Edge effects in graphene nanostructures: Semiclassical theory of spectral fluctuations and quantum transport

We investigate the effect of different edge types on the statistical properties of both the energy spectrum of closed graphene billiards and the conductance of open graphene cavities in the semiclassical limit. To this end, we use the semiclassical Green's function for ballistic graphene flakes [see J. Wurm, K. Richter, and İ. Adagideli, Phys. Rev. B 84, 075468 (2011)]. First, we study the spectral two-point correlation function or, more precisely, its Fourier transform the spectral form factor, starting from the graphene version of Gutzwiller's trace formula for the oscillating part of the density of states. We calculate the two leading-order contributions to the spectral form factor, paying particular attention to the influence of the edge characteristics of the system. Then, we consider transport properties of open graphene cavities. We derive generic analytical expressions for the classical conductance, the weak localization correction, the size of the universal conductance fluctuations, and the shot-noise power of a ballistic graphene cavity. Again, we focus on the effects of the edge structure. For both the conductance and the spectral form factor, we find that edge-induced pseudospin interference affects the results significantly. In particular, intervalley coupling mediated through scattering from armchair edges is the key mechanism that governs the coherent quantum interference effects in ballistic graphene cavities.