Dirac- Microwave Billiards , Photonic Crystals and Graphene

Dirac-Microwave Billiards, Photonic Crystals and Graphene

Supported by DFG within SFB 634

S. Bittner, C. Cuno, B. Dietz, T. Klaus, M. Masi, M. Miski-Oglu, A. R., F.Iachello, N. Pietralla, L. von Smekal, J. Wambach

Warsaw 2013

2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 1

• Graphene, Schrödinger-microwave billiards and photonic crystals • Band structure and relativistic Hamiltonian• Dirac-microwave billiards• Spectral properties• Periodic orbits• Edge states• Quantum phase transitions• Outlook

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 but low 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

Closed Flat Microwave Billiards: Model Systems for Quantum Phenomena

xyz

zeyxErEdcff

),()(

2max

0),(,0),(2 GyxEyxEk

scalar Helmholtz equation Schrödinger equation for quantum billiards

cf

krErEk G

2

,0)(n,0)(2 vectorial

Helmholtz equation

cylindrical resonators


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


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

conductionband

valenceband


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 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 7.2 K high Q value • Boundary does not violate the translational symmetry no edge states


• Measured S-matrix: |S21|2=P2 / P1

• Quality factors > 5∙105

• Altogether 5000 resonances observed• Pronounced stop bands and Dirac points

Transmission Spectrum at 4 K


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))

• Oscilations around the mean density finite size effect

stop band

stop band

stop band

Dirac point

Dirac point


• Level density

• Dirac point • Van Hove singularities of the bulk states • Next: TBM description of experimental DOS

Tight-Binding Model (TBM) for ExperimentalDensity of States (DOS)


Tight Binding 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

• Here the overlap is neglected

t1

t3t2


Fit of the Tight-Binding Model to Experiment

Numerical solution of Helmholtz equation

• Fit of the tight-binding model to the experimental frequencies , , yields the unknown coupling parameters f0,t1,t2,t3

ExperimentalDOS


Fit of the TBM to Experiment

obvious deviations good agreement

• Fluctuation properties of spectra


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: with w , where m is the “effective mass“ discribing the parabolic dispersion

• Dirac Regime: with , where is the group velocity at the Dirac frequency

• Unfolding is necessary in order to obtain the length spectra

Schrödinger regime

Dirac regime


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 (different opening angles of the upper and lower cone)


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 in the Schrödinger regime


Periodic Orbit Theory (POT)Gutzwiller‘s Trace Formula

• Description of quantum spectra in terms of classical periodic orbits

Periodic orbits

spectrum spectral density

Peaks at the lengths l of PO’s

wavenumbers length spectrum

FT

Dirac billiard

Effective description

around the Dirac point


Experimental Length Spectrum:Schrödinger regime

• Very good agreement

• Next: Dirac regime


Experimental Length Spectrum:Around the Dirac Point

• Some peak positions deviate from the

lengths of POs

• Comparison with semiclassical predictions

for a relativistic Dirac billiard

(J. Wurm et al., PRB 84, 075468 (2011))

• Possible reasons for deviations:

- Short sequence of levels (80 levels only)

- Anisotropic dispersion relation

around the Dirac point (trigonal

warping, i.e. deformation of the Dirac

cone)2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 22

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 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 24

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 including the overlap is needed


TB p

redi

ctio

n

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

• Evaluated the length spectra of periodic orbits around and away from the

Dirac point and made a comparison with semiclassical predictions

• Edge states are detected in the spectra

• Outlook: Do we see quantum phase transitions?


• Experimental density of states in the Dirac billiard

• Features of the DOS related to the topology of the isofrequency lines in k-space

• Van Hove singularities at saddle point: density of states diverges logarithmically for quasimomenta near the M point

• Topological phase transition

Spectroscopic Features of the DOS in a Dirac Billiard

saddle point

saddle point


Why is this a Topological Phase Transition?

phase transitionin two dimensions


• Consider real graphene with tunable Fermi energies, i.e. variable chemical potential → topology of the Fermi surface changes with

• Disruption of the “neck“ of the Fermi surface• This is called a Lifshitz topological phase transition with as a control

parameter (Lifshitz 1960)

• What happens when is close to the Van Hove singularity?

Finite-Size Scaling of DOS at the Van Hove Singularities

• TBM for infinitely large crystal yields

• Logarithmic behaviour as seen in

- transverse vibration of a hexagonal lattice (Hobson and Nierenberg, 1952)

- vibrations of molecules (Pèrez-Bernal, Iachello, 2008)

- two-level fermionic and bosonic pairing models (Caprio, Scrabacz, Iachello, 2011)

• Finite size photonic crystals or graphene flakes formed by hexagons

, i.e. logarithmic scaling of the VH peak

determined using Dirac billiards of varying size:


DOS, Static Susceptibility and Particle-Hole Excitations: Lindhard Function

• Polarization of the medium by a photon → bubble diagram

• Summation over all momenta of virtual electron-hole pairs → Lindhard function

• Static susceptibility 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 (Van Hove singularity)

• Divergence of at caused by the infinite degeneracy of ground state: ground state QPT


Spectral Distribution of Particle-Hole Excitations

• Spectral distribution of the particle-hole excitations

• Same logarithmic behavior as for the ground-state observed for the excited states: ESQPT

• Logarithmic singularity separates the relativistic excitations from the nonrelativistic ones

Diracregime

Schrödingerregime


• ”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

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