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Microwave Billiards , Photonic Crystals and Graphene

Microwave Billiards , Photonic Crystals and Graphene. KIT 2013. Classical-, quantum- and microwave billiards Photonic crystals Graphene and its modeling through photonic crystals Microwave Dirac billiards Density of states and the band structure Edge states

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Microwave Billiards , Photonic Crystals and Graphene

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  1. MicrowaveBilliards, Photonic Crystals andGraphene KIT 2013 • Classical-, quantum- and microwave billiards • Photonic crystals • Graphene and its modeling through photonic crystals • Microwave Dirac billiards • Density of states and the band structure • Edge states • Topology of the band structure • Outlook Supportedby DFG within SFB 634 C. Bouazza, C. Cuno, B. Dietz, T. Klaus, M. Miski-Oglu, A. Richter, F. Iachello, N. Pietralla, L. von Smekal, J. Wambach

  2. Classical-, quantum- and microwave billiards

  3. Quantum billiard Microwave billiard d eigenvalue Ewave number Schrödinger- andMicrowaveBilliards Analogy eigenfunction Y electric field strength Ez

  4. rf power out rf power in  d  Measurement Principle • Measurement of scattering matrix element S21 Resonance spectrum positions of the resonances fn=knc/2pyield eigenvalues

  5. 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

  6. Nobel Prize in Physics 2010

  7. 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 • with → at the Dirac point electrons behave like relativistic fermions • → → strongly interacting system (QCD) • Experimental realization of graphene in analog experiments of microwave photonic crystals conduction band valence band

  8. second band first 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 • The voids form a honeycomb lattice like atoms in graphene

  9. 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 an open photoniccrystalS. Bittner et al., PRB 82, 014301 (2010) • Next: experimental realizationof a relativisticbilliard

  10. 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) Armchairedge Zigzag edge

  11. Microwave Dirac BilliardswithandwithoutTranslationalSymmetry • Boundariesof B1 do not violatethetranslationalsymmetry→ coverthe plane withperfectcrystallattice • Boundariesof B2 violatethetranslationalsymmetry→ edgestatesalongthezigzagboundary • Almostthe same areafor B1 and B2 BilliardB2 BilliardB1

  12. 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 Tc=7.2 K  high Q value • Boundary does not violate the translational symmetry  no edge states

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

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

  15. Tight Binding Model (TBM) Description ofthePhotonic Crystal • The voids in a photonic crystal form a honeycomb lattice • resonancefrequencyof an “isolated“ void • nearestneighbourcontribution t1 • next-nearestneighbourcontribution t2 • second-nearestneighbourcontribution t3 t3 t2 t1 determinedfrom experimental frequencies f(), f() andf()

  16. Fit of the TBM to Experiment f(M) f(M) f() f() f(K) • Good agreement • Next: fluctuation properties of spectra

  17. Schrödinger and Dirac Dispersion Relation in the Photonic Crystal Dirac regime Schrödinger regime • Dispersion relationalongirreducibleBrillouinzone • Quadraticdispersionaroundthe  point Schrödingerregime • Linear dispersionaroundthe  point  Dirac regime

  18. Integrated Density of States Dirac regime Schrödinger regime • Schrödingerregime: • Dirac Regime: (J. • Fit of Weyl’s formula to the data  and

  19. Spectral Properties of a Rectangular Dirac Billiard: Nearest Neighbour Spacing Distribution Spacing between adjacent levels depends on DOS Unfolding procedure: such that 130 levels in the Schrödinger regime 159 levels in the Dirac regime Spectral properties around the Van Hove singularities?

  20. Ratio Distribution of Adjacent Spacings • DOS is unknown around Van Hove singularities • Ratio of two consecutive spacings • Ratios are independent of the DOS no unfolding necessary • Analytical prediction for Gaussian RMT ensembles (Y.Y. Atas, E. Bogomolny, O. Giraud and G. Roux, PRL, 110, 084101 (2013) )

  21. Ratio Distributions for Dirac Billiard • Poisson: ; GOE: • Poisson statistics in the Schrödinger and Dirac regime • GOE statistics to the left of first Van Hove singularity • Origin ? ;  e.m. waves “seethescatterers“

  22. Superconducting Dirac Billiard without Translational Symmetry • Boundaries violate the translational symmetry  edge states • Additional antennasclosetotheboundary Armchair edge Zigzag edge

  23. Transmission Spectra of B1 and B2 around the Dirac Frequency Noviolationoftranslational symmety • Accumulationofresonancesabovethe Dirac frequency • Resonanceamplitudeis proportional totheproductoffieldstrengthsatthepositionoftheantennas  detection of localized states Violation oftranslational symmety

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

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

  26. 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-bindingmodelyieldsperfectagreement Fluctuation properties of the spectrum agree with Poisson statistics both in the Schrödinger and the Dirac regime, but not around the Van Hove singularities Edge states are detected in the spectra Next: Do weseequantumphasetransitions?

  27. saddle point saddle point Experimental DOS and Topology of Band Structure • Eachfrequencyf in the experimental DOS r ( f )isrelatedto an isofrequencylineof band structure in kspace • Close to band edgesisofrequencylinesform circlesaroundtheGpoint • At the saddle point the isofrequency lines become straight lines which cross each other and lead to the Van Hove singularities • Parabolically shaped surface merges into the 6 Dirac cones around Dirac frequency → topological phase transition (“Neck DisruptingLifschitz Transition“) from non-relativistic to relativistic regime (B. Dietz et al., PRB 88, 1098 (2013)) neck r ( f )

  28. “Mother of all Graphitic Forms“(GeimandNovoselov (2007))

  29. 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

  30. Neck-Disrupting Lifshitz Transition • Gradually lift Fermi surface across saddle point, e.g., with a chemical potential m → topology of the Fermi surface changes • Disruption of the “neck“ of the Fermi surface at the saddle point • At Van Hove singularities DOS diverges logarithmically in infinite 2D systems → Neck-disrupting Lifshitz transition with m as a control parameter (Lifshitz 1960) topologicaltransition in twodimensions

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