Dirac-Microwave Billiards,
Photonic Crystals and Graphene
Warsaw
2013
• 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
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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 1
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
• 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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 2
valence
band
Closed Flat Microwave Billiards: Model
Systems for Quantum Phenomena
y
vectorial
Helmholtz equation
cylindrical resonators

z
x

 
2f 
  
  k E (r )  0, n  E (r ) G  0, k 
c
 

c
f  f max 
 E  ( r )  E  ( x, y )e z
2d

2

scalar
2


k
 E ( x, y)  0, E ( x, y) G  0
Helmholtz equation
 Schrödinger equation for quantum billiards
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 3
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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 4
Calculated Photonic Band Structure
• Dispersion relation
of a photonic crystal exhibits a band structure
analogous to the electronic band structure in a solid
conduction
band
valence
band
• The triangular photonic crystal possesses a conical dispersion relation
→ Dirac spectrum with a Dirac point where bands touch each other
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 5
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 crystal
S. Bittner et al., PRB 82, 014301 (2010)
• Next: experimental realization of a relativistic billiard
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 6
Armchair edge
Microwave Dirac Billiard:
Photonic Crystal in a Box→ “Artificial Graphene“
Zigzag 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)
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 7
Microwave Dirac Billiards with and without
Translational Symmetry
billiard B1
billiard B2
• 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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 8
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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 9
Transmission Spectrum at 4 K
• Measured S-matrix: |S21|2=P2 / P1
• Quality factors > 5∙105
• Altogether 5000 resonances observed
• Pronounced stop bands and Dirac points
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 10
Density of States of the Measured Spectrum
and the Band Structure
stop band
• Positions of stop bands are in
agreement with calculation
• DOS related to slope of a band
Dirac point
• Dips correspond to Dirac points
• High DOS at van Hove
singularities  ESQPT?
stop band
Dirac point
• Flat band has very high DOS
• Qualitatively in good agreement
with prediction for graphene
(Castro Neto et al., RMP 81,109 (2009))
stop band
• Oscilations around the mean
density  finite size effect
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 11
Tight-Binding Model (TBM) for Experimental
Density of States (DOS)
• Level density
• Dirac point 
• Van Hove singularities of the bulk states
• Next: TBM description of experimental DOS
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 12

Tight Binding Description of the Photonic
Crystal
• The voids in a photonic crystal form a honeycomb lattice
t3
t2
t1
• 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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 13
Fit of the Tight-Binding Model to Experiment
Experimental
DOS
Numerical solution
of Helmholtz equation
• Fit of the tight-binding model to the experimental frequencies
,
,
yields the unknown coupling parameters f0,t1,t2,t3
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 14
Fit of the TBM to Experiment
•
obvious deviations
• Fluctuation properties of spectra
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 15
good agreement
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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 16
Integrated Density of States
Dirac regime
Schrödinger regime
• Schrödinger regime:
with
w
the “effective mass“ discribing the parabolic dispersion
, where m is
• Dirac Regime:
velocity at the Dirac frequency
is the group
with
, where
• Unfolding is necessary in order to obtain the length spectra
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 17
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)
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 18
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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 19
Periodic Orbit Theory (POT)
Gutzwiller‘s Trace Formula
• Description of quantum spectra in terms of classical periodic
orbits
wave
length spectrum
spectral density
spectrum
numbers
FT
Peaks at the lengths l of PO’s
Dirac billiard
Periodic orbits
Effective description
around the Dirac
point
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 20
Experimental Length Spectrum:
Schrödinger regime
• Very good agreement
• Next: Dirac regime
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 21
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
Armchair edge
Superconducting Dirac Billiard without
Translational Symmetry
Zigzag edge
• Boundaries violate the translational symmetry  edge states
• Additional antennas close to the boundary
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 23
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 states
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 25
TB prediction
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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 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
• 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?
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 27
Spectroscopic Features of the DOS in a
Dirac Billiard
• Experimental density of states in the Dirac billiard
saddle point
saddle point
• 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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 28
Why is this a Topological Phase Transition?
• Consider real graphene with tunable Fermi energies, i.e. variable
chemical potential → topology of the Fermi surface changes with
phase transition
in two dimensions
• 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?
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 29
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
, i.e. logarithmic scaling of the VH peak

determined using Dirac billiards of varying size:
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 30
hexagons
DOS, Static Susceptibility and Particle-Hole
Excitations: Lindhard Function
• Polarization of the medium by a photon → bubble diagram
• Summation over all momenta
→ Lindhard function
• Static susceptibility
of virtual electron-hole pairs
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
• Divergence of
at
(Van Hove singularity)
caused by the infinite degeneracy of ground
state: ground state QPT
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 31
Spectral Distribution of Particle-Hole
Excitations
• Spectral distribution of the particle-hole excitations
Schrödinger
regime
Dirac
regime
• Same logarithmic behavior as for the ground-state observed for the
excited states: ESQPT
• Logarithmic singularity separates the relativistic excitations from the
nonrelativistic ones
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 32
Outlook
200 mm
• ”Artificial” Fullerene
• Superconducting quantum graphs
• Test of quantum chaotic scattering predictions
(Pluhař + Weidenmüller 2013)
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 33
50 mm
• Understanding of the measured spectrum in terms of TBM