Graphene is a new material consisting of a one-atom-thick layer of graphite. It has been discovered only a few years ago and since its discovery it has shown to possess many unique and fascinating properties. These properties are a consequence of the fact that electrons in graphene effectively behave as massless relativistic electrons, whose dynamics is described by the two-dimensional Dirac equation. In this talk I will first introduce the field, by explaining in some detail how the electronic properties of graphene originate, how they manifest themselves in the experiments, and how the experiments are actually done. I will then move to discuss more specific topics that we have investigated during the past few years. These include phase coherent transport of electrons (Aharonov-Bohm effect, weak-localization, Josephson supercurrent), finite conductivity at the charge neutrality point, and gate-tunable band structure. Taken together, these examples make clear that graphene (single-layers as well as materials consisting of a small number of layers) indeed is a unique material system, in which the discovery of new physical phenomena can be expected, and which could be of use in future electronic applications.
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