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Schrödinger- and Dirac-Microwave Billiards, Photonic Crystals and Graphene
Supported by DFG within SFB 634
C. Bouazza, C. Cuno, B. Dietz, T. Klaus, M. Masi, M. Miski-Oglu, A. Richter, F. Iachello, N. Pietralla, L. von Smekal, J. Wambach
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 1
• Microwave billiards, graphene and photonic crystals • Band structure and relativistic Hamiltonian• Dirac billiards• Spectral properties• Periodic orbits• Quantum phase transitions• Outlook
ECT* 2013
d
Microwave billiardQuantum billiard
eigenfunction Y electric field strength Ez
Schrödinger- and Microwave Billiards
Analogy
eigenvalue E wave number
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 2
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
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 structure2013 | 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
• 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
conductionband
valenceband
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 an open photonic crystalS. Bittner et al., PRB 82, 014301 (2010)
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 6
Reflection Spectrum of an Open Photonic Crystal
• Characteristic cusp structure around the Dirac frequency• Van Hove singularities at the band saddle point• Next: experimental realization of a relativistic (Dirac) billiard
• Measurement with a wire antenna a put through a drilling in the top plate → point like field probe
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 7
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 e
dge
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 Tc=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
• 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 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 10
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))
• Oscillations around the mean density finite size effect
stop band
stop band
stop band
Dirac point
Dirac point
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 11
• Level density
• Dirac point • Van Hove singularities of the bulk states at• Next: TBM description of experimental DOS
Tight-Binding Model (TBM) for ExperimentalDensity of States (DOS)
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 12
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
• Here the overlap is neglected
t1
t3
t2
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 13
determined from experiment
Fit of the TBM to Experiment
• Good agreement
• Fluctuation properties of spectra
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 14
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 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 15
Integrated Density of States
• Schrödinger regime:
• Dirac Regime: (J. Wurm et al., PRB 84, 075468 (2011))
• Fit of Weyl’s formula to the data and
Schrödinger regime
Dirac regime
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 16
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 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 17
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 ?
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 19
; e.m. waves “see the scatterers“
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
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 20
D:
S:
Effective description
Experimental Length Spectrum:Schrödinger Regime
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 21
• Effective description ( ) has a relative error of 5% at the
frequency of the highest eigenvalue in the regime
• Very good agreement
• Next: Dirac regime
Experimental Length Spectrum:Dirac Regime
upper Dirac cone (f>fD) lower Dirac cone (f<fD)
• Some peak positions deviate from the lengths of POs• Comparison with semiclassical predictions for a Dirac billiard
(J. Wurm et al., PRB 84, 075468 (2011))
• Effective description ( ) has a relative error of 20% at the
frequency of the highest eigenvalue in the regime
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 22
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
• Evaluated the length spectra of periodic orbits around and away from the Dirac
point and made a comparison with semiclassical predictions
• Next: Do we see quantum phase transitions?
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 23
• Each frequency f in the experimental DOS r ( f ) is related to an isofrequency line of band structure in k space
• Close to band gaps isofrequency lines form circles around G point • Sharp peaks at Van Hove singularities correspond to saddle points• Parabolically shaped surface merges into Dirac cones around Dirac frequency
→ topological phase transition from non-relativistic to relativistic regime
Experimental DOS and Topology of Band Structure
saddle point
saddle point r ( f )
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 24
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 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 25
Finite-Size Scaling of DOS at the Van Hove Singularities
• TBM for infinitely large crystal yields
• Logarithmic behaviour as seen in bosonic systems
- 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:
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter | 26
Particle-Hole Polarization Function: Lindhard Function
• Polarization in one loop calculated from bubble diagram
→ Lindhard function at zero temperature
with and the nearest-neighbor vectors•
• Overlap of wave functions for intraband (=l’) and interband (λ=-l’) transitions within, respectively, between cones
• Use TBM taking into account only nearest-neighbor hopping
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 27
• Static susceptibility at zero-momentum transfer
• Nonvanishing contributions only from real part of Lindhard function
• Divergence of at m = 1 caused by the infinite degeneracy of ground states when Fermi surface passes through Van Hove singularities → GSQPT
• Imaginary part of Lindhard function at zero-momentum transfer yields for spectral distribution of particle-hole excitations
Static Susceptibility and Spectral Distribution of Particle-Hole Excitations I
• Only interband contributions and excitations for w > 2 m (Pauli blocking)
• Same logarithmic behavior observed for ground and excited states → ESQPT
q
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 28
• Sharp peaks of at m=1, w=0 and for -1≤ ≤ 1, w=2 clearly visible
• Experimental DOS can be quantitatively related to GSQPT and ESQPT arising from a topological Lifshitz neck-disrupting phase transition
• Logarithmic singularities separate the relativistic excitations from the nonrelativistic ones
Static Susceptibility and Spectral Distribution of Particle-Hole Excitations II
Diracregime
Schrödingerregime
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 29
• Normalization is fixed due to charge
conservation via f - sum rule
f-Sum Rule as Quasi-Order Parameter
Z-
Z+
Z=Z++Z-
• Z const. in the relativistic regime < 1• At = 1 its derivative diverges logarithmically • Z decreases approximately linearly in non-relativistic regime > 1
• Transition due to change of topology of Fermi surface → no order parameter
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 30
• ”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
mm
2013 | Institute of Nuclear Physics | SFB 634 | Achim Richter| 31