Graphene billiards with fourfold symmetry

Abstract

We report on the realization of a graphene billiard with a fourfold rotational symmetry, which has the shape of a billiard with chaotic classical dynamics. The eigenstates are separated according to their transformation properties under rotation by π2 into four symmetry classes. These subspectra can be divided into regions around the band edges, where they are governed by the nonrelativistic Schrödinger equation, and a region of low energy-excitations around zero energy, that exhibit a linear dispersion relation and are described by the relativistic Dirac equation for a spin-1/2 quasiparticle. We analyze the spectral properties in these parts and compare them with those for nonrelativistic quantum billiards and relativistic neutrino billiards. In both regions the spectral properties conform with those of nonrelativistic quantum billiards of corresponding shape. Furthermore, we compute wave functions in configurational space and momentum distributions in quasimomentum space and find that the first Brillouin zone is formed by two hexagonal ones that are rotated by 30∘ with respect to each other. Namely, in the relativistic region the momentum distributions are localized at or near the 12 Dirac points at the corners of these hexagons. At the Van Hove singularities, which separate the nonrelativistic and relativistic regions, they are localized along the isofrequency lines that connect the six saddle points at the centers between two corners for each hexagon. We propose a design of a microwave photonic crystal for the experimental modeling of graphene billiards with fourfold symmetry and present numerical results for its properties. © 2023 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.