Edge effects in graphene nanostructures: From multiple reflection expansion to density of states

We study the influence of different edge types on the electronic density of states of graphene nanostructures. To this end we develop an exact expansion for the single-particle Green's function of ballistic graphene structures in terms of multiple reflections from the system boundary, which allows for a natural treatment of edge effects. We first apply this formalism to calculate the average density of states of graphene billiards. While the leading term in the corresponding Weyl expansion is proportional to the billiard area, we find that the contribution that usually scales with the total length of the system boundary differs significantly from what one finds in semiconductor-based, Schrodinger-type billiards: The latter term vanishes for armchair and infinite-mass edges and is proportional to the zigzag edge length, highlighting the prominent role of zigzag edges in graphene. We then compute analytical expressions for the density of states oscillations and energy levels within a trajectory-based semiclassical approach. We derive a Dirac version of Gutzwiller's trace formula for classically chaotic graphene billiards and further obtain semiclassical trace formulas for the density of states oscillations in regular graphene cavities. We find that edge-dependent interference of pseudospins in graphene crucially affects the quantum spectrum.