Microwave Billiards , Photonic Crystals and Graphene

Microwave Billiards, Photonic Crystals and Graphene

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

2013 | SFB 634 | Achim Richter| 1

• 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

KIT 2013

Classical-, quantum- and microwave billiards

2013 | SFB 634 | Achim Richter | 2

d

Microwave billiardQuantum billiard

eigenfunction Y electric field strength Ez

Schrödinger- and Microwave Billiards

Analogy

eigenvalue E wave number


Measurement Principle

Resonance spectrum

221

1,

2, SPP

in

out

rf power in

rf power out

d

• Measurement of scattering matrix element S21

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

• 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


Nobel Prize in Physics 2010


Graphene

• 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

• “What makes graphene so attractive for research is that the spectrum closely resembles the Dirac spectrum for massless fermions.”M. Katsnelson, Materials Today, 2007

conductionband

valenceband


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

secondband

firstband


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 crystalS. Bittner et al., PRB 82, 014301 (2010)

• Next: experimental realization of a relativistic billiard


Microwave Dirac Billiard: Photonic Crystal in a Box→ “Artificial Graphene“

• 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)

Zigzag edge

Arm

chai

r edg

e


Microwave Dirac Billiards with and without Translational Symmetry

• 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

Billiard B2Billiard B1


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


• Measured S-matrix: |S21|2=P2 / P1

• Pronounced stop bands and Dirac points• Quality factors > 5∙105

• Altogether 5000 resonances observed

Transmission Spectrum at 4 K


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

stop band

stop band

Dirac point

Dirac point

2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 14

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

t1

t3t2

determined from experimental frequenciesf(), f() and f()


Fit of the TBM to Experiment

• Good agreement• Next: fluctuation properties of spectra


f()

f(M)

f(K) f()

f(M)

Schrödinger and Dirac Dispersion Relation in the Photonic Crystal

Dirac regimeSchrödinger 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

• Schrödinger regime:

• Dirac Regime: (J.

• Fit of Weyl’s formula to the data and

Schrödinger regime

Dirac regime


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“2013 | SFB 634 | Achim Richter | 21

Superconducting Dirac Billiard without Translational Symmetry

• Boundaries violate the translational symmetry edge states

• Additional antennas close to the boundary

Zigzag edge

Arm

chai

r edg

e


Transmission Spectra of B1 and B2 around the Dirac Frequency

• 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 states2013 | SFB 634 | Achim Richter | 23

No violation of translational symmety

Violation of translational symmety

Comparison of Spectra Measured with Different Antenna Combinations

• Modes living in the inner part (black lines)• Modes localized at the edge (red lines) have higher amplitudes

Antenna positions


Smoothed Experimental Density of States

• 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

TBM

pre

dict

ion


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?


Experimental DOS and Topology of Band Structure

saddle point

saddle pointr ( f )


neck

• 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 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))


• ”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

Outlook

50 m

m


Neck-Disrupting Lifshitz Transition

topological transitionin two dimensions

• 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)


Microwave Billiards , Photonic Crystals and Graphene

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