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 Topology of the band structure Outlook Supported by 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

Classical-, quantum- and microwave billiards

Schrödinger- and Microwave Billiards Quantum billiard d Microwave billiard Analogy Part of the subproject C4 focuses on the spectral properties of classically regular or fully chaotic systems and of systems with mixed regular and chaotic dynamics. Billiards are well suited for such studies as their classical dynamics is solely determined by their shape. The eigenvalues and eigenfunctions of these so-called quantum billiards are determined experimentally by using the analogy between them and flat microwave resonators. Below a certain frequency, that is, for wave lengths larger than double the height of the resonator, the electric field strength is perpendicular to the top and bottom of the resonator. There the underlying Helmholtz equation is mathematically equivalent to the Schrödinger equation of the quantum billiard of corresponding shape. In both systems the solutions of the wave equation fulfill Dirichlet boundary conditions along the wall of the billiard. Accordingly the eigenvalues of the quantum billiard are directly related to the resonance frequencies, the eigenfunctions to the electric field strengths of the microwave resonator, which is also called microwave billiard. The eigenvalues are obtained in superconducting measurements of resonance spectra. eigenvalue E  wave number eigenfunction Y  electric field strength Ez 3

Measurement Principle Measurement of scattering matrix element S21 rf power in out   d Resonance spectrum For this two antennas are attached to the microwave billiard and microwave power is coupled into the resonator via one antenna and coupled out via the same or the other antenna. and the scattering matrix describing the scattering process with the antennas acting as single scattering channels is measured. The squared modulus of the scattering matrix element from antenna 1 to antenna 2 is given by the ratio of the power coupled out at antenna 2 and that coupled into the resonator at antenna 1. The resonance spectrum is obtained by plotting this ratio versus the excitation frequency f. The resonance frequencies of the microwave billiard and thus the eigenvalues are obtained from the positions of the resonances. positions of the resonances fn=knc/2p yield eigenvalues

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 Ein photonische Kristall ist eine Gebilde deren elektromagnetische Eigenschaften ändern sich periodisch, zum Beispiel die periodische Anordnung der Metall Zylindern die zwischen zwei Kupfer platten eingespannt ist. Wie es wurde in dem Vortrag von Dr. Barbara Dietz gezeigt in diesem flachen Kristall unter einen bestimmten Frequenz nur die Moden existieren mit dem elektrischen Feld senkrecht zu metal Platten. Die propagierenden Moden sind Lösungen der skalaren Helmholz Gleichung, die mathematisch equivalent zu der Srödinger Gleichung für das multiple Streuproblem. 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 21/04/2017 |

Nobel Prize in Physics 2010

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 conduction band valence Two triangular sublattices of carbon atoms Near each corner of the first hexagonal Brillouin zone the electron energy E has 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 Ich beginne mein Vortrag mit einen Zitat aus der Arbeit …… Graphen ist eine flache periodische Anordnung der Kohlenstoff Atomen. Den Graphene hat eine Gitterstruktur die aus zwei überlappenden dreieckigen Untergittern besteht. Durch blau und rot sind die Atomen aus diesen Untergittern gekenzeichnet. Das Bild zeigt die Energiespektrum des Elektrons als Funktion von Quasimomentum. Der Hexagon zeigt die erste Brilliouine zone. Das untere Band ist eine Valenzband, die obere ist die Leitungsband. Die beider Bänder berühren sich an der Ecken der Ersten Brilloiune Zone und formen zeigen konische Dispersionrelation. Die Dispersionrelation ist dem der von massenlosen fermionen Ähnlich ist. 21/04/2017 |

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 second band first

Effective Hamiltonian around Dirac 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 an open photonic crystal S. Bittner et al., PRB 82, 014301 (2010) Next: experimental realization of a relativistic billiard

Microwave Dirac Billiard: Photonic Crystal in a Box→ “Artificial Graphene“ Zigzag edge Armchair edge Graphene flake: the electron cannot escape → Dirac billiard Photonic crystal: electromagnetic waves can escape from it → microwave Dirac billiard: “Artificial Graphene“ Relativistic massless spin-one half particles in a billiard (Berry and Mondragon,1987)

Microwave Dirac Billiards with and without Translational Symmetry Billiard B2 Billiard B1 Boundaries of B1 do not violate the translational symmetry → cover the plane with perfect crystal lattice Boundaries of B2 violate the translational symmetry → edge states along the zigzag boundary Almost the same area for B1 and B2

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

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

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 Neto et al., RMP 81,109 (2009)) Oscillations around the mean density  finite size effect of the crystal stop band Dirac point stop band Dirac point stop band

Tight Binding Model (TBM) Description of the Photonic Crystal The voids in a photonic crystal form a honeycomb lattice resonance frequency of an “isolated“ void nearest neighbour contribution  t1 next-nearest neighbour contribution  t2 second-nearest neighbour contribution  t3 t3 t2 t1 determined from experimental frequencies f(), f() and f() 21/04/2017 |

Fit of the TBM to Experiment f(M) f(M) f() f() f(K) Good agreement Next: fluctuation properties of spectra 21/04/2017 |

Schrödinger and Dirac Dispersion Relation in the Photonic Crystal Dirac regime Schrödinger regime Dispersion relation along irreducible Brillouin zone  Quadratic dispersion around the  point Schrödinger regime Linear dispersion around the  point  Dirac regime

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

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?

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) )

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 “see the scatterers“

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

Transmission Spectra of B1 and B2 around the Dirac Frequency No violation of translational symmety Violation of translational symmety Accumulation of resonances above the Dirac frequency Resonance amplitude is proportional to the product of field strengths at the position of the antennas  detection of localized states

Comparison of Spectra Measured with Different Antenna Combinations Antenna positions Modes living in the inner part (black lines) Modes localized at the edge (red lines) have higher amplitudes

Smoothed Experimental Density of States TBM prediction Clear evidence of the edge states Position of the peak for the edge states deviates from the theoretical prediction (K. Sasaki, S. Murakami, R. Saito (2006)) Modification of tight-binding model is needed

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 of the experimental DOS with a tight-binding model yields perfect agreement 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 we see quantum phase transitions? 21/04/2017 | | 26

Experimental DOS and Topology of Band Structure saddle point neck saddle point r ( f ) Each frequency f in the experimental DOS r ( f ) is related to an isofrequency line of band structure in k space Close to band edges isofrequency lines form circles around the G point 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 Disrupting Lifschitz Transition“) from non-relativistic to relativistic regime (B. Dietz et al., PRB 88, 1098 (2013))

“Mother of all Graphitic Forms“ (Geim and Novoselov (2007))

Outlook ”Artificial” Fullerene 200 mm 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

Neck-Disrupting Lifshitz Transition Gradually lift Fermi surface across saddle point, e.g., with a chemical potential m → topology of the Fermi surface changes topological transition in two dimensions 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)