Dirac cones for point scatterers on a honeycomb lattice

Abstract

We investigate the spectrum and the dispersion relation of the Schrödinger operator with point scatterers on a triangular lattice and a honeycomb lattice. We prove that the low-level dispersion bands have conic singularities near Dirac points, which are the vertices of the first Brillouin Zone. The existence of such conic dispersion bands plays an important role in various electronic properties of honeycomb-structured materials such as graphene. We then prove that for a honeycomb lattice, the spectra generated by higher-level dispersion relations are all connected so the complete spectrum consists of at most three bands. Numerical simulations for dispersion bands with various parameters are also presented.

30 pages, 13 figures

References in corpus (5)

• Electric Field Effect in Atomically Thin Carbon Films
• The electronic properties of graphene
• On the spectra of carbon nano-structures
• Statistics of wave functions for a point scatterer on the torus
• Quantum Chaos for point scatterers on flat tori

Cited by in corpus (5)

• Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene
• Honeycomb-lattice Minnaert bubbles
• A high-frequency homogenization approach near the Dirac points in bubbly honeycomb crystals
• Characterization of edge states in perturbed honeycomb structures
• Dirac cones for a mean-field model of graphene