Quantum transport via evanescent waves in undoped graphene

Charge carriers in graphene are chiral quasiparticles ("massless Dirac fermions"). Graphene provides therefore an amazing opportunity to study subtle quantum relativistic effects in condensed matter experiment. Here I review a theory of one of these unusual features of graphene, a "pseudodiffusive" transport in the limit of zero charge carrier concentration, which is related to existence of zero-modes of the Dirac operator and to the Zitterbewegung of unltrarelativistic particles. A conformal mapping technique is a powerful mathematical tool to study these phenomena, as demonstrated here, using the Aharonov-Bohm effect in graphene rings with Corbino geometry as an example.