graphene in analog experiments of microwave photonic crystals conduction band valence band

  3. z x y Closed Flat Microwave Billiards: Model Systems for Quantum Phenomena vectorial Helmholtz equation cylindrical resonators scalar Helmholtz equation  Schrödinger equation for quantum billiards

  4. Open Flat Microwave Billiard:Photonic Crystal • A photonic crystal is a structure, whose electromagnetic properties vary periodically in space, e.g. an array of metallic cylinders→ open microwave resonator • Flat “crystal” (resonator) → E-field is perpendicular to the plates (TM0 mode) • Propagating modes are solutions of the scalar Helmholtz equation → Schrödinger equation for a quantum multiple-scattering problem→ Numerical solution yields the band structure

  5. conduction band valence band Calculated Photonic Band Structure • Dispersion relation of a photonic crystal exhibits a band structure analogous to the electronic band structure in a solid • The triangular photonic crystal possesses a conical dispersion relation → Dirac spectrum with a Dirac point where bands touch each other

  6. Effective Hamiltonian around Dirac Point • Close to Dirac point the effective Hamiltonian is a 2x2 matrix • Substitution and leads to the Dirac equation • Experimental observationof a Dirac spectrum in open photoniccrystalS. Bittner et al., PRB 82, 014301 (2010) • Next: experimental realizationof a relativisticbilliard

  7. Microwave Dirac Billiard: Photonic Crystal in a Box→ “ArtificialGraphene“ • Grapheneflake: theelectroncannotescape → Dirac billiard • Photoniccrystal: electromagneticwavescanescapefromit → microwave Dirac billiard: “ArtificialGraphene“ • Relativistic massless spin-one half particles in a billiard (Berry and Mondragon,1987) Armchair edge Zigzag edge

  8. Microwave Dirac BilliardswithandwithoutTranslationalSymmetry • Boundariesof B1 do not violatethetranslationalsymmetry→ coverthe plane withperfectcrystallattice • Boundariesof B2 violatethetranslationalsymmetry→ edgestatesalongthezigzagboundary • Almostthe same areafor B1 and B2 billiardB2 billiardB1

  9. Superconducting Dirac Billiard with Translational Symmetry • The Dirac billiard is milled out of a brass plate and lead plated • 888 cylinders • Height h=3 mm fmax = 50 GHz for 2D system • Lead coating is superconducting below 7.2 K  high Q value • Boundary does not violate the translational symmetry  no edge states

  10. Transmission Spectrum at 4 K • Measured S-matrix: |S21|2=P2 / P1 • Quality factors > 5∙105 • Altogether 5000 resonancesobserved • Pronounced stop bands and Dirac points

  11. Density of States of the Measured Spectrum and the Band Structure Positions of stop bands are in agreement with calculation DOS related to slope of a band Dips correspond to Dirac points High DOS at van Hove singularities  ESQPT? Flat band has very high DOS Qualitatively in good agreement with prediction for graphene(Castro Neto et al., RMP 81,109 (2009)) Oscilations around the mean density  finite size effect stop band Dirac point stop band Dirac point stop band

  12. Tight-Binding Model (TBM) for ExperimentalDensity of States (DOS) • Level density • Dirac point  • Van Hove singularities of the bulk states  • Next: TBM description of experimental DOS

  13. Tight Binding Description of the Photonic Crystal • The voids in a photonic crystal form a honeycomb lattice • resonancefrequencyof an “isolated“ void • nearestneighbourcontribution t1 • nextnearestneighbourcontribution t2 • second-nearestneighbourcontribution t3 • Heretheoverlapisneglected t3 t2 t1 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 13

  14. Fit of the Tight-Binding Model to Experiment Experimental DOS Numericalsolution ofHelmholtz equation • Fit of the tight-binding model to the experimental frequencies , , yields the unknown coupling parameters f0,t1,t2,t3 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 14

  15. Fit of the TBM to Experiment obvious deviations good agreement • Fluctuation properties of spectra 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 15

  16. Schrödingerand Dirac Dispersion Relation in thePhotonic Crystal Dirac regime Schrödinger regime • Dispersion relationalongirreducibleBrillouinzone • Quadraticdispersionaroundthe  point Schrödingerregime • Linear dispersionaroundthe  point Dirac regime 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 16

  17. Integrated Density of States Dirac regime Schrödinger regime • Schrödingerregime: with w , wheremisthe“effectivemass“ discribingtheparabolicdispersion • Dirac Regime: with, whereisthegroupvelocityatthe Dirac frequency • Unfoldingisnecessary in ordertoobtainthelengthspectra 2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 17

  18. Integrated DOS near Dirac Point • Weyl’s law for Dirac billiard: (J. Wurmet al., PRB 84, 075468 (2011)) • group velocity is a free parameter • Same area A for two branches, but different group velocities  electron-hole asymmetry like in graphene (different opening angles of the upper and lower cone)

  19. Spectral Properties of a Rectangular Dirac Billiard: Nearest Neighbour Spacing Distribution 159 levels around Dirac point Rescaled resonance frequencies such that Poisson statistics Similar behavior in the Schrödinger regime

  20. Periodic Orbit Theory (POT) Gutzwiller‘s Trace Formula • Description of quantum spectra in terms of classical periodic orbits wave numbers spectrum length spectrum spectral density FT Peaks atthelengthslof PO’s Dirac billiard Periodicorbits Effectivedescription aroundthe Dirac point

  21. Experimental Length Spectrum: Schrödingerregime • Very good agreement • Next: Dirac regime

  22. Experimental Length Spectrum: Around the Dirac Point • Some peak positions deviate from the lengths of POs • Comparison with semiclassical predictions for a relativistic Dirac billiard(J. Wurm et al., PRB 84, 075468 (2011)) • Possible reasons for deviations: • - Short sequence of levels (80 levels only) • - Anisotropic dispersion relation around the Dirac point (trigonal warping, i.e. deformation of the Dirac cone)

  23. Superconducting Dirac Billiard without Translational Symmetry • Boundaries violate the translational symmetry  edge states • Additional antennas close to the boundary Armchair edge Zigzag edge

  24. Transmission Spectra of B1 and B2 around the Dirac Frequency • Accumulationofresonancesabovethe Dirac frequency • Resonanceamplitudeis proportional totheproductoffieldstrengthsatthepositionoftheantennas  detection of localized states

  25. Comparison of Spectra Measured with Different Antenna Combinations • Modes living in theinnerpart (blacklines) • Modes localizedattheedge (redlines) havehigheramplitudes Antenna positions

  26. Smoothed Experimental Density of States • Clear evidence of the edge states • Position ofthepeakfortheedgestatesdeviatesfromthetheoreticalprediction (K. Sasaki, S. Murakami, R. Saito (2006)) • Modificationoftight-bindingmodelincludingtheoverlapisneeded TB prediction

  27. Summary I Measured the DOS in a superconducting Dirac billiard with high resolution Observation of two Dirac points and associated van Hove singularities:qualitative agreement with the band structure for graphene Description ofthe experimental DOS with a Tight-Binding Model yieldsperfectagreement Fluctuation properties of the spectrum agree with Poisson statistics Evaluated the length spectra of periodic orbits around and away from the Dirac point and made a comparison with semiclassical predictions Edge states are detected in the spectra Outlook: Do weseequantumphasetransitions?

  28. saddle point saddle point Spectroscopic Features of the DOS in a Dirac Billiard • Experimental density of states in the Dirac billiard • Features of the DOS related to the topology of the isofrequency lines in k-space • Van Hove singularities at saddle point: density of states diverges logarithmically for quasimomenta near the M point • Topological phase transition

  29. Why is this a Topological Phase Transition? • Consider real graphene with tunable Fermi energies, i.e. variable chemical potential → topology of the Fermi surface changes with • Disruption of the “neck“ of the Fermi surface • This is called a Lifshitz topological phase transition with as a control parameter (Lifshitz 1960) • What happens when is close to the Van Hove singularity? phase transition in two dimensions

  30. Finite-Size Scaling of DOS at the Van Hove Singularities • TBM forinfinitely large crystalyields • Logarithmicbehaviourasseen in - transversevibrationof a hexagonal lattice(Hobsonand Nierenberg, 1952) - vibrations of molecules (Pèrez-Bernal, Iachello, 2008) - two-level fermionic and bosonic pairing models (Caprio, Scrabacz, Iachello, 2011) • Finite sizephotoniccrystalsorgrapheneflakesformedbyhexagons , i.e. logarithmicscalingofthe VH peak determinedusing Dirac billiardsofvaryingsize: 

  31. DOS, Static Susceptibility and Particle-Hole Excitations: Lindhard Function • Polarizationofthe medium by a photon → bubblediagram • Summation over all momentaofvirtualelectron-hole pairs→ Lindhardfunction • Staticsusceptibilitydefinedas • Itcanbeshownwithinthetight-bindingapproximationthat , i.e.evolvesasfunctionofthechemical potential likethe DOS→ logarithmicdivergenceat (Van Hovesingularity) • Divergenceofatcausedbythe infinite degeneracyofgroundstate: groundstate QPT

  32. Spectral Distribution of Particle-Hole Excitations • Spectraldistributionoftheparticle-hole excitations • Same logarithmicbehaviorasfortheground-state observedfortheexcitedstates: ESQPT • Logarithmicsingularity separates therelativisticexcitationsfromthenonrelativisticones Schrödinger regime Dirac regime

  33. Outlook • ”Artificial” Fullerene • Understanding of the measured spectrum in terms of TBM • Superconducting quantum graphs • Test of quantum chaotic scattering predictions(Pluhař + Weidenmüller 2013) 200 mm 50 mm

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