CONIC DISPERSION SURFACES FOR POINT SCATTERERS ON A HONEYCOMB LATTICE

Abstract. We investigate the spectrum and dispersion relation of Schrödinger operator with point scatterers on a triangular lattice and a honeycomb lattice, respectively. We prove that the low-level dispersion surfaces have conic sin-gularities near Dirac points, which are the vertices of the first Brillouin Zone. The existence of such conic dispersion surface plays an important role in vari-ous electronic properties of honeycomb-structured materials such as graphene. We then prove that for the honeycomb lattice, the band spectra generated by higher-level dispersion surfaces are all connected so the complete spectrum consists of at most three bands. Numerical simulations for dispersion surfaces with various coupling constant are also presented. 1