QED in Graphene*

Electronic properties of materials are commonly described by quasiparticles that behave as non-relativistic electrons with a finite mass and obey the Schr\"{o}dinger equation. I will describe our experimental studies of graphene (a free-standing single layer of carbon atoms) in which electron transport is essentially governed by Dirac's (relativistic) equation and charge carriers mimic relativistic particles with zero rest mass and an effective ``speed of light'' of $\approx $10$^{6}$m/s. We have found a variety of unusual quantum phenomena characteristic of two-dimensional Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; b) graphene's conductivity never falls below a minimum value corresponding to the conductance quantum, even when carrier concentrations tend to zero; c) the cyclotron mass of massless carriers in graphene is described by Einstein's equation $E $=\textit{mc}$^{2}$; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of $\pi $ due to Berry's phase. I will also explain another, third type of the integer quantum Hall effect that happens in bilayer graphene and accompanied by Berry's phase of 2$\pi $.