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
2013 | SFB 634 | Achim Richter | 3
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
2013 | SFB 634 | Achim Richter | 4
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 | SFB 634 | Achim Richter | 5
Nobel Prize in Physics 2010
2013 | SFB 634 | Achim Richter | 6
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
2013 | SFB 634 | Achim Richter | 7
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
2013 | SFB 634 | Achim Richter | 8
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
2013 | SFB 634 | Achim Richter | 9
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
2013 | SFB 634 | Achim Richter | 10
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
2013 | SFB 634 | Achim Richter | 11
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
2013 | SFB 634 | Achim Richter | 12
• 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
2013 | SFB 634 | Achim Richter | 13
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()
2013 | SFB 634 | Achim Richter | 15
Fit of the TBM to Experiment
• Good agreement• Next: fluctuation properties of spectra
2013 | SFB 634 | Achim Richter | 16
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
2013 | SFB 634 | Achim Richter | 17
Integrated Density of States
• Schrödinger regime:
• Dirac Regime: (J.
• Fit of Weyl’s formula to the data and
Schrödinger regime
Dirac regime
2013 | SFB 634 | Achim Richter | 18
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?
2013 | SFB 634 | Achim Richter | 19
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) )
2013 | SFB 634 | Achim Richter | 20
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
2013 | SFB 634 | Achim Richter | 22
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
2013 | SFB 634 | Achim Richter | 24
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
2013 | SFB 634 | Achim Richter | 25
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?
2013 | SFB 634 | Achim Richter | 26
Experimental DOS and Topology of Band Structure
saddle point
saddle pointr ( f )
2013 | SFB 634 | Achim Richter | 27
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))
2013 | SFB 634 | Achim Richter | 28
• ”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
2013 | SFB 634 | Achim Richter | 29
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)
2013 | SFB 634 | Achim Richter | 30