1. 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

2. Classical-, quantum- and microwave billiards

3. 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

4. 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

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
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
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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
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.
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8. 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

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 observation of a Dirac spectrum in an open photonic crystal S. Bittner et al., PRB 82, (2010)
Next: experimental realization of a relativistic billiard

10. 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)

11. 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

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 resonances observed

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 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

15. 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()
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16. Fit of the TBM to Experiment
f(M)
f(M)
f()
f()
f(K)
Good agreement
Next: fluctuation properties of spectra
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17. 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

18. Integrated Density of States
Dirac regime
Schrödinger regime
Schrödinger regime:
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, (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 “see the scatterers“

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

23. 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

24. 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

25. 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

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 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?
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27. 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))

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

29. 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

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
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)