Graphene
Graphene can be considered as an infinite diameter carbon nanotube, or as an infinitely large aromatic molecule, the limiting case of the family of flat polycyclic aromatic hydrocarbons (PAH) or perhaps a single sheet of graphite. For many years graphene was only considered as a hypothetical limit, the theoreticians' tool in determining physical properties of carbon nanotubes or other systems. The turning point was inevitably 2004, when physicists from the University of Manchester and Institute for Microelectronics Technology, Chernogolovka found a way to isolate individual graphene planes by peeling them off from graphite with Scotch tape.
Intrinsic graphene is a semi-metal or zero-gap semiconductor. Understanding the electronic structure of graphene is the starting point for finding the band structure of other carbon materials like graphite or nanotubes. It was realized early on that the electronic dispersion relation is linear for low energies near the six corners of the two-dimensional hexagonal Brillouin zone, leading to zero effective mass for electrons and holes. Due to this linear (or “conical") dispersion relation at low energies, electrons and holes near these six points, two of which are inequivalent, behave like relativistic particles described by the Dirac equation for spin-half particles. Hence, the electrons and holes are called Dirac fermions, and the six corners of the Brillouin zone are called the Dirac points. The equation describing the electronic dispersion relation around the Fermi energy is E = ℏ vF |k| where the Fermi velocity is approximately 1/300 smaller than the speed of light (vF ~ 106 m/s).
The animation next to this text shows the dispersion relation of graphene (calculated within tight-binding approximation). Note that it is not conical anywhere. The Fermi energy lies between the two sheets, therefore the Fermi surface is only six points in the k-space. The low energy approximation (around the Fermi energy) is linear, so (6) cones (so-called Dirac cones) can 'be fitted to' that parts of the dispersion. This linear dispersion is responsible for the delicate effects of graphene, e.g. Klein paradox or minimal conductivity.
The Nobel Prize in Physics 2010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene".
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