Graphene and Playing Relativistic Billards with microwaves

1. Graphene and Playing Relativistic Billards with microwaves Dresden 2012 • Something about graphene and microwave billiards • Dirac spectrum in a photonic crystal • Experimental setup • Reflection spectra • Photonic crystal in a box: Dirac billiards • Measured spectra • Density of states • Spectral properties • Outlook Supported by DFG within SFB 634 S. Bittner, B. Dietz, C. Cuno, J.Isensee, T. Klaus, M. Miski-Oglu, A. R., C. Ripp N.Pietralla, L. von Smekal, J. Wambach

2. Nobel Prize in Physics 2010

3. conductance band valence band 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

4. Quantum billiard Microwave billiard d Analogy for eigenvalue Ewave number Non-Relativistic Quantum Billiards and Microwave Billiards eigenfunction Y electric field strength Ez

5. rf power out rf power in  d  Measurement Principle • Measurement of scattering matrix element S21 Resonance spectrum Length spectrum (Gutzwiller) Resonance density FT

6. 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

7. Calculated Photonic Band Structure conductance band valence band • 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

8. Effective Hamiltonian around theDirac Point Close to Dirac point the effective Hamiltonian is a 2x2 matrix Substitution and leads to the Dirac equation Experimental observation of a Dirac spectrum in open photonic crystal(S. Bittner et al., PRB 82, 014301 (2010)) Scattering experiment

9. Single Antenna Reflection Spectrum • Measurement with a wire antennaaput through a drilling in the top plate → point like field probe • Characteristic cusp structure around the Dirac frequency • Next: comparison with calculated band structure cusp

10. Dirac point Projected Band Diagram • The density plot of the 1st frequency band • The projected band diagram along the irreducible Brillouin zone  • The 1st and 2nd frequency bands touch each other at the corners (K points) of the Brillouin zone → Dirac point

11. Single Antenna Reflection Spectrum:Comparison with Calculated Banddiagram • Characteristic cusp structure around the Dirac frequency • Van Hove singularities at the band saddle point where • Next: analysis of the measured spectrum

12. fit parameters Local Density of States and Reflection Spectrum • The scattering matrix formalism relates the reflection spectra to the local density of states (LDOS) • LDOS • LDOS around the Dirac point (Wallace,1947) • Three parameter fit formula

13. antenna a in the middle of the crystal antenna a6 rows apart from the boundary Reflection Spectra • Description of experimental reflection spectra • Experimental Dirac frequencies agree with calculated one, fD =13.81 GHz, within the standard error of the fit • Oscillations around the mean intensity  origin?

14. photonic crystal graphene flake, Li et al. (2009) Comparison with STM Measurements • Tunneling conductance is proportional to LDOS • Similarity with measured reflection spectrum of the photonic crystal • Oscillations in STM are not as pronounced due to the large sample size • Finestructure in the photonic crystal shows fluctuations (RMT)

15. Summary I • Connection between reflection spectra and LDOS is established • Cusp structure in the reflection spectra is identified with the Dirac point • Photonic crystal simulates one particle properties of graphene • Results are published in Phys. Rev. B 82, 014301 (2010) • Measured also transmission near the Dirac Point ( published in Phys. Rev. B 85, 064301 (2012)) • Relativistic quantum billiard: Dirac billiard

16. Dirac Billiard Photonic crystal  box: bounded area = billiard 888 cylinders (scatterers) milled out of a brass plate Height d = 3 mm = 50 GHz for 2D system Lead plated  superconducting below 7.2 K  high Q value Boundary does not violate the translation symmetry  no edge states Relativistic massless spin-one half particles in a billiard(Berry and Mondragon,1987)

17. Transmission Spectrum at 4 K Pronounced stop bands Quality factors > 5∙105  complete spectrum Altogether 5000 resonances observed

18. 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)) stop band Dirac point stop band Dirac point stop band

19. Integrated Density of States: 1st and 2nd Bands Does not follow Weyl’s law for 2D resonators: Small slope at the Dirac frequency  DOS nearly zero Nearly symmetric with respect to the Dirac frequency Two parabolic branches

20. Integrated DOS near Dirac Point Weyl’s law for Dirac billiard: (J. Wurm et 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

21. 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 at second Dirac point

22. Periodic Orbit Theory (POT) • 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

23. Experimental Length Spectrum: Around the Dirac Point • Some peak positions deviate from the lengths of POs • Comparison with semiclassical predictions for Dirac billiard(J. Wurmet al., PRB 84, 075468 (2011)) • Possible reasons for deviations: - Short sequenceoflevels (80 levelsonly) - Anisotropicdispersionrelationaroundthe Dirac point (trigonal warping)

24. Experimental Length Spectrum: Away from Dirac Point Dirac regime Schrödinger regime • Very good agreement

25. Dirac Billard with and without Translational Symmetry: Edge States Billiard 2 Billiard 1 Armchair edge Zigzag edge • Areas: AB1AB2 • Two different edges • Localized states expected along zigzag edge → “edge states“

26. Resonance Spectra of B1 and B2 around the Dirac Point • B1: Antennas A1 and A7 placed in the interior • B2: Antennas A6 and A2 placedattheedges → accumulationofresonances

27. Experimental Mean DOS as Function of Quasimomentum • Quasimomentumandisthelatticeconstant • Additional localmaximumabovethe Dirac point • Semiclassicalprediction(J. Wurmet al., PRB 84, 075468 (2011)) describes the asymmetry around the Dirac point but not the peak due to the edge states

28. Summary II Measured the DOS in a superconduction Dirac billiard with high resolution Observation of two Dirac points and associated van Hove singularities:qualitative agreement with the band structure for graphene 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 Observation of predicted edge states in a Dirac billiard without translational symmetry Comparison of the mean DOS with a semiclassical prediction Next: Do we see an excited state quantum phase transition?

29. saddle point saddle point Excited State Quantum Phase Transition in Dirac Billiard • Experimental density of states in the Dirac billiard • Van Hove singularities at saddle point: density of states diverges logarithmicaly for quasi-momenta near the M point • Lifshitz topological phase transition

30. What is a Lifshitz Topological Phase Transition? • Consider real Graphene with tunable Fermi energies, i.e. variable chemical potential → topology of the Fermi surface changes with • This is called a Lifshitz topological phase transition with as a control parameter • What happens when is close to the van Hove singularity? phase transition

31. Lindhardt Function • Polarization of the medium by a photon → bubble diagram • Summation over all momenta of virtual electron-hole pairs → Lindhardt function • Static susceptibiity defined as • It can be shown within the tight binding approximation that i.e. evolves as function of the chemical potential like the DOS→ logarithmic divergence at

32. Density of States for Particle-Hole Excitations • Density of states • Logarithmic singularity separates the relativistic excitations from the nonrelativistic ones

33. f-Sum Rule • f-sum rule: • Possible order parameter: • Dirac billiard: experimentally only the divergence of DOS is observed → see poster session infinite derivative